Global homotopy theory
Stefan Schwede






equivariant homotopy theory, compact Lie group, spectrum



Global homotopy theory
    Stefan Schwede

Mathematisches Institut, Universität Bonn

schwede@math.uni-bonn.de

    December 30, 2023
========================================================================================






CHAPTER: PREFACE



Equivariant stable homotopy theory has a long tradition,
starting from geometrically motivated questions about symmetries of manifolds. 
The homotopy theoretic foundations of the
subject were laid by tom Dieck, Segal and May and their students and
collaborators in the 70's, and during the last decades 
equivariant stable homotopy theory 
has been very useful to solve computational and conceptual problems 
in algebraic topology, geometric topology and algebraic K-theory. 
Various important equivariant theories naturally exist not just 
for a particular group, but in a uniform way for all groups in a specific class. 
Prominent examples of this are
equivariant stable homotopy, equivariant K-theory or equivariant bordism.
Global equivariant homotopy theory studies such uniform phenomena,
i.e., the adjective `global' refers to simultaneous 
and compatible actions of all compact Lie groups. 

This book introduces a new context for global homotopy theory.
Various ways to provide a home for global stable homotopy types have 
previously been
explored in <cit.>, <cit.>, 
<cit.> and <cit.>.
We use a different approach: we work with the well-known category
of orthogonal spectra, but use a much finer notion of equivalence,
the global equivalences, than what is traditionally considered.
The basic underlying observation is that an orthogonal spectrum
gives rise to an orthogonal G-spectrum for every compact Lie group G,
and the fact that all these individual equivariant objects come from
one orthogonal spectrum implicitly encodes strong compatibility conditions
as the group G varies. 
An orthogonal spectrum thus has G-equivariant homotopy groups 
for every compact Lie group,
and a global equivalence is a morphism of orthogonal spectra that induces
isomorphisms for all equivariant homotopy groups for all compact Lie groups
(based on `complete G-universes', compare Definition <ref>).

The structure on the equivariant homotopy groups of an orthogonal spectrum
gives an idea of the information encoded in a global homotopy type
in our sense: the equivariant homotopy groups π_k^G(X) are contravariantly
functorial for continuous group homomorphisms (`restriction maps'), and they are
covariantly functorial for inclusions of closed subgroups (`transfer maps').
The restriction and transfer maps enjoy various transitivity properties
and interact via a double coset formula. This kind of algebraic structure has
been studied before under different names, 
e.g., `global Mackey functor', `inflation functor', ….
From a purely algebraic perspective, there are various parameters here than one can vary,
namely the class of groups to which a value is assigned and the classes
of homomorphisms to which restriction maps respectively transfer maps are assigned,
and lots of variations have been explored.
However, the decision to work with orthogonal spectra and equivariant
homotopy groups on complete universes dictates a canonical choice: we prove
in Theorem <ref> 
that the algebra of natural operations between the equivariant homotopy groups 
of orthogonal spectra is freely generated by restriction maps 
along continuous group homomorphisms and transfer maps along closed subgroup inclusion, 
subject to explicitly understood relations.

We define the global stable homotopy category 
by localizing the category of orthogonal spectra at the class of
global equivalences.
Every global equivalence is in particular a non-equivariant stable equivalence,
so there is a `forgetful' functor U:→ on localizations,
where  denotes the traditional non-equivariant stable homotopy category.
By Theorem <ref> this forgetful functor
has a left adjoint L and a right adjoint R, both fully faithful,
that participate in a recollement of triangulated categories:

    @C=15mm^+ @<-.3ex>[r]^-i_*    @<-.3ex>[r]^-U   @<.4ex>@/^1pc/[l]^-i^!@<-.4ex>@/_1pc/[l]_-i^*    @<.4ex>@/^1pc/[l]^-R @<-.4ex>@/_1pc/[l]_-L

Here ^+ denotes the full subcategory of the global stable homotopy category
spanned by the orthogonal spectra that are stably contractible in the traditional,
non-equivariant sense.

The global sphere spectrum and suspension spectra are
in the image of the left adjoint 
(Example <ref>).
Global Borel cohomology theories are the image of the right adjoint
(Example <ref>).
The `natural' global versions of 
Eilenberg-Mac Lane spectra (Construction <ref>),
Thom spectra (Section <ref>), 
or topological K-theory spectra 
(Sections <ref> and <ref>)
are not in the image of either of the two adjoints.
Periodic global K-theory, however, is right induced from finite cyclic groups,
i.e., in the image of the analogous right adjoint from an intermediate 
global homotopy category _ based on finite cyclic groups
(Example <ref>).

Looking at orthogonal spectra through the eyes of global equivalences
is like using a prism: the latter breaks up white light into a spectrum
of colors, and  global equivalences split a traditional, non-equivariant homotopy
type into many different global homotopy types. 
The first example of this phenomenon that we will encounter
refines the classifying space of a compact Lie group G.
On the one hand, there is the constant orthogonal space with value
a non-equivariant model for B G; 
and there is the global classifying space B_ G 
(see Definition <ref>). 
The global classifying space is analogous to
the geometric classifying space of a linear algebraic group
in motivic homotopy theory <cit.>,
and it is the counterpart to the stack of G-principal bundles
in the world of stacks.

Another good example is the splitting up of the non-equivariant homotopy
type of the classifying space of the infinite orthogonal group.
Again there is the constant orthogonal space with value B O, 
the Grassmannian model  (Example <ref>),
a different Grassmannian model  (Example <ref>),
the bar construction model ^∘ (Example <ref>), 
and finally a certain `cofree' orthogonal space R(B O).
The orthogonal space  is also a homotopy colimit, 
as n goes to infinity, of the global classifying spaces B_ O(n).
We discuss these different global forms of B O 
is some detail in Section <ref>,
and the associated Thom spectra in Section <ref>.

In the stable global world, every non-equivariant homotopy type
has two extreme global refinements, the `left induced'
(the global analog of a constant orthogonal space,
see Example <ref>)
and the `right induced' global homotopy type (representing Borel cohomology theories,
see Example <ref>).
Many important stable homotopy types have other natural global forms.
The non-equivariant Eilenberg-Mac Lane spectrum of the integers has a 
`free abelian group functor' model (Construction <ref>),
and another incarnation as the Eilen­berg-Mac Lane spectrum
of the constant global functor with value 
(Remark <ref>).
These two global refinements of the integral Eilenberg-Mac Lane spectrum
agree on finite groups, but differ for compact Lie groups of positive dimensions.

As already indicated, there is a great variety of orthogonal Thom spectra,
in real (or unoriented) flavors as  and ,
as complex (or unitary) versions  and ,
and there are periodic versions , ,  and  of these;
we discuss these spectra in Section <ref>.
The theories represented by  and  have the closest ties to geometry;
for example, the equivariant homotopy groups of 
receive Thom-Pontryagin maps from equi­variant bordism rings,
and these are isomorphisms for products of finite groups and tori
(compare Theorem <ref>).
The theories represented by  are tom Dieck's homotopical equivariant bordism,
isomorphic to `stable equivariant bordism'.

Connective topological K-theory also has two
fairly natural global refinements, in addition to the left and right induced ones.
The `orthogonal subspace' model  
(Construction <ref>)
represents connective equivariant K-theory on the class of finite groups;
on the other hand, global connective K-theory 
(Construction <ref>)
is the global synthesis of equivariant connective K-theory
in the sense of Greenlees <cit.>.
The periodic global K-theory spectrum  is introduced in 
Construction <ref>; as the name suggests, 
is Bott periodic and represents equivariant K-theory.


The global equivalences are part of a closed model structure
(see Theorem <ref>), 
so the methods of homotopical algebra can be used to study the stable global homotopy
category. This works more generally relative to a class ℱ
of compact Lie groups, where we define ℱ-equivalences
by requiring that π_k^G(f) is an isomorphism for all integers and all
groups in ℱ.
We call a class ℱ of compact Lie groups a global family
if it is closed under isomorphisms, subgroups and quotients.
For global families we refine the ℱ-equivalences to a stable model structure,
the -global model structure, see Theorem <ref>.
Besides all compact Lie groups,
interesting global families are the classes of all finite groups,
or all abelian compact Lie groups. The class of trivial groups
is also admissible here, but then we just recover the `traditional' stable category.
If the family  is multiplicative, then the 
 -global model structure is monoidal with respect to
the smash product of orthogonal spectra and satisfies the monoid axiom
(Proposition <ref>).
Hence this model structure lifts to modules over an orthogonal ring
spectrum and to algebras over an ultra-commutative ring spectrum
(Corollary <ref>).




Ultra-commutativity.
A recurring theme throughout this book is a phenomenon 
that I call ultra-commutativity.
I use this term in the unstable and stable context
for the homotopy theory of strictly commutative objects under global equivalences.
An ultra-commutative multiplication is significantly more structure
than just a coherently homotopy-commutative product
(usually called an E_∞-multiplication). 
For example, the extra structure gives rise to power operations
that can be turned into transfer maps (in additive notation) 
respectively norm  maps (in multiplicative notation).
Another difference is that an unstable global E_∞-structure 
would give rise to naive deloopings (i.e., by trivial representations). 
As I hope discuss elsewhere, a global ultra-commutative multiplication,
in contrast, gives rise to `genuine' deloopings (i.e., by non-trivial representations). 
As far as the objects are concerned, ultra-commutative monoids and
ultra-commutative ring spectra are not at all new and have been much studied before;
so one could dismiss the name 'ultra-commutativity' as a mere marketing maneuver.
However, the homotopy theory of ultra-commutative monoids and 
ultra-commutative ring spectra
with respect to global equivalences is new and, in the authors opinion,
important. And important concepts deserve catchy names.




Global homotopy types as orbifold cohomology theories. subjectorbifold|(
I would like to briefly mention another reason for
why one might be interested in global stable homotopy theory.
In short, global stable homotopy types represent 
genuine cohomology theories on stacks, orbifolds, and orbispaces.
Stacks and orbifolds are concepts from algebraic geometry respectively
geometric topology that allow us to talk about objects
that locally look like the quotient of a smooth object by a group action,
in a way that remembers information about the isotropy groups of the action.
Such `stacky' objects can behave like smooth objects
even if the underlying spaces have singularities.
As for spaces, manifolds and schemes, cohomology theories
are important invariants also for stacks and orbifolds,
and examples such as ordinary cohomology or K-theory lend themselves
to generalization. Special cases of orbifolds are `global quotients',
often denoted M G, for example for a smooth action of a compact
Lie group G on a smooth manifold M. In such examples, the orbifold
cohomology of M G is supposed to be the G-equivariant cohomology of M.
This suggests a way to define orbifold cohomology theories by
means of equivariant stable homotopy theory, via suitable G-spectra E_G. 
However, since the group G is not intrinsic and can vary, one needs
equivariant cohomology theories for all groups G, with some compatibility.

Part of the compatibility can be deduced from the fact that 
the same orbifold can be presented in different ways; for example, if G is a closed
subgroup of K, then the global quotients
M G and (M×_G K) K describe the same orbifold.
So if the orbifold cohomology theory is represented by equivariant spectra
{E_G}_G as indicated above, then necessarily E_G≃^K_G(E_K), i.e., 
the equivariant homotopy types are consistent under restriction.
This is the characteristic feature of global equivariant homotopy types, 
and it suggest that the latter ought to define orbifold cohomology theories.

The approach to global homotopy theory presented in this book
in particular provides a way to turn the above outline into rigorous mathematics.
There are different formal frameworks for 
stacks and orbifolds (algebro-geometric, smooth, topological), 
and these objects can be studied with respect to various notions of `equivalence'.
The approach that most easily feeds into our present context 
are the notions of topological stacks
respectively orbispaces as developed by Gepner and Henriques
in their paper <cit.>. Their homotopy theory of topological stacks
is rigged up so that the derived mapping spaces out of 
the classifying stacks for principal G-bundles detect equivalences.
In our setup, the global classifying spaces of compact Lie groups
(see Definition <ref>)
play exactly the same role, and this is another hint of a deeper connection.
In fact, the global homotopy theory of orthogonal spaces
as developed in Chapter <ref>
is a model for the homotopy theory of orbispaces in the sense of
Gepner and Henriques. 
For a formal comparison of the two models we refer the reader to
the author's paper <cit.>.
The comparison proceeds through yet another model, the
global homotopy theory of `spaces with an action
of the universal compact Lie group'. 
Here the universal compact Lie group (which is neither compact nor a Lie group)
is the topological monoid  of linear isometric self-embeddings
of ^∞, and in <cit.> 
we establish a global model structure on the
category of -spaces.

If we now accept that one can pass between stacks, orbispaces and
orthogonal spaces in homotopically meaningful way, a consequence is
that every global stable homotopy type (i.e., every orthogonal spectrum)
gives rise to a cohomology theory on stacks and orbifolds.
Indeed, by taking the unreduced suspension spectrum,
every unstable global homotopy type is transferred into the triangulated
global stable homotopy category . In particular, taking morphisms
in  into an orthogonal spectrum E defines  -graded 
E-cohomology groups. 
The counterpart of a global quotient M G in the global homotopy 
theory of orthogonal spaces is the semifree orthogonal space _G,V M
introduced in Construction <ref>.
By the adjunction relating the global and G-equivariant stable homotopy 
categories (see Theorem <ref>),
the morphisms Σ^∞_+ _G,VM, E in the global stable
homotopy category biject with the G-equi­variant E-co­homo­logy groups of M.
In other words, when evaluated on a global quotient M G, our recipe
for generating an orbifold cohomology theory from a global stable homotopy type
precisely returns the G-equivariant cohomology of M,
which was the original design criterion.

The procedure sketched so far actually applies to more general objects 
than our global stable homotopy types: indeed, all that was needed to produce
the orbifold cohomology theory was a sufficiently exact functor from 
the homotopy theory of orbispaces to a triangulated category.
If we aim for a stable homotopy theory (as opposed to its triangulated
homotopy category), then there is a universal example, namely
the stabilization of the homotopy theory of orbispaces, obtained by formally inverting 
suspension. Our global theory is, however, richer than 
this `naive' stabilization. Indeed, the global stable homotopy category forgets to
the G-equivariant stable homotopy category based on a complete G-universe;
the equivariant cohomology theories represented by such objects are
usually called `genuine' (as opposed to `naive').
Genuine equivariant cohomology theories have much more structure
than naive ones; this structure manifests itself in different forms, 
for example as transfer maps, stability under `twisted suspension' 
(i.e., smash product with linear representation spheres), 
an extension of the -graded cohomology groups to an R O(G)-graded theory,
and an equivariant refinement 
of additivity (the so called Wirthmüller isomorphism).
Hence global stable homotopy types in the sense of this book
represent genuine (as opposed to `naive') orbifold cohomology theories.
subjectorbifold|)



Organization.
In Chapter <ref> we set up unstable global homotopy theory
using orthogonal spaces, i.e., continuous functors from 
the category of finite-dimensional inner product spaces 
and linear isometric embeddings to spaces.
We introduce global equivalences (Definition <ref>),
discuss global classifying spaces of compact Lie groups
(Definition <ref>),
and set up the global model structures on the category of orthogonal spaces
(Theorem <ref>). 
In Section <ref>
we investigate the box product of orthogonal spaces 
from a global equivariant perspective.
Section <ref>
introduces a variant of unstable global homotopy
theory based on a global family, i.e., a class  of 
compact Lie groups with certain closure properties. 
We discuss the -global model structure 
and record that for multiplicative global families, it lifts to category of modules and
algebras (Corollary <ref>).
In Section <ref> we discuss 
the G-equivariant homotopy sets of orthogonal spaces
and identify the natural structure between them
(restriction maps along continuous group homomorphisms).
The study of natural operations on π_0^G(Y) 
is a recurring theme throughout this book;
in the later chapters we return to it in the contexts of
ultra-commutative monoids, orthogonal spectra and ultra-commutative ring spectra.

Chapter <ref> is devoted to ultra-commutative monoids
(a.k.a. commutative mo­noids with respect to the box product, 
or lax symmetric monoidal functors), which we want to advertise as 
a rigidified notion of `global E_∞-space'.
In Section <ref>
we establish a global model structure for ultra-commutative monoids
(Theorem <ref>). 
Section <ref> introduces and studies
global power monoids, the algebraic structure that an ultra-commutative multiplication
gives rise to on the homotopy group -functor _0(R).
Section <ref> contains a large collection
of examples of ultra-commutative monoids and interesting morphisms between them.
In Section <ref> 
we discuss and compare different global refinements of
the non-equivariant homotopy type B O, 
the classifying space for the infinite orthogonal group.
Section <ref>
discusses `units' and `group completions' of ultra-commutative monoids.
As an application of this technology we formulate and prove 
a global, highly structured version of Bott periodicity,
see Theorem <ref>.

Chapter <ref> is a largely self-contained 
exposition of many basics about equivariant stable homotopy theory
for a fixed compact Lie group, modeled by orthogonal G-spectra.
In Section <ref> 
we recall orthogonal G-spectra and equivariant homotopy groups
and prove their basic properties, such as the suspension isomorphism 
and long exact sequences of mapping cones and homotopy fibers, 
and the additivity of equivariant homotopy groups on sums and products.
Section <ref> discusses the Wirthmüller isomorphism
and the closely related transfers.
In Section <ref> we introduce and study
geometric fixed point homotopy groups, an alternative invariant
to characterize equivariant stable equivalences. 
Section <ref> contains a proof of 
the double coset formula for the composite of a transfer followed by
the restriction to a closed subgroup.
We review Mackey functors for finite groups and show that
after inverting the group order, the category of G-Mackey functors
splits as a product, indexed by conjugacy classes of subgroups,
of module categories over the Weyl groups (Theorem <ref>).
A topological consequence is that after inverting the group order,
equivariant homotopy groups and geometric fixed point homotopy groups
determine each other in a completely algebraic fashion,
compare Proposition <ref> 
and Corollary <ref>.
Section <ref> is devoted to multiplicative aspects of
equivariant stable homotopy theory.

Chapter <ref> sets the stage for stable 
global homotopy theory, based on the notion of global equivalences
for orthogonal spectra (Definition <ref>).
We discuss semifree orthogonal spectra
and identify certain morphisms between semifree orthogonal spectra
as global equivalences (Theorem <ref>).
In Section <ref>
we investigate global functors, the natural algebraic structure
on the collection of equivariant homotopy groups of a global stable homotopy type.
Among other things, we explicitly calculate the 
algebra of natural operations on equivariant homotopy groups
(Theorem <ref>).
In Section <ref> we complement 
the global equivalences of orthogonal spectra by a stable model structure
that we call the global model structure.
Its fibrant objects are the `global Ω-spectra' 
(Definition <ref>),
the natural concept of a `global infinite loop space' in our setting.
Here we work more generally relative to a global family  and consider the 
-equivalences (i.e., equivariant stable equivalences for all compact
Lie groups in the family ).
We follow the familiar outline: a certain -level model structure
is Bousfield localized to an -global model structure
(see Theorem <ref>).
In Section <ref> we develop some basic theory 
around the global stable homotopy category;
since it comes from a stable model structure, this category is naturally
triangulated and we show that the suspension spectra of
global classifying spaces form a set of compact generators 
(Theorem <ref>).
In Section <ref> we vary the global family:
we construct and study left and right adjoints
to the forgetful functors associated to a change of global family
(Theorem <ref>).
As an application of Morita theory for stable model categories
we show that rationally the global homotopy category for finite groups
has an algebraic model, namely the derived category of rational global functors
(Theorem <ref>).

Chapter <ref> focuses on
ultra-commutative ring spectra, i.e.,  commutative orthogonal ring spectra
under multiplicative global equivalences. 
Section <ref> introduces `global power functors',
the algebraic structure on the equivariant homotopy
groups of ultra-commutative ring spectra.
Roughly speaking, global power functors are
global Green functors equipped with additional power operations,
satisfying various properties reminiscent
of those of the power maps x↦ x^m in a commutative ring.
The power operations give rise to norm maps
(`multiplicative transfers') along finite index inclusions,
and in our global context, the norm maps conversely determine the power operations,
compare Remark <ref>.
As we show  in Theorem <ref>,
the 0-th equivariant homotopy groups of an ultra-commutative ring spectrum
form a global power functor.
In Section <ref> 
we develop a description of the category of global power functors
via the comonad of `exponential sequences' 
(Theorem <ref>)
and discuss localization of global power functors
at a multiplicative subset of the underlying ring
(Theorem <ref>).
In Section <ref> we give various examples of 
global power functors, such as the Burnside ring global power functor, 
the global functor represented by an abelian compact Lie group,
free global power functors, constant global power functors,
and the complex representation ring global functor.
In Section <ref> we establish the global model structure
for ultra-­commu­tative ring spectra (Theorem <ref>)
and show that every global power functor is realized 
by an ultra-commutative ring spectrum
(Theorem <ref>).

Chapter <ref> is devoted to interesting examples 
of ultra-commutative ring spectra.
Section <ref> discusses two orthogonal 
Thom spectra  and .
The spectrum  is globally connective
and closely related to equivariant bordism. 
The global functor _0() admits a short and elegant algebraic presentation:
it is obtained from the Burnside ring global functor by imposing the 
single relation _e^C_2=0, compare Theorem <ref>.
The Thom spectrum  was first considered by tom Dieck
and it represents `stable' equivariant bordism; it is periodic
for orthogonal representations of compact Lie groups, and admits Thom isomorphisms
for equivariant vector bundles.
The equivariant homology theory represented by  can be obtained
from the one represented by  in an algebraic fashion, by inverting
the collection of `inverse Thom classes', 
compare Corollary <ref>.
Section <ref> recalls the geometrically defined
equivariant bordism theories.
The Thom-Pontryagin construction 
maps the unoriented G-equivariant bordism ring 𝒩_*^G
to the equivariant homotopy ring π_*^G(),
and that map is an isomorphism when G is a product of a finite group and a torus,
see Theorem <ref>. 
We discuss global K-theory in Sections <ref> 
and <ref>,
which comes in three interesting flavors as
connective global K-theory ,
global connective K-theory  and 
periodic global K-theory  (and in the real versions
, ^c and ).

We include three appendices where we collect material that is mostly
well-known, but that is either scattered through the literature
or where we found the existing expositions too sketchy.
Appendix <ref> is a self-contained review of
compactly generated spaces, our basic category to work in.
Appendix <ref> deals with fundamental properties
of equivariant spaces, including the basic model structure 
in Proposition <ref>.
We also provide an exposition of the
equivariant -space machinery, culminating in a version
of the Segal-Shimakawa delooping machine.
In Appendix <ref> we review the basic definitions, 
properties and constructions involving categories of enriched functors.

While most of the material in the appendices is well-known, 
there are a few results I could not find in the literature.
These results include the fact that compactly generated spaces
are closed under geometric realization 
(Proposition <ref> (iii)),
fixed points commute with geometric realization and latching objects
(Proposition <ref> (iv)),
and compactly generated spaces are closed under prolongation of -spaces
(Proposition <ref>).
Also apparently new are the results that prolongation of
G-cofibrant -G-spaces to finite G-CW-complexes 
is homotopically meaningful (Proposition <ref>),
and that prolongation of G-cofibrant -G-spaces 
to spheres gives rise to G-Ω-spectra
(for very special -G-spaces, see Theorem <ref>)
respectively positive G-Ω-spectra (for special -G-spaces,
see Theorem <ref>).
Here the key ideas all go back to Segal <cit.> 
and Shimakawa <cit.>; however, we formulate our results for the
prolongation (i.e., categorical Kan extension), whereas Segal and Shimakawa
work with a bar construction (also known as a homotopy coend
or homotopy Kan extension) instead.
We also give a partial extension of the machinery to compact Lie groups,
whereas previous papers on the subject restrict attention to finite groups.
As we explain in Remark <ref>, there is no hope
to obtain a G-Ω-spectrum by evaluation on spheres
for compact Lie groups of positive dimension.
However, we do prove in Theorem <ref> 
that evaluating a G-cofibrant special -G-space
on spheres yields a `G^∘-trivial positive G-Ω-spectrum',
where G^∘ is the identity component of G.
Our Appendix <ref> substantially overlaps
with the paper <cit.> by May, Merling and Osorno that
provides comparisons of prolongation, 
bar construction and the operadic approach to equivariant deloopings.




Relation to other work.
The idea of global equivariant homotopy theory is not at all new and has 
previously been explored in different contexts. For example,
in Chapter II of <cit.>, Lewis and May define coherent families
of equivariant spectra; these consist of collections of equivariant 
coordinate free spectra in the sense Lewis, May and Steinberger,
equipped with comparison maps involving change of groups
and change of universe functors.

The approach closest to ours are the 
global ℐ_*-functors 
introduced by Greenlees and May in <cit.>.
These objects are `global orthogonal spectra' in that they are indexed on pairs
(G,V) consisting of a compact Lie group and a G-representation V.
The corresponding objects with commutative multiplication are called
global ℐ_*-functors with 
smash products in <cit.> 
and it is for these that Greenlees and May define and study multiplicative
norm maps. Clearly, an orthogonal spectrum gives
rise to a  global ℐ_*-functors in the sense of Greenlees and May.
In the second chapter of her thesis <cit.>,
Bohmann compares the approaches of Lewis-May and Greenlees-May;
in the paper <cit.> she also relates these to orthogonal spectra.

Symmetric spectrasubjectsymmetric spectrumsubjectspectrum!symmetric|seesymmetric spectrum 
in the sense of Hovey, Shipley and Smith <cit.>
are another prominent model for the (non-equivariant) stable homotopy category.
Much of what we do here with orthogonal spectra can also be done 
with symmetric spectra, if one is willing to restrict to finite groups
(as opposed to general compact Lie groups). 
This restriction arises because only finite groups embed into symmetric groups,
while every compact Lie group embeds into an orthogonal group.
Hausmann <cit.> 
has established  a global model structure on the category of symmetric spectra, 
and he showed that the forgetful functor is a right Quillen equivalence
from the category of orthogonal spectra with the -global model structure
to the category of symmetric spectra with the global model structure.
While some parts of the symmetric and orthogonal theories are similar,
there are serious technical complications arising from
the fact that for symmetric spectra the naively defined equivariant homotopy groups 
are not `correct', a phenomenon that is already present non-equivariantly.





Prerequisites. 
This book assumes a solid background in algebraic topology
and (non-equivariant) homotopy theory, including topics such as
singular homology and cohomology, CW-complexes, homotopy groups, 
mapping spaces, loop spaces, fibrations and fiber bundles,
Eilenberg-Mac Lane spaces, smooth manifolds, Grassmannian
and Stiefel manifolds. Two modern references that contain
all we need (and much more) are the textbooks by Hatcher <cit.>
and tom Dieck <cit.>.
Some knowledge of non-equivariant stable homotopy theory
is helpful to appreciate the equivariant and global features of the structures
and examples we discuss; from a strictly logical perspective, however,
the non-equivariant theory is a degenerate special case of the global theory for
the global family of trivial Lie groups.
In particular, by simply ignoring all group actions, 
the examples presented in this book give models for many interesting 
and prominent non-equivariant stable homotopy types.

Since actions of compact Lie groups are central to this book,
some familiarity with the structure and representation theory of
compact Lie groups is obviously helpful, but we give references
to the literature whenever we need any non-trivial facts.
Many of our objects of study organize themselves into model categories
in the sense of Quillen <cit.>, so some background on model
categories is necessary to understand the respective sections.
The article <cit.> by Dwyer and Spalinski is a good introduction,
and Hovey's book <cit.> is still the definitive reference.
Some acquaintance with unstable equivariant homotopy theory is useful
(but not logically necessary). In contrast, we do not assume any prior knowledge
of equivariant stable homotopy theory, and Chapter <ref>
is a self-contained introduction based on equivariant orthogonal spectra.
The last two sections of Chapter <ref> study the global stable
homotopy category, and here we freely use the language of triangulated categories.
The first chapter of Neeman's book <cit.>
is a possible reference for the necessary background.

Throughout the book we work in the category
of compactly generated spaces in the sense of McCord <cit.>,
i.e., a k-spaces (also called Kelley spaces)
that satisfy the weak Hausdorff condition, see Definition <ref>.
Since the various useful properties of
compactly generated spaces are scattered through the literature,
we include a detailed discussion in Appendix <ref>.



Acknowledgments. 
I would like to thank John Greenlees for being a reliable consultant
on matters of equivariant stable homotopy theory; Johannes Ebert and Michael Joachim
for tutorials on the C^*-algebraic approach to equivariant K-theory;
and Markus Hausmann and Irakli Patchkoria 
for careful reading of large parts of this book, along with
numerous suggestions for improvements. 
I am indebted to Benjamin Böhme, Jack Davies, Lars Hesselholt, Richard Hitzl,
Matthias Kreck, Jacob Lurie, Cary Malkiewich, Peter May,
Mona Merling, Luca Pol, Steffen Sagave, Albert Schulz, Neil Strickland, 
Karol Szumiło, Peter Teichner, Andreas Thom, Jorge Vitória,
Christian Wimmer and Mahmoud Zeinalian for various helpful discussions,
comments and suggestions.









CHAPTER: UNSTABLE GLOBAL HOMOTOPY THEORY


In this chapter we develop a framework for unstable global homotopy theory
via orthogonal spaces, i.e., continuous functors from the
linear isometries category  to spaces. 
In Section <ref> we define
global equivalences of orthogonal spaces and establish many basic properties
of this class of morphisms. 
We also introduce global classifying spaces of compact Lie groups, 
the basic building blocks of global homotopy types.
In Section <ref> we complement the
global equivalences by a global model structure 
on the category of orthogonal spaces.
The construction follows a familiar pattern, by Bousfield localization
of an auxiliary `strong level model structure'.
Section <ref> also contains
a discussion of cofree orthogonal spaces, i.e., global homotopy types
that are `right induced' from non-equivariant homotopy types.
In Section <ref> we recall the box product of orthogonal spaces, 
a Day convolution product based on orthogonal direct sum of inner product spaces.
The box product is a symmetric monoidal product,
fully homotopical under global equivalences, and globally equivalent to
the cartesian product.
Section <ref>
introduces an important variation of our theme, where we discuss
unstable global homotopy theory for a `global family',
i.e., a class of compact Lie groups with certain closure properties.
In Section <ref> we introduce
the G-equivariant homotopy set π_0^G(Y) of an orthogonal space
and identify the natural structure on these sets 
(restriction maps along continuous group homomorphisms).
The study of natural operations on the sets π_0^G(Y) 
is a recurring theme throughout this book,
and we will revisit and extend the results in the later chapters
for ultra-commutative monoids, orthogonal spectra and ultra-commutative ring spectra.

Our main reason for working with orthogonal spaces
is that they are the direct unstable analog of orthogonal spectra,
and in this unstable model for global homotopy theory
the passage to the stable theory in Chapter <ref>
is especially simple. However, there are other models for 
unstable global homotopy theory, most notably the topological stacks
and the orbispaces as developed by Gepner and Henriques
in their paper <cit.>. 
For a comparison of these two models to our orthogonal space model 
we refer to the author's paper <cit.>.
The comparison proceeds through yet another model, the
global homotopy theory of `spaces with an action
of the universal compact Lie group'. 
Here the universal compact Lie group (which is neither compact nor a Lie group)
is the topological monoid  of linear isometric self-embeddings
of ^∞, and in <cit.> 
we establish a global model structure on the
category of -spaces.





§ ORTHOGONAL SPACES AND GLOBAL EQUIVALENCES



In this section we introduce orthogonal spaces, 
along with the notion of global equivalences,
our setup to rigorously formulate the idea of 
`compatible equivariant homotopy types for all compact Lie groups'.
We introduce various basic techniques to manipulate global equivalences
of orthogonal spaces, such as recognition criteria by 
homotopy or strict colimits over representations 
(Propositions <ref> and <ref>),
and a list of standard constructions that preserve global equivalences
(Proposition <ref>).  
Theorem <ref> is a cofinality result
for orthogonal spaces, showing that fairly general changes in the 
linear isometries indexing category do not affect the global homotopy type.
Definition <ref> introduces
global classifying spaces of compact Lie groups,
the basic building blocks of global homotopy theory.
Proposition <ref> 
justifies the name by explaining in which sense the global classifying space B_ G
`globally classifies' principal G-bundles.






Before we start, let us fix some notation and conventions.
By a `space' we mean a compactly generated space in the sense of <cit.>,
i.e., a k-space (also called Kelley space)
that satisfies the weak Hausdorff condition,
see Definition <ref>.
We denote the category of compactly generated spaces
by  and review the basic properties in Appendix <ref>.

An inner product spacesubjectinner product space
is a finite-dimensional real vector space 
equipped with a scalar product, i.e., a positive definite symmetric bilinear form.
We denote by symbol - category of finite-dimensional inner product
spaces  
the category with objects the 
inner product spaces and morphisms the linear isometric embeddings.
The category  is a topological category in the sense
that the morphism spaces come with a preferred topology:
if φ:V→ W is one linear isometric embedding,
then the action of the orthogonal group O(W), by postcomposition,
induces a bijectionsymbol(V,W) - space of linear isometric embeddings 

    O(W)/O(φ^⊥)   (V,W)  ,   
    A · O(φ^⊥)  ⟼ A∘φ ,

where φ^⊥=W-φ(V) is the orthogonal complement of the image of φ.
We topologize (V,W) so that this bijection is a homeomorphism, and
this topology is independent of φ.
If (v_1,…,v_k) is an orthonormal basis of V, then for every 
linear isometric embedding φ:V→ W the tuple
(φ(v_1),…,φ(v_k)) is an orthonormal k-frame of W.
This assignment is a homeomorphism from (V,W) to the Stiefel manifold
of k-frames in W.


An example of an inner product spaces is the vector space ^n 
with the standard scalar product

    x,y =  x_1 y_1  +  … +   x_n y_n  .

In fact, every inner product space V is isometrically isomorphic
to the inner product space ^n, for n the dimension of V.
So the full topological subcategory with objects
the ^n is a small skeleton of . 


 
An orthogonal spacesubjectorthogonal space 
is a continuous functor Y:→ to the category of spaces.
A morphism of orthogonal spaces is a natural transformation.
We denote by symbol   - category of orthogonal spaces 
the category of orthogonal spaces.subjectmorphism!of orthogonal spaces 





The use of continuous functors from the category  to spaces has a long history
in homotopy theory. 
The systematic use of inner product spaces (as opposed to numbers)
to index objects in stable homotopy theory seems to go back to Boardman's 
thesis <cit.>.
The category  (or its extension that
also contains countably infinite dimensional inner product spaces)
is denoted ℐ by Boardman and Vogt <cit.>,
and this notation is also used in <cit.>;
other sources <cit.> use the symbol ℐ.
Accordingly, orthogonal spaces are sometimes referred to as ℐ-functors,
ℐ-spaces or ℐ-spaces.
Our justification for using yet another name is twofold: on the one hand,
our use of orthogonal spaces comes with a shift in emphasis, away from 
a focus on non-equivariant homotopy types, and towards viewing an
orthogonal space as representing compatible equivariant homotopy types for
all compact Lie groups. Secondly, we want to stress the analogy between
orthogonal spaces and orthogonal spectra, the former being an unstable global
world with the latter the corresponding stable global world.

Now we define our main new concept, the notion of `global equivalence'
between orthogonal spaces.
We let G be a compact Lie group. By a G-re­presen­tationsubjectG-representation@G-representation
we mean a finite-dimensional orthogonal representation, 
i.e., a real inner product space
equipped with a continuous G-action by linear isometries.
In other words, a G-representation consists of an inner product space V
and a continuous homomorphism ρ:G→ O(V).
In this context, and throughout the book,
we will often use without explicit mentioning that
continuous homomorphisms between Lie groups are automatically smooth,
see for example <cit.>.
For every orthogonal space Y and every G-representation V,
the value Y(V) inherits a G-action from the
G-action on V and the functoriality of Y. 
For a G-equivariant linear isometric embedding φ:V→ W
the induced map Y(φ):Y(V)→ Y(W) is G-equivariant.

We denote by

    D^k  =  {x∈^k  :  x,x≤ 1}   and  ∂ D^k  =  {x∈^k  :  x,x= 1}

the unit disc in ^k respectively
its boundary, a sphere of dimension k-1.
In particular, D^0={0} is a one-point space and ∂ D^0=∅
is empty.


A morphism f:X→ Y
of orthogonal spaces is a global equivalencesubjectglobal equivalence!of orthogonal spaces 
if the following condition holds: for every compact Lie group G,
every G-representation V, every k≥ 0 
and all continuous maps α:∂ D^k→ X(V)^G
and β:D^k→ Y(V)^G such that β|_∂ D^k=f(V)^G∘α,
there is a G-representation W, a G-equivariant linear
isometric embedding φ:V→ W and a continuous map λ:D^k→ X(W)^G
such that λ|_∂ D^k=X(φ)^G∘α and
such that f(W)^G∘λ
is homotopic, relative to ∂ D^k, to Y(φ)^G∘β.


In other words, for every commutative square on the left

    @C=10mm∂ D^k[r]^-α[d]_incl    X(V)^G [d]^f(V)^G   ∂ D^k[r]^-α[d]_incl    X(V)^G [r]^-X(φ)^G    
    X(W)^G [d]^f(W)^G
    
    D^k[r]_-β    Y(V)^G     
    D^k[r]_-β@–>[urr]^-(.4)λ    Y(V)^G [r]_-Y(φ)^G    Y(W)^G

there exists the lift λ on the right hand side that makes the upper left
triangle commute on the nose, and the lower right triangle up to homotopy
relative to ∂ D^k. In such a situation we will often 
refer to the pair (α,β) as a `lifting problem' and we will say that
the pair (φ,λ) solves the lifting problem.


 If X=A and Y=B 
are the constant orthogonal spaces with 
values the spaces A respectively B, and f=g the constant 
morphism associated to a continuous map g:A→ B,
then g is a global equivalence if and only if
for every commutative square 

    @C=10mm∂ D^k[r] [d]_incl    A  [d]^g 
    
    D^k[r] @–>[ur]^-(.4)λ    B

there exists a lift λ that makes the upper left
triangle commute, and the lower right triangle up to homotopy
relative to ∂ D^k. But this is one of the equivalent ways of
characterizing weak equivalences of spaces,
compare <cit.>. 
So g is a global equivalence
if and only if g is a weak equivalence.  




The notion of global equivalence is meant to capture the idea that
for every compact Lie group G,
some induced morphism 

    _V f(V) : _V X(V) → _V Y(V)

is a G-weak equivalence, where `_V' is a suitable homotopy
colimit over all G-representations V along all 
equivariant linear isometric embeddings.
This is a useful way to think about global equivalences, 
and it could be made precise by letting V run over the poset of
finite-dimensional subrepresentations of a complete G-universe and
using the Bousfield-Kan construction of a homotopy colimit over this poset.
Since the `poset of all G-representations' has a cofinal
subsequence, called an exhaustive sequence 
in Definition <ref>,
we can also model the `homotopy colimit over all G-representations'
as the mapping telescope over an exhaustive sequence.
However, the actual definition we work with has the advantage that
it does not refer to universes
and we do not have to define or manipulate homotopy colimits.

In many examples of interest, all the structure maps of an orthogonal space Y
are closed embeddings. When this is the case, the actual colimit
(over the subrepresentations of a complete universe) of the G-spaces Y(V)
serves the purpose of a `homotopy colimit over all representations',
and it can be used to detect global equivalences, 
compare Proposition <ref> below.





We will now establish some useful criteria for detecting global equivalences.
We call a continuous map f:A→ B 
an h-cofibrationsubjecth-cofibration 
if it has the homotopy extension property, 
i.e., given a continuous map φ:B→ X and a homotopy
H:A× [0,1]→ X starting with φ f,
there is a homotopy H̅:B× [0,1]→ X starting with φ
such that H̅∘(f× [0,1])=H.
Below we will write H_t=H(-,t):A→ X.
All h-cofibrations in the category of compactly generated spaces
are closed embeddings, compare Proposition <ref>.
The following somewhat technical lemma should be well known, but I was unable
to find a reference.


 
Let A be a subspace of a space B such that the 
inclusion A→ B is an h-cofibration.
Let f:X→ Y be a continuous map and

    H :  A× [0,1] → X      and    
    K  :  B× [0,1] → Y

homotopies such that K|_A× [0,1]=f  H.
Then the lifting problem (H_0,K_0)  has a solution if and only if
the lifting problem (H_1,K_1) has a solution. 


The problem is symmetric, so we only show one direction.
We suppose that the lifting problem (H_0,K_0)  has a solution 
consisting of a continuous map λ:B→ X 
such that λ|_A=H_0 and a homotopy G:B×[0,1]→ Y
such that

    G_0  =  f∘λ ,    G_1 =  K_0      and    
    (G_t)|_A  =  f∘ H_0

for all t∈[0,1]. 
The homotopy extension property provides
a homotopy H':B×[0,1]→ X such that

    H'_0 =  λ     and     H'|_A×[0,1] =  H  .

Then the map λ'=H'_1:B→ X satisfies

    λ'|_A  =  ( H'_1)|_A  =  H_1 .

We define a continuous map J:B×[0,3]→ Y by

    J_t  = 
      f∘ H'_1-t    for 0≤ t≤ 1,
     G_t-1    for 1≤ t≤ 2, and
     K_t-2    for 2≤ t≤ 3.

In particular,

    J_0  =  f∘λ'     and     
    J_3 =  K_1  ;

so J almost witnesses the fact that λ' solves the 
lifting problem (H_1,K_1), except that J is not a relative homotopy.

We improve J to a relative homotopy from f∘λ' to K_1.
We define a continuous map L:A×[0,3]×[0,1]→ Y by

    L(-,t,s)  = 
    f∘ H_1-t    for 0≤ t≤ s, 
    
    f∘ H_1-s    for s≤ t≤ 3-s, and
    
    f∘ H_t-2    for 3-s≤ t≤ 3.

Then L(-,-,0) is the constant homotopy at the map f∘ H_1, and  

    L(-,-,1)  =  J|_A× [0,3] : A×[0,3] →  Y .

Since the inclusion of A into B is an h-cofibration, 
the inclusion
of B×{0}∪_A×{0} A× [0,1] into B×[0,1]
has a continuous retraction; hence the inclusion

    B×{0}× [0,1] ∪_A×{0}× [0,1] A× [0,1]× [0,1] → 
    B×[0,1]× [0,1]

also has a continuous retraction.
We abbreviate D=[0,3]×{1}∪{0,3}× [0,1];
the pair of spaces ([0,3]× [0,1],  D)
is pair homeomorphic to ([0,1]×[0,1], {0}× [0,1]).
So the inclusion

    B× D ∪_A× D A× [0,3]× [0,1] → B× [0,3]× [0,1]

has a continuous retraction.
The map L and the map

    J ∪const_fλ∪const_K_1 :  
    B× D  =  B× ( [0,3]×{1}∪{0,3}× [0,1]) → Y

agree on A× D, so there is a continuous map
L̅:B×[0,3]× [0,1]→ Y such that

    L̅(-,-,1) =  J  ,     L̅|_A×[0,3]× [0,1] =  L  ,

and

    L̅(-,0,s) =  f∘λ     and    L̅(-,1,s) =  K_1
 
for all s∈[0,1].
The map J̅=L̅(-,-,0):B×[0,3]→ Y then satisfies

    J̅|_A× [0,3] =  L̅(-,-,0)|_A× [0,3] =  
    L(-,-,0) ,

which is the constant homotopy at the map f∘ H_1;
so J̅ is a homotopy (parametrized by [0,3] instead of [0,1]) 
relative to A.
Because

    J̅_0  =  L̅(-,0,0) =  f∘λ     and    J̅_3  =  L̅(-,3,0) =  K_1 ,

the homotopy J̅ witnesses that λ' solves the lifting problem (H_1,K_1).





Let G be a compact Lie group. 
An exhaustive sequencesubjectexhaustive sequence!of representations
is a nested sequence

    V_1  ⊂ V_2  ⊂ … ⊂ V_n  ⊂ …

of finite-dimensional G-representations such that every finite-dimensional
G-representation admits a linear isometric G-embedding into some V_n.


Given an exhaustive sequence {V_i}_i≥ 1 of G-representations 
and an orthogonal space Y, the values at the representations
and their inclusions form a sequence of G-spaces and G-equivariant continuous maps

    Y(V_1)  → Y(V_2)  → ⋯ → Y(V_i)  → ⋯ .

We denote by

    _i  Y(V_i)

the mapping telescope of this sequence of G-spaces; this telescope inherits
a natural G-action.

We recall that a G-equivariant continuous map f:A→ B between G-spaces
is a G-weak equivalencesubjectG-weak equivalence@G-weak equivalence
if for every closed subgroup H of G
the map f^H:A^H→ B^H of H-fixed points is a weak homotopy equivalence
(in the non-equivariant sense).

 
For every morphism of orthogonal spaces f:X→ Y, the following
three conditions are equivalent.


  * The morphism f is a global equivalence.

  * For every compact Lie group G,
every G-representation V, every finite G-CW-pair (B,A)
and all continuous G-maps α:A→ X(V)
and β:B→ Y(V) such that β|_A=f(V)∘α,
there is a G-representation W, a G-equivariant linear
isometric embedding φ:V→ W and a continuous G-map λ:B→ X(W)
such that λ|_A=X(φ)∘α and
such that f(W)∘λ
is G-homotopic, relative to A, to Y(φ)∘β.

  * For every compact Lie group G and every exhaustive sequence {V_i}_i≥ 1
of G-representations the induced map

    _i f(V_i) :  _i X(V_i) → _i Y(V_i)

is a G-weak equivalence.



At various places in the proof we use without explicit mentioning that
taking G-fixed points commutes with formation of the mapping telescopes;
this follows from the fact that taking G-fixed points
commutes with pushouts along closed embeddings and sequential colimits
along closed embeddings, compare Proposition <ref>.

(i)⟹(ii)  
We argue by induction over the number of the relative G-cells in (B,A).
If B=A, then λ=α solves the lifting problem, and there is nothing to show.
Now we suppose that A is a proper subcomplex of B. We choose a
G-CW-subcomplex B' that contains A and such that (B,B') has
exactly one equivariant cell. Then (B',A) has strictly fewer cells,
and the restricted equivariant lifting problem 
(α:A→ X(V),β'=β|_B':B'→ Y(V)) 
has a solution (φ:V→ U, λ':B'→ X(U))
by the inductive hypothesis.

We choose a characteristic map for the last cell, i.e., a pushout square of G-spaces

    G/H ×∂ D^k [r]^-χ[d]_incl    B' [d]^incl
    
    G/H × D^k [r]_-χ    B

in which H is a closed subgroup of G.
We arrive at the non-equivariant lifting problem on the left:

    @C=13mm∂ D^k[r]^-(λ')^H∘χ̅[d]_incl    X(U)^H [d]^f(U)^H   ∂ D^k[r]^-(λ')^H∘χ̅[d]_incl    
    X(U)^H [r]^-X(ψ)^H    
    X(W)^H [d]^f(W)^H
    
    D^k[r]_-Y(φ)^H∘β^H∘χ̅    Y(U)^H     
    D^k[r]_-Y(φ)^H∘β^H∘χ̅@–>[urr]^-(.4)λ    
    Y(U)^H [r]_-Y(ψ)^H    Y(W)^H

Here χ̅=χ(e H,-):D^k→ B^H.
Since f is a global equivalence, there is an H-equivariant
linear isometric embedding ψ:U→ W and a continuous map 
λ:D^k→ X(W)^H such that 
λ|_∂ D^k=X(ψ)^H∘(λ')^H∘χ̅
and f(W)^H∘λ is homotopic, relative ∂ D^k, to
Y(ψ)^H∘ Y(φ)^H∘β^H∘χ̅, 
as illustrated by the diagram on the right above.
By enlarging W, if necessary, we can assume without loss of generality
that W is underlying a G-representation and ψ is even G-equivariant.

The G-equivariant extension of λ 

    G/H× D^k→ X(W)  ,    
    (g H,x) ⟼ g·λ(x)

and the map X(ψ)∘λ':B'→ X(W)
then agree on G/H×∂ D^k, so they glue to a G-map
λ̃: B → X(W).
The pair (ψφ:V→ W,λ̃:B→ X(W)) then
solves the original lifting problem (α,β).

(ii)⟹(iii)  
We suppose that f satisfies (ii), and we let G be any compact Lie group
and {V_i}_i≥ 1 an exhaustive sequence of G-representations.
We consider an equivariant lifting problem, i.e., a finite G-CW-pair (B,A) and
a commutative square:

    @C=15mm
    A[r]^-α[d]_incl   _i  X(V_i)  [d]^_i f(V_i)
    
    B[r]_-β   _i  Y(V_i)

We show that every such lifting problem has an equivariant solution.
Since B and A are compact, there is an n≥ 0
such that α has image in the truncated telescope _[0,n] X(V_i)
and β has image in the truncated telescope _[0,n] Y(V_i)
(see Proposition <ref> (i)).
There is a natural equivariant homotopy from the identity of the truncated  
telescope _[0,n] X(V_i) to the composite

    _[0,n] X(V_i)  proj X(V_n)  incl_[0,n] X(V_i)  .

Naturality means that this homotopy is compatible with the 
same homotopy for the telescope of the G-spaces Y(V_i).
Lemma <ref>
(or rather its G-equivariant generalization)
applies to these homotopies, so instead of the original lifting problem
we may solve the homotopic lifting problem

    @C=15mm
    A[r]^-α'[d]_incl     X(V_n)[r]^-i_n[d]_f(V_n)   _i  X(V_i)  [d]^_i f(V_i)
    
    B[r]_-β'    Y(V_n) [r]_-i_n   _i  Y(V_i)

where α' is the composite of the projection 
_[0,n] X(V_i) → X(V_n)
with α, viewed as a map into the truncated telescope,
and similarly for β'.

Since f satisfies (ii), the lifting problem
(α':A→ X(V_n), β':B→ Y(V_n))
has a solution after enlarging V_n along some linear isometric G-em­bedding.
Since the sequence {V_i}_i≥ 1 is exhaustive, we can take this embedding as
the inclusion i:V_n→ V_m for some m≥ n, i.e., there is 
a continuous G-map λ:B→ X(V_m) such that 
λ|_A=X(i)^G∘α'
and such that f(V_m)^G∘λ is G-homotopic, relative A,
to Y(i)^G∘β', compare the diagram:

    @C=15mm
    A[r]^-α'[d]_incl      
    X(V_n)  [r]^-X(i)    X(V_m)  [d]^f(V_m)
    
    B[r]_-β'@–>[urr]_-(.7)λ    Y(V_n)  [r]_-Y(i)    Y(V_m)

The composite

    @C=10mm
      X(V_n)[r]^-X(i)    X(V_m) [r]^-i_m   _i  X(V_i)

does not agree with i_n:X(V_n)→_i  X(V_i);
so the composite i_n∘λ:B→_i X(V_i) does not quite
solve the (modified) lifting problem (i_n ∘α', i_n∘β').
But there is a G-equivariant homotopy H:X(V_n)× [0,1]→_i X(V_i)
between i_m∘ X(i) and i_n, and a similar homotopy 
K:Y(V_n)× [0,1]→_i Y(V_i) for Y instead of X.
These homotopies satisfy

    K∘ (f(V_n)×[0,1])  =  (_i f(V_i))∘ H  ,

so Lemma <ref> implies that the modified
lifting problem, and hence the original lifting problem,
has an equivariant solution.

(iii)⟹(i)  
We let G be a compact Lie group,
V a G-representation, k≥ 0
and (α:∂ D^k→ X(V)^G, β:D^k→ Y(V)^G) 
a lifting problem, i.e., such that β|_∂ D^k=f(V)^G∘α.
We choose an exhaustive sequence {V_i} of G-representations;
then we can embed V into some V_n by a linear isometric G-map 
and thereby assume without loss of generality that V=V_n.
 
We let i_n:X(V_n)→_i X(V_i) and i_n:Y(V_n)→_i Y(V_i)
be the canonical maps. 
Since _i f(V_i):_i X(V_i)→_i Y(V_i)
is a G-weak equivalence, there is a continuous map
λ:D^k→ (_i X(V_i))^G such that λ|_∂ D^k=i_n^G∘α
and (_i f(V_i))^G∘λ is homotopic, relative ∂ D^k, to
i_n^G∘β.
Since fixed points commute with mapping telescopes and since D^k is compact,
there is an m≥ n such that λ and the relative homotopy
that witnesses the relation (_i f(V_i))^G∘λ≃ i_n^G∘β
both have image in
_[0,m] X(V_i)^G, the truncated telescope of the G-fixed points.
The following diagram commutes

    @C=14mm 
    X(V_n)^G [r]_-can[d]_f(V_n)^G@/^1pc/[rrr]^(.3)X(incl)^G   _[0,n] X(V_i)^G [r]_-incl[d]_ f(V_i)^G   _[0,m] X(V_i)^G [r]_-proj[d]^ f(V_i)^G   
    X(V_m)^G [d]^f(V_m)^G
    
    Y(V_n)^G [r]^-can@/_1pc/[rrr]_(.3)Y(incl)^G   _[0,n] Y(V_i)^G [r]^-incl   _[0,m] Y(V_i)^G [r]^-proj    Y(V_m)^G

where the right horizontal maps are the projections of 
the truncated telescope to the last term.
So projecting from _[0,m] X(V_i)^G to X(V_m)^G 
and from _[0,m] Y(V_i)^G to Y(V_m)^G 
produces the desired solution to the lifting problem. 




We establish some more basic facts about the class of global equivalences.
A homotopy between two morphisms of orthogonal spaces f,f':X→ Y is a morphism

    H  :  X× [0,1]  → Y

such that H(-,0)=f and H(-,1)=f'. 


  A morphism f:X→ Y of orthogonal spaces is a homotopy equivalence
  if there is a morphism g:Y→ X such that g f and f g
  are homotopic to the respective identity morphisms.
  The morphism f is a 
  strong level equivalencesubjectstrong level equivalence!of orthogonal spaces 
  if for every compact Lie group G and every G-representation V the map
  f(V)^G:X(V)^G→ Y(V)^G is a weak equivalence.
  The morphism f is a strong level fibrationsubjectstrong level fibration!of orthogonal spaces 
  if for every compact Lie group G and every G-representation V the map
  f(V)^G:X(V)^G→ Y(V)^G is a Serre fibration.
  

If f,f':X→ Y are homotopic morphisms of orthogonal spaces, 
then the maps f(V)^G, f'(V)^G:X(V)^G→ Y(V)^G
are homotopic for every compact Lie group G and every G-representation V.
So if f is a homotopy equivalence of orthogonal spaces,
then the map f(V)^G:X(V)^G→ Y(V)^G
is a non-equivariant homotopy equivalence for every G-representation V.
So every homotopy equivalence is in particular a strong level equivalence.
By the following proposition, strong level equivalences are global equivalences.

A continuous map φ:A→ B is a 
closed embeddingsubjectclosed embeddingsubjectembedding!closed|seeclosed embedding
if it is injective and a closed map.
Such a map is then a homeomorphism of A onto the closed subspace φ(A) of B.
If a compact Lie group G acts on two spaces A and B and
φ:A→ B is a G-equivariant closed embedding,
then the restriction φ^G:A^G→ B^G to G-fixed points 
is also a closed embedding.

We call a morphism f:A→ B of orthogonal spaces
an h-cofibrationsubjecth-cofibration!of orthogonal spaces 
if it has the homotopy extension property, 
i.e., given a morphism of orthogonal spaces φ:B→ X and a homotopy
H:A× [0,1]→ X starting with φ f,
there is a homotopy H̅:B× [0,1]→ X starting with φ
such that H̅∘(f× [0,1])=H.


 
  
  
  * Every strong level equivalence is a global equivalence.
  
  * The composite of two global equivalences is a global equivalence.
  
  * If f, g and h are composable morphisms of orthogonal spaces 
    such that h g and g f are global equivalences, then f, g, h and h g f
    are also global equivalences.
  
  * Every retract of a global equivalence is a global equivalence.
  
  * A coproduct of any set of global equivalences is a global equivalence.
  
  * A finite product of global equivalences is a global equivalence.
  
  * For every space K the functor -× K preserves global equivalences
    of orthogonal spaces.
  
  * Let e_n:X_n→ X_n+1 and f_n:Y_n→ Y_n+1 be morphisms 
    of orthogonal spaces that are objectwise closed embeddings, for n≥ 0. 
    Let ψ_n:X_n→ Y_n be global equivalences of orthogonal spaces
    that satisfy ψ_n+1∘ e_n=f_n∘ψ_n for all n≥ 0.
    Then the induced morphism ψ_∞:X_∞→ Y_∞ 
    between the colimits of the sequences is a global equivalence.
  
  * Let f_n:Y_n→ Y_n+1 be a global equivalence of orthogonal spaces
   that is objectwise a closed embedding, for n≥ 0. 
   Then the canonical morphism 
   f_∞:Y_0→ Y_∞ to the colimit of the sequence {f_n}_n≥ 0
   is a global equivalence.
 
  * Let 
   
    C  [d]_γ    A [l]_-g [d]^α[r]^-f     B [d]^β
    
         C'      A' [l]^-g'[r]_-f'    B'

   be a commutative diagram of orthogonal spaces such that f and f' are
   h-cofibrations.
   If the morphisms α,β and γ are global equivalences,
   then so is the induced morphism of pushouts
    
    γ∪β :  C∪_A B  → C'∪_A' B' .

  
  * Let 
    
    A [r]^-f [d]_g     B [d]^h
    
          C [r]_-k     D

    be a pushout square of orthogonal spaces such that f is a global equivalence.
    If in addition f or g is an h-cofibration, 
    then the morphism k is a global equivalence.
  
  * Let 
    
    P[r]^-k [d]_g     X [d]^f 
     Z [r]_-h     Y

    be a pullback square of orthogonal spaces in which f is a global equivalence.
    If in addition f or h is a strong level fibration, 
    then the morphism g is also a global equivalence.
 


(i)  We let f:X→ Y be a strong level equivalence, G a compact Lie group,
V a G-representation and α:∂ D^k→ X(V)^G
and β:D^k→ Y(V)^G continuous maps 
such that f(V)^G∘α=β|_∂ D^k.
Since f is a strong level equivalence, the map f(V)^G:X(V)^G→ Y(V)^G
is a weak equivalence, so there is a continuous map λ:D^k→ X(V)^G such that
λ|_∂ D^k=α and f(V)^G∘λ is homotopic to
β relative ∂ D^k. So the pair (_V,λ) solves the
lifting problem, and hence f is a global equivalence.


(ii) We let f:X→ Y and g:Y→ Z be global equivalences, G a compact Lie group,
(B,A) a finite G-CW-pair, V a G-representation and α:A→ X(V)
and β:B→ Z(V) continuous G-maps 
such that (g f)(V)∘α=β|_A.
Since g is a global equivalence, the equivariant lifting problem 
(f(V)∘α,β)
has a solution (φ:V→ W, λ:B→ Y(W)) such that

    λ|_A = Y(φ)∘ f(V)∘α =  
    f(W)∘ X(φ)∘α ,

and g(W)∘λ is homotopic to Z(φ)∘β relative A. 
Since f is a global equivalence,
the equivariant lifting problem (X(φ)∘α,λ)
has a solution (ψ:W→ U, λ':B→ X(U)) such that

    λ'|_A = X(ψ)∘ X(φ)∘α

and such that f(U)∘λ'
is G-homotopic to Y(ψ)∘λ relative A.
Then (g f)(U)∘λ' is G-homotopic, relative A, to

    g(U)∘ Y(ψ)∘λ =  Z(ψ)∘ g(W)∘λ

which in turn is G-homotopic to Z(ψφ)∘β,
also relative A. So the pair (ψφ,λ') solves
the original lifting problem for the morphism g f:X→ Z.

(iii) 
We let f:X→ Y, g:Y→ Z and h:Z→ Q be the three composable morphisms
such that g f:X→ Z and h g:Y→ Q are global equivalences.
We let G be a compact Lie group and {V_i}_i≥ 1 an exhaustive sequence
of G-representations. Evaluating everything in sight on the
representations and forming mapping telescopes yields three composable
continuous G-maps

    _i X(V_i) _i f(V_i)_i Y(V_i) _i g(V_i)_i Z(V_i) _i h(V_i)_i Q(V_i) .

Proposition <ref> shows that the G-maps

    (_i g(V_i))∘(_i f(V_i))    =  _i ( g f)(V_i)      and  
    
    (_i h(V_i))∘(_i g(V_i))    =  _i (h g)(V_i)

are G-weak equivalences.
Since G-weak equivalences satisfy the 2-out-of-6-property,
we conclude that the G-maps
_i f(V_i), _i g(V_i), _i h(V_i) 
and

    (_i h(V_i))∘(_i g(V_i))∘(_i f(V_i)) =  _i (h g f)(V_i)

are G-weak equivalences.
Another application of Proposition <ref> then shows that 
f, g, h and h g f are global equivalences.

(iv) Let g be a global equivalence and f a retract of g. So there is a
commutative diagram

    X[r]^-i [d]_f    X̅[r]^-r[d]^g     X[d]^f 
    
    Y [r]_-j    Y̅[r]_-s     Y

such that r i=_X and s j=_Y.
We let G be a compact Lie group, V a G-representation,
(B,A) a finite G-CW-pair and α:A→ X(V)
and β:B→ Y(V) continuous G-maps 
such that f(V)∘α=β|_A.
Since g is a global equivalence and

    g(V)∘ i(V)∘α =  j(V)∘ f(V)∘α =  
     ( j(V)∘β)|_A  ,

there is a G-equivariant linear isometric embedding φ:V→ W 
and a continuous G-map λ:B→X̅(W) such that
λ|_A=X̅(φ)∘ i(V)∘α 
and g(W)∘λ is G-homotopic to 
Y̅(φ)∘ j(V)∘β relative A. 
Then

    ( r(W)∘λ )|_A    =  
     r(W)∘X̅(φ)∘ i(V)∘α 
    =   X(φ)∘ r(V)∘ i(V)∘α =   
     X(φ)∘α

and

    f(W)∘ r(W)∘λ =   s(W)∘ g(W)∘λ

is G-homotopic to

    s(W)∘Y̅(φ)∘ j(V)∘β =  
    Y(φ)∘ s(V)∘ j(V)∘β =  Y(φ)∘β

relative A. 
So the pair (φ, r(W)∘λ) solves
the original lifting problem for the morphism f:X→ Y;
thus f is a global equivalence.

Part (v) holds because the disc D^k is connected, so any lifting problem for
a coproduct of orthogonal spaces is located in one of the summands.

For part (vi) it suffices to consider a product of two global equivalences f:X→ Y
and f':X'→ Y'.
Because global equivalences are closed under composition (part (ii))
and f× f'=(f× Y')∘(X× f'), it suffices to show that for
every global equivalence f:X→ Y and every orthogonal space Z the morphism
f× Z:X× Z→ Y× Z is a global equivalence.
But this is straightforward:
we let G be a compact Lie group, V a G-representation,
(B,A) a finite G-CW-pair and α:A→ (X× Z)(V)
and β:B→ (Y× Z)(V) continuous G-maps 
such that (f× Z)(V)∘α=β|_A.
Because

    (X× Z)(V)  =  X(V) × Z(V)

and similarly for (Y× Z)(V), we have
α=(α_1,α_2) and β= (β_1,β_2)
for continuous G-maps α_1:A→ X(V), α_2:A→ Z(V),
β_1:B→ Y(V) and β_2:B→ Z(V).
The relation (f× Z)(V)∘(α_1,α_2)=(β_1,β_2)|_A
shows that α_2=(β_2)|_A.
Since f is a global equivalence, 
the equivariant lifting problem (α_1,β_1)
for f(V) has a solution (φ:V→ W, λ:B→ X(W)) such that
λ|_A=X(φ)∘α_1 
and f(W)∘λ is G-homotopic to 
Y(φ)∘β_1 relative A. 
Then the pair (φ,(λ,Z(φ)∘β_2)) solves
the original lifting problem, so f× Z is a global equivalence.

(vii) If X is an orthogonal space and K a space, then
X× K is the product of X with the constant orthogonal space
with values K. So part (vii) is a special case of (vi).

(viii) We let G be a compact Lie group,
V a G-representation, (B,A) a finite G-CW-pair
and α:A→ X_∞(V)
and β:B→ Y_∞(V) continuous G-maps 
such that ψ_∞(V)∘α=β|_A:A→ Y_∞(V).
Since A and B are compact and X_∞(V) respectively Y_∞(V) 
are colimits of sequences of closed embeddings,
the maps α and β factor through  maps

    α̅ : A → X_n(V)     respectively    β̅ : B → Y_n(V)

for some n≥ 0, see Proposition <ref> (i).
Since the canonical maps X_n(V)→ X_∞(V) and Y_n(V)→ Y_∞(V) are injective,
α̅ and β̅ are again G-equivariant.
Moreover, the relation ψ_n(V)∘α̅=β̅|_A:A→ Y_n(V)
holds because it holds after composition with the injective map Y_n(V)→ Y_∞(V). 

Since ψ_n is a global equivalence, 
there is a G-equivariant linear isometric embedding φ:V→ W 
and a continuous G-map  λ:B→ X_n(W) such that
λ|_A=X_n(φ)∘α̅ and 
ψ_n(W)∘λ is G-homotopic to
Y_n(φ)∘β̅ relative A. 
We let λ':B→ X_∞(W) be the composite of λ and
the canonical map X_n(W)→ X_∞(W). 
Then the pair (φ,λ') is a solution for the original lifting problem,
and hence ψ_∞:X_∞→ Y_∞ is a global equivalence.

(ix) This is a special case of part (viii) where we set X_n=Y_0,
e_n=_Y_0 and ψ_n=f_n-1∘…∘ f_0:Y_0→ Y_n.
The morphism ψ_n is then a global equivalence by part (ii), 
and Y_0 is a colimit of the constant first sequence.
Since the morphism ψ_∞ induced on the colimits of the two sequences is
the canonical map Y_0→ Y_∞, part (viii) proves the claim.

(x)
Let G be a compact Lie group. 
We consider an exhaustive sequence {V_i}_i≥ 1 
of finite-dimensional G-representations.
Since α, β and γ are global equivalences, the three
vertical maps in the following commutative diagram of G-spaces
are G-weak equivalences, by Proposition <ref>:
    
    @C=15mm_i C(V_i)  [d]_γ(V_i)   _i A(V_i) [l]_- g(V_i)[d]^α(V_i)[r]^- f(V_i)   _i B(V_i) [d]^β(V_i)
    _i  C'(V_i)     _i A'(V_i) [l]^- g'(V_i)[r]_- f'(V_i)   _i B'(V_i)

Since mapping telescopes commute with product with [0,1] and retracts,
the maps _i f(V_i) and _i f'(V_i) are h-cofibrations of G-spaces.
The induced map of the horizontal pushouts is thus a G-weak equivalence
by Proposition <ref>.
Since formation of mapping telescopes commutes with pushouts, 
the map
    
    _i (γ∪β)(V_i) :  _i (C∪_A B)(V_i)  → _i (C'∪_A' B')(V_i)

    is a G-weak equivalence. The claim thus
follows by another application of the telescope criterion for 
global equivalences, Proposition <ref>.


(xi)
In the commutative diagram
    
    C  @=[d]     A [l]_-g @=[d] @=[r]     A [d]^f
    
          C      A [l]^-g[r]_-f    B

all vertical morphisms are global equivalences.
If f is an h-cofibration, we apply part (x) to this diagram
to get the desired conclusion.
If g is an h-cofibration, we apply part (x) after interchanging the roles
of left and right horizontal morphisms.


(xii) We let G be a compact Lie group,
V a G-representation, (B,A) a finite G-CW-pair
and α:A→ P(V) and β:B→ Z(V) continuous G-maps 
such that g(V)∘α=β|_A.
Since f is a global equivalence, there is 
a G-equivariant linear isometric embedding φ:V→ W 
and a continuous G-map λ:B→ X(W)
such that 
λ|_A=X(φ)∘ k(V)∘α and
such that f(W)∘λ
is G-homotopic, relative to A, to Y(φ)∘ h(V)∘β.  
We let H:B×[0,1]→ Y(W) be a relative G-homotopy from 
Y(φ)∘ h(V)∘β =  h(W)∘ Z(φ)∘β
to f(W)∘λ. 
Now we distinguish two cases.

Case 1: The morphism h is a strong level fibration.
We can choose a lift H̅ in the square

    @C25mm 
    B× 0∪_A× 0 A× [0,1][d]_∼[r]^-(Z(φ)∘β)∪ K   
    Z(W) @->>[d]^h(W)
    
    B× [0,1] [r]_-H @–>[ur]_(.6)H̅    Y(W)

where K:A× [0,1]→ Z(W) is the constant homotopy
from g(W)∘ P(φ)∘α to itself.
Since the square is a pullback and
h(W)∘H̅(-,1)= H(-,1)=f(W)∘λ,
there is a unique continuous G-map λ̅: B→ P(W)
that satisfies

    g(W) ∘λ̅ =  H̅(-,1)
         and      
    k(W) ∘λ̅ =  λ .

The restriction of λ̅ to A satisfies

    g(W) ∘λ̅|_A     =  H̅(-,1)|_A  =  
    g(W)∘ P(φ)∘α     and
     
    k(W) ∘λ̅|_A      =  λ|_A  =  
    X(φ)∘ k(V)∘α =  
    k(W)∘ P(φ)∘α .

The pullback property thus implies that
λ̅|_A =P(φ)∘α. 

Finally, the composite g(W) ∘λ̅ 
is homotopic, relative A and via H̅,
to H̅(-,0)= Z(φ)∘β. 
This is the required lifting data, and
we have verified the defining property of a global equivalence for the
morphism g.

Case 2: The morphism f is a strong level fibration. 
The argument is similar as in the first case.
Now we can choose a lift H' in the square

    @C30mm 
    B× 1∪_A× 1 A× [0,1][d]_∼[r]^-λ∪ K'   
    X(W) @->>[d]^f(W)
    
    B× [0,1] [r]_-H @–>[ur]_H'    Y(W)

where K':A× [0,1]→ X(W) is the constant homotopy
from X(φ)∘ k(V)∘α to itself.
Since the square is a pullback and
f(W)∘ H'(-,0)= H(-,0)=h(W)∘ Z(φ)∘β,
there is a unique continuous G-map λ̅: B→ P(W) that satisfies

    g(W) ∘λ̅ =  Z(φ)∘β     and      
    k(W) ∘λ̅ =  H'(-,0)  .

The restriction of λ̅ to A satisfies

    g(W) ∘λ̅|_A     =  
    Z(φ)∘ g(V)∘α =  
    g(W)∘ P(φ)∘α     and
     
    k(W) ∘λ̅|_A      =  H'(-,0)|_A  =  
    X(φ)∘ k(V)∘α =  
    k(W)∘ P(φ)∘α .

The pullback property thus implies that
λ̅|_A =P(φ)∘α. 
Since g(W) ∘λ̅= Z(φ)∘β,
this is the required lifting data, and
we have verified the global equivalence criterion 
of Proposition <ref> (ii) for the morphism g.



The restriction to finite products is essential in part (vi)
of the previous Proposition <ref>, i.e.,
an infinite product of global equivalences need not be a global equivalence.
The following simple example illustrates this. 
We let Y_n denote the orthogonal space with

    Y_n(V)  =  ∅    if (V)< n, and
    {∗}    if (V)≥ n.

The unique morphism Y_n→ Y_0 
is a global equivalence for every n≥ 0.
However, the product ∏_n≥ 0  Y_n is
the empty orthogonal space, whereas the product
∏_n≥ 0 Y_0 is a terminal orthogonal space.
The unique morphism from the initial (i.e., empty)
to a terminal orthogonal space is not a global equivalence.




The following proposition provides a lot of flexibility
for changing an orthogonal space into a globally equivalent one by modifying
the input variable. We will use it multiple times in this book.


  Let F:→ be a continuous endofunctor of the linear isometries
  category and i:→ F a natural transformation.
  Then for every orthogonal space Y the morphism
  
    Y∘ i :  Y → Y∘ F

  is a global equivalence of orthogonal spaces.


In a first step we show an auxiliary statement. We let V be an inner product
space and z∈ F(V) an element that is orthogonal to the subspace i_V(V),
the image of the linear isometric embedding i_V:V→ F(V).
We claim that for every linear isometric embedding φ:V→ W
the element F(φ)(z) of F(W) is orthogonal to the subspace i_W(W).
To prove the claim we write any given element of W as φ(v)+ y 
for some v∈ V and y∈ W orthogonal to φ(V). Then 

    F(φ)(z), i_W(φ(v))  =   F(φ)(z), F(φ)(i_V(v))  =   z,  i_V(v)  =  0

by the hypotheses on z. 
Now we define A∈ O(W) as the linear isometry that is the identity on φ(V)
and the negative of the identity on the orthogonal complement of φ(V).
Then A∘φ=φ and

    F(φ)(z), i_W(y)     =   F(A)( F(φ)(z)) , F(A)(i_W(y)) 
       =   F(A φ)(z) , i_W(A(y))  =  -  F(φ)(z) , i_W(y)  ,

and hence F(φ)(z), i_W(y) =  0.
Altogether this shows that F(φ)(z), i_W(φ(v)+y) =  0,
which establishes the claim.

Now we consider a compact Lie group G, a G-representation V,
a finite G-CW-pair (B,A) and a lifting problem α:A→ Y(V) 
and β:B→ Y(F(V)) for (Y∘ i)(V). 
Then β|_A=Y(i_V)∘α by hypothesis, 
and we claim that Y(i_F(V))∘β
is G-homotopic to Y(F(i_V))∘β = (Y∘ F)(i_V)∘β,
relative A; granting this for the moment, we conclude that
the pair (i_V:V→ F(V), β) solves the lifting problem.

It remains to construct the relative homotopy.
The two embeddings
  
    F(i_V)  ,  i_ F(V) :  F(V) →  F( F(V))

are homotopic, relative to i_V:V→ F(V), 
through G-equivariant isometric embeddings, via

    H  :   F(V) × [0,1]    →      F(F(V)) 
    
    (v + w, t)       ⟼  F(i_V)(v)  +  t· i_F(V)(w) +  √(1-t^2)·  F(i_V)(w)  ,

where v∈ i_V(V) and w is orthogonal to i_V(V).
The verification that H(-,t):F(V)→ F(F(V)) is indeed
a linear isometric embedding for every t∈[0,1] uses that
i_F(V)=F(i_V) on the subspace  i_V(V) of F(V), and 
that i_F(V)(w) is orthogonal to F(i_V)(w), by the claim proved above.
The continuous functor Y takes this homotopy of equivariant linear isometric
embeddings to a G-equivariant homotopy Y(H(-,t))
from Y(F(i_V)) to Y(i_F(V)),  relative to Y(i_V).
Composing with β gives the required relative G-homotopy
from Y(F(i_V))∘β to Y(i_F(V))∘β.



[Additive and multiplicative shift]
Here are some typical examples where the previous theorem applies. 
Every inner product space W defines an `additive shift functor'subjectshift!of an orthogonal space!additive
and a `multiplicative shift functor'subjectshift!of an orthogonal space!multiplicative
on the category of orthogonal spaces,
defined by precomposition with the continuous endofunctors

    -⊕ W  :   →      respectively    
     - W  :   →  .

In other words, the additive respectively multiplicative W-shift 
of an orthogonal space Y have values

    (^W_⊕ Y)(V) =  Y(V⊕ W)     respectively    
     (^W_ Y)(V) =  Y(V W) .

Here, and in the rest of the book, we endow
the tensor product V W of two inner product spaces V and W 
with the inner product characterized by

    v w, v̅w̅ =  v, v̅·w, w̅

for all v,v̅∈ V and w,w̅∈ W.
Another way to say this is that for every orthonormal basis {b_i}_i∈ I
of V and every orthonormal basis {d_j}_j∈ J of W the family
{b_i d_j}_(i,j)∈ I× J forms an orthonormal basis of V W.
Theorem <ref>  then shows that the morphism
Y→^W_⊕ Y given by applying Y 
to the first summand embedding V→ V⊕ W
is a global equivalence.
To get a similar statement for the multiplicative shift we have to assume
that W 0; then for every vector w∈ W of length 1 the map

    V  → V W  ,   v ⟼ v w

is a natural linear isometric embedding. 
So Theorem <ref> shows that the morphism
Y(- w):Y→^W_ Y is a global equivalence.





For the following discussion of universes we recall that 
a finite group has finitely many isomorphism classes of
irreducible orthogonal representations, and a compact Lie group of
positive dimension has countably infinitely many such isomorphism classes.
I have not yet found an explicit reference with a proof of this well-known fact,
but one can argue as follows. 

We first consider unitary representations of G.
If G is finite, then the characters of irreducible unitary representations form 
a basis of the -vector space of conjugation invariant -valued functions
on G. So the number of isomorphism classes of irreducible
unitary representations agrees with the number of conjugacy classes of elements
of G, and is thus finite.
If G is of positive dimension, 
then the characters of irreducible unitary representations form 
an orthonormal basis of the complex Hilbert space of square integrable 
(with respect to the Haar measure), conjugation invariant functions on G, 
see for example <cit.>.
Since G is compact and of positive dimension, 
this Hilbert space is infinite dimensional and separable, so there are
countably infinitely many isomorphism classes of irreducible unitary representations.

To treat the case of orthogonal representations of G,
we recall from <cit.>
that complexification can be used to construct a map from
the set of isomorphism classes of irreducible orthogonal representations
to the set of isomorphism classes of irreducible unitary representations of G.
There is a caveat, however, namely that the complexification of
an irreducible orthogonal representation need not be irreducible.
More precisely, the reducibility behavior under complexification
depends on the `type' of the irreducible orthogonal representation λ.
By Schur's lemma, the endomorphism algebra _^G(λ,λ) 
is a finite-dimensional skew-field extension of , hence isomorphic
to ,  or . 


  * If _^G(λ,λ) is isomorphic to , 
then λ is of real type.
In this case the complexification λ_
is irreducible as a unitary G-representation.

  * If _^G(λ,λ) is isomorphic to  or , 
then λ is of complex type respectively of quaternionic type.
In this case there is an irreducible unitary G-representation ρ
such that λ_ is isomorphic to the direct sum of ρ and its
conjugate ρ̅.
If λ is of complex type, then ρ is not isomorphic to its conjugate;
if λ is of quaternionic type, then ρ is self-conjugate,
i.e., isomorphic to its conjugate.

Since the underlying orthogonal representation 
of λ_ is isomorphic to the direct sum of two copies of λ,
two non-isomorphic irreducible orthogonal representations
cannot become isomorphic after complexification. 
So the above construction gives an injective map from 
the set of irreducible orthogonal representations
to the set of irreducible unitary representations.
Altogether this shows that there at most countably many
isomorphism classes of irreducible orthogonal representations
of a compact Lie group.




 
Let G be a compact Lie group. 
A G-universesubjectG-universe@G-universesubjectuniverse|seeG-universe
is an orthogonal G-representation  
of countably infinite dimension with the following two properties:


  * the representation  has non-zero G-fixed points,

  * if a finite-dimensional G-representation V embeds into ,
then a countable infinite direct sum
of copies of V also embeds into .

A G-universe is completesubjectG-universe@G-universe!complete 
if every finite-dimensional G-representation embeds into it.


A G-universe is characterized, up to equivariant linear isometry,
by the set of irreducible G-represen­ta­tions that embed into it.
We let Λ={λ} be a complete set of 
pairwise non-isomorphic irreducible G-representations that embed into .
The first condition says that Λ contains
a trivial 1-dimensional representation, and the second condition is equivalent
to the requirement that

    ⊕_λ∈Λ⊕_λ .

Moreover,  is complete if and only if Λ contains
(representatives of) all irreducible G-representations.
Since there are only countably many isomorphism classes of irreducible orthogonal 
G-representations, a complete G-universe exists.



 
We let H be a closed subgroup of a compact Lie group G.
We will frequently use that the underlying H-representation of a complete
G-universe  is a complete H-universe.
Indeed, if U is an H-representation, 
then there is a G-representation V 
and an H-equivariant linear isometric embedding U→ V,
see for example <cit.> 
or <cit.>.
Since V embeds G-equivariantly into ,
the original representation U embeds H-equivariantly into .


In the following we fix, for every compact Lie group G,
a complete G-universe _G.symbol_G - complete G-universe
We let s(_G)symbols(_G) - poset of G-subrepresentation of _G
denote the poset, under inclusion,
of finite-dimensio­nal G-subrepresentations of _G.


For an orthogonal space Y and a compact Lie group G we define the 
underlying G-spacesubjectunderlying G-space!of an orthogonal space as

    Y(_G) =  _V∈ s(_G)  Y(V) ,

the colimit of the G-spaces Y(V). 



The underlying G-space Y(_G) can always be written as a sequential colimit
of values of Y. Indeed, we can choose a nested sequence of
finite-dimensional G-subrepresentations

    V_0  ⊆ V_1 ⊆ V_2 ⊆ ⋯

whose union is all of _G.
This is then in particular an exhaustive sequence in the sense of
Definition <ref>.
Since the subposet
{V_n}_n≥ 0 is cofinal in s(_G), the colimit of the functor
V↦ Y(V) over s(_G) is also a colimit over the 
subsequence Y(V_n).

If the group G is finite, then we can define a complete universe as 

    _G  =  ⊕_ ρ_G ,

a countably infinite sum of copies of the regular representation 
ρ_G=[G], with G as orthonormal basis.
Then _G is filtered by the finite sums n·ρ_G, and we get

    Y(_G) =  _n   Y(n·ρ_G)  ,

where the colimit is taken along the inclusions 
n·ρ_G→(n+1)·ρ_G that miss the last summand.






An orthogonal space Y is closedsubjectorthogonal space!closed 
if it takes every linear isometric embedding φ:V→ W of inner product spaces
to a closed embedding Y(φ):Y(V)→ Y(W).


In particular, for every closed orthogonal space Y 
and every G-equivariant linear isometric embedding φ:V→ W of
G-representations, the induced map on G-fixed points
Y(φ)^G:Y(V)^G→ Y(W)^G is also a closed embedding.


 
Let f:X→ Y be a morphism between closed orthogonal spaces.
Then f is a global equivalence if and only if for every compact Lie
group G the map

    f(_G)^G :  X(_G)^G  → Y(_G)^G

is a weak equivalence.


The poset s(_G) has a cofinal subsequence, so all colimits
over s(_G) can be realized as sequential colimits.
The claim is then a straightforward consequence of the fact that
fixed points commute with sequential colimits along closed embeddings
(see Proposition <ref> (ii))
and continuous maps from compact spaces such as D^k and ∂ D^k 
to sequential colimits along closed embeddings factor through a finite stage 
(see Proposition <ref> (i)).






Now we turn to semifree orthogonal spaces.
Important special cases of this construction are the global classifying spaces
of compact Lie groups (Definition <ref>),
the basic building blocks of global homotopy theory.
Free and semifree orthogonal spaces are made from
spaces of linear isometric embeddings, so we start by recalling
various properties of certain spaces of linear isometric embeddings.
We consider two compact Lie groups G and K,
a finite-dimensional G-representation V, 
and a K-representation , possibly of countably infinite dimension. 
If  is infinite dimensional, we topologize the space (V,) 
of linear isometric embeddings as the filtered colimit of the 
spaces (V,U), taken over the poset of finite-dimensional subspaces U of .
The space (V,) 
inherits a continuous left K-action and a compatible continuous right G-action
from the actions on the target and source, respectively.
We turn these two actions into a single left action of the group K× G
by defining

    ((k,g)·φ)(v)  =  k·φ(g^-1· v)

for φ∈(V,) and (k,g)∈ K× G.
We recall that a continuous (K× G)-equivariant map is a
(K× G)-cofibration if it has the right lifting
property with respect to all morphisms of (K× G)-spaces
f:X→ Y such that the map f^Γ:X^Γ→ Y^Γ is a weak equivalence and
Serre fibration for every closed subgroup Γ of K× G.

 
Let G and K be compact Lie groups,
V a finite-dimensional G-representation, and  a K-representation 
of finite or countably infinite dimension.


  * For every finite-dimensional K-subrepresentation U of , 
the inclusion induces a (K× G)-cofibration

    (V,U) → (V,)

and a K-cofibration of orbit spaces

    (V,U)/G → (V,)/G  .


  * The (K× G)-space (V,) is (K× G)-cofibrant.
The K-space (V,)/G is K-cofibrant.



(i) 
We consider two natural numbers m,n≥ 0. 
The space (V,^m+n) is homeomorphic to the Stiefel manifold of   
(V)-frames in ^m+n, hence a compact smooth manifold, and the
action of O(m)× O(n)× G is smooth.
Illman's theorem <cit.> thus provides an 
(O(m)× O(n)× G)-CW-structure on (V,^m+n).
In particular, (V,^m+n) is cofibrant as an 
(O(m)× O(n)× G)-space.
The group N= e× O(n)× e is a closed normal subgroup of 
O(m)× O(n)× G, so the inclusion of the N-fixed points
into (V,^m+n) is an (O(m)× O(n)× G)-cofibration
(compare Proposition <ref>).
The map

    (V,^m) → (V,^m+n)    ,

induced by the embedding ^m→^m+n 
as the first m coordinates,
is a homeomorphism from (V,^m) onto the N-fixed points
of (V,^m+n); so the map (<ref>)
is an (O(m)× O(n)× G)-cofibration.

Now we can prove the proposition when  is finite-dimensional. 
We can assume that  is ^m+n with the standard scalar product, 
and that U is the subspace where the last n coordinates vanish.
The K-action on  is given by a continuous homomorphism ψ:K→ O(m+n).
Since U is a K-subrepresentation, the image of ψ must be contained in the
subgroup O(m)× O(n).
The (K× G)-action on the map (<ref>)
is then obtained by restriction of the (O(m)× O(n)× G)-action
along the homomorphism

    ψ× :   K× G  → O(m)× O(n)× G .

Restriction along any continuous homomorphism between compact Lie groups
preserves cofibrations by Proposition <ref> (i),
so the map (<ref>) is a (K× G)-cofibration by the first part. 

Now we treat the case when the dimension of  is infinite.
We choose an exhausting nested sequence of K-subrepresentations

    U = U_0  ⊂ U_1  ⊂ U_2 ⊂ … .

Then all the morphisms (V,U_n-1)→(V,U_n) are (K× G)-cofibrations
by the above. Since cofibrations are closed under sequential composites,
the morphism 

    (V,U_0) → _n (V,U_n)  =  (V,)

is also a (K× G)-cofibration.

Applying Proposition <ref> (iii)
to the normal subgroup e× G of K× G shows that
the functor

    (e× G)-  :  (K× G) → K

takes (K× G)-cofibrations to K-cofibrations.
This proves the second claim.

(ii) This is the special case U={0}. The space (V,0) is either empty
or consists of a single point; in any case (V,{0}) 
is (K× G)-cofibrant. Part (i) then implies that
(V,) is (K× G)-cofibrant
and (V,)/G is K-cofibrant.



The following fundamental contractibility property goes back, 
at least, to Boardman and Vogt <cit.>.
The equivariant version that we need can be found in <cit.>.

 
Let G be a compact Lie group, V a G-representation and  a G-universe
such that V embeds into . Then the space (V,), 
equipped with the conjugation action by G, is G-equivariantly contractible.


We start by showing that the space ^G(V,)
of G-equivariant linear isometric embeddings is weakly contractible.
We let  be a G-representation of finite or countably infinite dimension. 
Then the map

    H  :  [0,1]×^G(V,) → ^G(V,⊕ V)

defined by

    H(t,φ)(v) =  ( t·φ(v), √(1-t^2)· v)

is a homotopy from the constant map with value i_2:V→⊕ V
to the map i_1∘ - (postcomposition with i_1:→⊕ V).

Since V embeds into  and  is a G-universe, it contains
infinitely many orthogonal copies of V. In other words, we can assume
that 

    =  '⊕ V^∞

for some G-representation '.
Then

    ^G(V,)  =  ^G(V, '⊕ V^∞)  =  _n≥ 0^G(V, '⊕ V^n)  ;

the colimit is formed along the postcomposition maps
with the direct sum embedding '⊕ V^n→'⊕ V^n+1.
Every map in the colimit system is a closed embedding and homotopic
to a constant map, by the previous paragraph. 
So the colimit is weakly contractible.

Applying the previous paragraph to a closed subgroup H of G
shows that the fixed point space ^H(V,) is weakly contractible;
in other words, (V,) is G-weakly contractible.
The space (V,) comes with a (G× G)-action
as in (<ref>), and it is 
(G× G)-cofibrant by Proposition <ref> (ii).  
Then (V,) is also cofibrant as a G-space for the
diagonal action, by Proposition <ref> (i). 
Since (V,) is G-cofibrant and weakly G-contractible,
it is actually equivariantly contractible.



Now we turn to semifree orthogonal spaces.


Given a compact Lie group G and a G-representation V, the functor

    _G,V :   → G

that sends an orthogonal space Y to the G-space Y(V)
has a left adjoint

    _G,V :  G →  .

To construct the left adjoint we note that G acts from the right on (V,W) by

    (φ· g)(v)  =  φ(g v)

for φ∈(V,W), g∈ G and v∈ V.
Given a G-space A, the value of _G,V A at an inner product space W is

    (_G,VA)(W)  =  (V,W)×_G A =  ( (V,W)× A ) / (φ g,a)∼ (φ, g a) .

We refer to _G,V A as
the semifree orthogonal spacesubjectorthogonal space!semifree
subjectsemifree orthogonal spacesymbol_G,V - semifree orthogonal space generated by (G,V)
generated by A at (G,V).
We also denote by _G,V the orthogonal space with

    _G,V(W) =  (V,W)/G  .

So _G,V is isomorphic to the semifree orthogonal space 
generated at (G,V) by a one-point G-space.


The `freeness' property of _G,V A
is a consequence of the enriched Yoneda lemma,subjectYoneda lemma!enriched
see Remark <ref> or <cit.>;
it means explicitly that
for every orthogonal space Y and every continuous G-map f:A→ Y(V) 
there is a unique morphism 
f^♭:_G,V A→ Y of orthogonal spaces such that the composite

    A  [, -]  (V,V) ×_G A = (_G,V A)(V)  f^♭(V) Y(V)

is f.
Indeed, the map f^♭(W) is the composite

    (V,W) ×_G A  ×_G f(V,W) ×_G Y(V)  [φ,y]↦ Y(φ)(y) Y(W) .





For every compact Lie group G, every G-representation V
and every G-space A the semifree orthogonal space _G,V A 
is closed.subjectorthogonal space!closed
To see that we let φ:U→ W be a linear isometric embedding. 
Since (V,U) is compact, the continuous injection

    (V,φ) :  (V,U) → (V,W)

is a closed embedding. 
So the map (V,φ)× A is a closed embedding as well.
The orbit map 

    (V,φ)×_G A :  (V,U)×_G A → (V,W)×_G A

is then a closed embedding
by Proposition <ref> (iii).



The next proposition identifies the fixed point spaces 
of a semifree orthogonal space _G,V.
A certain family (K;G) of subgroups of K× G arises naturally,
which we call `graph subgroups'.


Let K and G be compact Lie groups. The family
(K;G)symbol(K;G) - family of graph subgroups of K× G
of graph subgroupssubjectgraph subgroup
consists of those closed subgroups Γ of K× G
that intersect 1× G only in the neutral element (1,1).


The name `graph subgroup' stems from the fact that
(K;G) consists precisely of the graphs of all `subhomomorphisms',
i.e., continuous homomorphisms α:L→ G from a closed subgroup L of K.
Clearly, the graph Γ(α)={(l,α(l)) | l∈ L} 
of every such homomorphism belongs to (K;G).
Conversely, for Γ∈(K;G) we let L≤ K be the
image of Γ under the projection K× G→ K.
Since Γ∩ (1× G)={(1,1)}, every element l∈ L
then has a unique preimage (l,α(l)) under the projection,
and the assignment l↦α(l) is a continuous homomorphism
from L to G whose graph is Γ.

We recall that a 
universal G-spacesubjectuniversal space!for a set of subgroups
for a family  of closed subgroups 
is a cofibrant G-space E such that


  * all isotropy groups of E belong to the family , and

  * for every H∈ the fixed point space E^H is weakly contractible.



If V and W are G-representations, then restriction of a linear isometry
from V⊕ W to V defines a G-equivariant morphism of orthogonal spacessymbolρ_G,V,W - fundamental global equivalence of orthogonal spaces

    ρ_V,W :  _V⊕ W → _V .

If U is a K-representation, then we combine the left K-action
and the right G-action on (V,U) into a left action
of K× G as in (<ref>).



Let G and K be compact Lie groups and V a faithful G-representation.


  * The (K× G)-space _V(_K)=(V,_K) is a universal space 
for the family (K;G) of graph subgroups.subjectuniversal space!for the family of graph subgroups

  * If W is another G-representation, then the restriction map

    ρ_V,W(_K) :  (V⊕ W, _K)  → (V, _K)

is a (K× G)-homotopy equivalence.
For every G-space A, the map

    (ρ_V,W×_G A)(_K) :  
    (_G,V⊕ W A) (_K)  → ( _G,V A)(_K)

is a K-homotopy equivalence and the morphism of orthogonal spaces

    ρ_V,W×_G A :  _G,V⊕ W A  → _G,V A

is a global equivalence.



(i) We let Γ be any closed subgroup of K× G.
Since the G-action on V is faithful, the induced right G-action
on  (V,_K) is free.
So if Γ intersects 1× G non-trivially, then
(V,_K)^Γ is empty.
On the other hand, if Γ∩ (1× G)={(1,1)},
then Γ is the graph of a unique continuous homomorphism α:L→ G,
where L is the projection of Γ to K. Then

    (V,_K)^Γ =  ^L(α^* V, _K)

is the space of L-equivariant linear isometric embeddings from
the L-represen­tation α^* V to the underlying L-universe of _K.
Since _K is a complete K-universe, the underlying L-universe is also
complete (Remark <ref>), 
and so the space ^L(α^* V, _K)  is contractible
by Proposition <ref>.
The space (V,_K) is cofibrant as a
(K× G)-space by Proposition <ref> (ii). 


(ii)
Since G acts faithfully on V, and hence also on V⊕ W, the
(K× G)-spaces (V⊕ W,_K) and (V,_K)
are universal spaces for the same family (K;G), by part (i).
So the map ρ_V,W(_K):(V⊕ W,_K)→(V,_K)
is a (K× G)-equivariant homotopy equivalence, 
see Proposition <ref> (ii).
The functor -×_G A preserves homotopies, so 
the restriction map (ρ_V,W×_G A)(_K) is a K-homotopy equivalence.

The orthogonal spaces _G,V⊕ W A and _G,V A are closed
by Example <ref>, 
so Proposition <ref> applies 
and shows that ρ_V,W×_G A is a global equivalence.



subjectglobal classifying space|(


The global classifying spacesubjectglobal classifying spacesubjectclassifying space!global|seeglobal classifying spacesymbolB_ G - global classifying space of the compact Lie group G
B_ G of a compact Lie group G is the semifree orthogonal space

    B_ G  =  _G,V =  (V,-)/G ,

where V is any faithful G-representation.


The global classifying space B_G is well-defined up to preferred 
zigzag of global equivalences of orthogonal spaces. Indeed, if V and
V̅ are two faithful G-representations, then V⊕V̅ is
yet another one, and the two restriction morphisms

    _G,V  _G,V⊕V̅    _G,V̅

are global equivalences by 
Proposition <ref> (ii).

subjectglobal classifying space!of C_2  
We make the global classifying space more explicit in 
the smallest non-trivial example, i.e., for the cyclic group
C_2 of order 2. The sign representation σ 
of C_2 is faithful, so we can take B_ C_2 
to be the semifree orthogonal space generated by
(C_2,σ); its value at an inner product space W  is

    (B_C_2)(W)  =  _C_2,σ(W)  =  (σ,W)/C_2 .

Evaluation at any of the two unit vectors in σ 
is a homeomorphism (σ,W) S(W)
to the unit sphere of W, and the C_2-action on the left
becomes the antipodal action on S(W). So the map descends to a homeomorphism
between (σ,W)/C_2 and P(W), the projective space of W,
and hence

    (B_C_2)(W)   P(W) .

So for a compact Lie group K, the underlying K-space of
B_ C_2 is P(_K), the projective space of a complete K-universe. 
In particular, the underlying non-equivariant space is
homeomorphic to ^∞.



[B_G globally classifies principal G-bundles]
  The term `glo­bal classifying space' is justified by the fact that B_ G
  `globally classifies principal G-bundles'. 
  We recall that a (K,G)-bundle,subjectK,G-bundle@(K,G)-bundle
  also called a K-equivariant G-principal bundle,
  is a principal G-bundle in the category of K-spaces, i.e., a G-principal bundle 
  p:E→ B that is also a morphism of K-spaces and such that
  the actions of G and K on the total space E commute (see for example <cit.>).
  For every compact Lie group K, the quotient map
  
    q :  (V,_K)  → (V,_K)/G  =  _G,V(_K) 
       =  (B_G)(_K)

  is a principal (K,G)-bundle. Indeed, the total space (V,_K)
  is homeomorphic to the Stiefel manifold of (V)-frames in ^∞,
  and hence it admits a CW-structure. Every CW-complex is a normal Hausdorff space
  (see for example <cit.> or <cit.>),
  hence completely regular. So (V,_K) is completely regular.
  Since the G-action on (V,_K) is free, the quotient map q
  is a G-principal bundle by <cit.>
  or <cit.>. 
  Moreover, this bundle is universal in the sense of the following proposition.
  Every G-space that admits a G-CW-structure is paracompact,
  see <cit.> 
  (this reference is rather sketchy, but one can follow the non-equivariant
  argument spelled out in more detail in <cit.>).
  So the next proposition applies in particular to all G-CW-complexes.


   
    Let V be a faithful representation of a compact Lie group G.
    Then for every paracompact K-space A the map
    
    [A, (V,_K)/G]^K  → _(K,G)(A) ,  
        [f] ⟼ [f^*(q)]

    from the set of equivariant homotopy classes of K-maps to the
    set of isomorphism classes of (K,G)-bundles is bijective.
  
  
    I do not know a reference for the result in precisely this form,
    so I sketch how to deduce it from various results in the
    literature about equivariant fiber bundles.
    A principal (K,G)-bundle p:E→ B is equivariantly trivializable 
    if there is a closed subgroup L of K, a continuous homomorphism
    α:L→ G, an L-space X and an isomorphism of (K,G)-bundles
    between p and the projection
    
    (K× G)×_L X  → K×_L X  ;

    here the source is the quotient space of
    K× G× X by the equivalence relation
    (k,g,l x)∼ (k l, gα(l),x) 
    for all (k,g,l,x)∈ K× G× L× X.
    A principal (K,G)-bundle p:E→ B is numerable if B 
    has a trivializing (in the above sense) open cover
    by K-invariant subsets, such that moreover the cover admits 
    a subordinate partition of unity by K-invariant functions.

    A universal (K,G)-principal bundle
    is a numerable (K,G)-bundle p^u:E^u→ B^u 
    such that for every K-space A the map
    
    [A, B^u]^K  → ^num_(K,G)(A) ,   
        [f] ⟼ [f^*(p^u)]

    is bijective, where now the target is the set of isomorphism classes of
    numerable principal (K,G)-bundles.
    Over a paracompact base, every principal (K,G)-bundle is numerable
    by <cit.>. So we are done if we can show
    that q:(V,_K)→(V,_K)/G is a universal (K,G)-principal bundle 
    in the above sense.

    Universal (K,G)-principal bundles can be built in different ways; 
    the most common construction is a version of
    Milnor's infinite join <cit.>, 
    see for example <cit.>
    or <cit.>.
    Another method is via bar construction, 
    compare <cit.>.
    I do not know of a reference that explicitly identifies the projection
    q:(V,_K)→(V,_K)/G as a universal bundle in the present sense,
    so we appeal to Lashof's criterion <cit.>. 
    The base space (V,_K)/G is the union, along h-cofibrations,
    of the compact spaces (V,W_i)/G, where {W_i}_i≥ 1
    is any exhausting sequence of subrepresentations of _K.
    Since compact spaces are paracompact, the union (V,_K)/G 
    is paracompact, see for example <cit.>.
    Since (V,_K)/G is also normal, hence completely regular,
    the bundle q is numerable by Corollaries 1.5 and 1.13 
    of <cit.>.
    Moreover, the fixed points of (V,_K)
    under any graph subgroup of (K× G) are contractible by
    Proposition <ref> (i),
    so Theorem 2.14 of <cit.>
    applies and shows that q:(V,_K)→(V,_K)/G is strongly universal, 
    and hence a universal principal (K,G)-bundle.
    

    As another example we look at the case G=O(n), the n-th orthogonal group. 
    The category of principal O(n)-bundles is equivalent to the category 
    of euclidean vector bundles of rank n, via the associated frame bundle.
    By the same construction, principal (K,O(n))-bundles can be identified
    with K-equivariant euclidean vector bundles of rank n over K-spaces.
    The space (^n,_K)/O(n) is homeomorphic to G r_n(_K),
    the Grassmannian of n-planes in _K. When K is a trivial group,
    the fact that G r_n(^∞) is a classifying space for rank n
    vector vector bundles over paracompact spaces is proved in various textbooks.
    Since O(1) is a cyclic group of order 2, this gives another perspective on
    Example <ref>. 
subjectglobal classifying space|)





§ GLOBAL MODEL STRUCTURE FOR ORTHOGONAL SPACES




In this section we establish the global model structure on the category of orthogonal spaces, see Theorem <ref>.
Towards this aim we first discuss a `strong level model structure', 
which we then localize.
In Proposition <ref>
we use the global model structure to relate unstable global homotopy
theory to the homotopy theory of K-spaces for a fixed compact Lie group K.
We also show that global classifying spaces of compact Lie groups 
with abelian identity component are `cofree', i.e.,
right induced from non-equivariant classifying spaces, see
Theorem <ref>. 
At the end of this section we briefly discuss the realization of
simplicial orthogonal spaces in Construction <ref>; 
we show that the realization takes levelwise global equivalences 
to global equivalences, under a certain `Reedy flatness' condition
(Proposition <ref>).



There is a functorial way to write an orthogonal space as a 
sequential colimit of orthogonal spaces which are made from the information
below a fixed dimension.
We refer to this as the skeleton filtration of an orthogonal
space. The word `filtration' should be used with caution
because the maps from the skeleta to the orthogonal space
need not be injective.

The skeleton filtration is in fact a special case of 
a more general skeleton filtration on certain enriched functor categories
that we discuss in Appendix <ref>.
Indeed, if we specialize the base category to =,
the category of spaces under cartesian product, and the index category
to =, then the functor category ^∗ becomes the
category  of orthogonal spaces. The dimension function needed in
the construction and analysis of skeleta is the vector space dimension.

We denote by ^≤ m the full topological subcategory of the linear isometries
category  with objects all inner product spaces of dimension at most m.
We denote by ^≤ m the category of continuous functors from ^≤ m
to . The restriction functor 

    → ^≤ m ,    Y  ⟼ Y^≤ m =  Y|_^≤ m

has a left adjoint 

    l_m  :   ^≤ m →

given by an enriched Kan extension as follows. 
The extension l_m(Z) of a continuous functor Z:^≤ m→ 
is a coequalizer of the two morphisms of orthogonal spaces

    @C=7mm∐_0≤ j≤ k≤ m(^k,-)×(^j,^k)× Z(^j) 
     @<-.4ex>[r]  @<.4ex>[r]      ∐_0≤ i≤ m(^i,-)× Z(^i)

One morphism arises from the composition morphisms

    (^k,-)×(^j,^k)  → (^j,-)

and the identity on Z(^j);
the other morphism arises from the action maps

    (^j,^k)× Z(^j)  → Z(^k)

and the identity on the free orthogonal space (^k,-).
Colimits in the category of orthogonal spaces are created objectwise, 
so the value l_m(Z)(V) at an inner product space can be calculated 
by plugging V into the variable slot 
in the coequalizer diagram (<ref>).

It is a general property of Kan extensions along a fully faithful functor 
(such as the inclusion ^≤ m→) that the values do not
change on the given subcategory,
see for example <cit.>. 
More precisely, the adjunction unit

    Z  → (l_m(Z))^≤ m

is an isomorphism for every continuous functor Z:^≤ m→. 


The m-skeleton,subjectskeleton!of an orthogonal space
for m≥ 0, of an orthogonal space Y is the orthogonal space

    ^m Y =  l_m(Y^≤ m)  ,

the extension of the restriction of Y to ^≤ m.
It comes with a natural morphism i_m:^m Y→ Y, the counit of
the adjunction (l_m,(-)^≤ m).
The m-th latching spacesubjectlatching space!of an orthogonal space
of Y is the O(m)-space

    L_m Y  =  (^m-1 Y)(^m)  ;

it comes with a natural O(m)-equivariant map

    ν_m=i_m-1(^m) : L_m Y → Y(^m)  ,

the m-th latching map.


We agree to set ^-1 Y=∅, the empty orthogonal space,
and L_0 Y=∅, the empty space. The value

    i_m(V) : (^m Y)(V) → Y(V)

of the morphism i_m is an isomorphism for all inner product spaces V of dimension
at most m.

The two morphisms i_m-1:^m-1 Y→ Y and
i_m:^m Y→ Y both restrict to isomorphisms on ^≤ m-1, so there is a 
unique morphism j_m:^m-1 Y→^m Y such that i_m∘ j_m=i_m-1.
The sequence of skeleta stabilizes to Y in a 
very strong sense. For every inner product space V, the maps
j_m(V) and i_m(V) are homeomorphisms as soon as m >(V).
In particular, Y(V) is a colimit, with respect to the morphisms i_m(V), 
of the sequence of maps j_m(V).
Since colimits in the category of orthogonal spaces are created objectwise,
we deduce that the orthogonal space Y is a colimit, 
with respect to the morphisms i_m, of the sequence of morphisms j_m. 


We denote by

    G_m  :  O(m) →

the left adjoint to the functor Y↦ Y(^m).
So G_m is a shorthand notation for _O(m),^m,
the semifree functor (<ref>) indexed by the tautological
O(m)-representation.
Proposition <ref>
specializes to:


For every orthogonal space Y and every m≥ 0 the commutative square

    @C=15mm G_m L_m Y [r]^G_mν_m[d]     G_m Y(^m) [d]
    ^m-1 Y [r]_-j_m   ^m Y

is a pushout of orthogonal spaces.
The two vertical morphisms are adjoint to the identity of L_m Y 
respectively Y(^m).





As an illustration of the definition, we describe the skeleta and  latching objects
for small values of m. We have 

    ^0 Y  =  const(Y(0))  ,

the constant orthogonal space with value Y(0); the latching map

    ν_1  :  L_1 Y  =  (^0 Y)()  =  Y(0)  Y(u) Y()

is the map induced by the unique linear isometric embedding u:0→.
Now we evaluate the pushout square (<ref>)
for m=1 at an inner product space V; the result is a pushout square
of O(1)-spaces

    @C=15mm P(V)× Y(0) [r] [d]_proj   (,V)×_O(1) Y() [d]
    
    Y(0)  [r]     (^1 Y)(V)

where P(V) is the projective space of V.
Here we exploit that O(1) acts trivially on L_1 Y=Y(0)
and we can thus identify

    (G_1 L_1 Y)(V) =  (,V)×_O(1) Y(0)  P(V)× Y(0) ,   
    [φ, y] ⟼ (φ(),y)  .

The upper horizontal map sends (φ(),y) to [φ, Y(u)(y)].



[Latching objects of free orthogonal spaces]
  We let V be an n-dimensional representation of compact Lie group G,
  and A a G-space. 
  Then the semifree orthogonal space (<ref>)
  generated by A in level V is `purely n-dimensional' in the following sense.
  The evaluation functor
  
    _G,V :   → G

  factors through the category ^≤ n as the composite
  
    → ^≤ n _G,V G .

  So the left adjoint semifree functor _G,V can be chosen as the composite
  of the two individual left adjoints
  
    _G,V =  l_n ∘ l_G,V .

  Here l_G,V:G→^≤ n 
  is given at a G-space A and an inner product
  space W of dimension at most n by
  
    (l_G,V A)(W)  = (V,W)×_G A     if (W)= n,
        ∅    if (W)< n.

  The space (_G,VA)_m is trivial for m < n, and hence the 
  latching space L_m (_G,VA) is trivial for m≤ n.
  For m>n the latching map ν_m:L_m(_G,VA) → (_G,VA)_m 
  is an isomorphism. So for m<n the skeleton ^m (_G,VA) is trivial, 
  and for m ≥ n the skeleton ^m (_G,VA)=_G,VA is the entire orthogonal space. 




Now we work our way towards the strong level model structure of orthogonal spaces.
Proposition <ref> 
is a fairly general recipe for constructing 
level model structures on a category such as orthogonal spaces.
We specialize the general construction to the situation at hand.
We recall from Definition <ref>
that a morphism f:X→ Y of orthogonal spaces is a
strong level equivalencesubjectstrong level equivalence!of orthogonal spaces 
(respectively  strong level fibration)subjectstrong level fibration!of orthogonal spaces 
if for every compact Lie group G and every G-representation V the map
 f(V)^G:X(V)^G→ Y(V)^G is a weak equivalence (respectively a Serre fibration).



 
For every morphism f:X→ Y of orthogonal spaces,
the following are equivalent.
  

  * The morphism f is a strong level equivalence.

  * For every compact Lie group G and every faithful G-representation V the map
    f(V):X(V)→ Y(V) is a G-weak equivalence.

  * The map f(^m):X(^m)→ Y(^m) is an O(m)-weak equivalence
  for every m≥ 0.
  
  
  
Clearly, condition (i) implies condition (ii), and that implies condition (iii)
(because the tautological action of O(m) on ^m is faithful).
So we suppose that f(^m) is an O(m)-weak equivalence
for every m≥ 0, and we show that f is a strong level equivalence.
Given a G-representation V of dimension m, 
we choose a linear isometry φ:V^m; conjugation by φ turns 
the G-action on V into a homomorphism ρ:G→ O(m), i.e.,

    ρ(g) =  φ∘(g·-)∘φ^-1 .

The homeomorphism X(φ):X(V)→ X(^m) then restricts to a homeomorphism

    X(V)^G   X(^m)^ρ(G) .

This homeomorphism is natural for morphisms of orthogonal spaces, so the hypothesis that
f(^m)^ρ(G): X(^m)^ρ(G)→ Y(^m)^ρ(G) is a weak equivalence
implies that also the map f(V)^G:X(V)^G→ Y(V)^G is a weak equivalence.


The same kind of reasoning as in Lemma <ref>
shows:



  
For every morphism f:X→ Y of orthogonal spaces,
the following are equivalent.
  

  * The morphism f is a strong level fibration.

  * For every compact Lie group G and every faithful G-representation V the map
    f(V):X(V)→ Y(V) is a fibration in the projective model structure of G-spaces.

  * The map f(^m):X(^m)→ Y(^m) is an O(m)-fibration
  for every m≥ 0.
  
  




  A morphism of orthogonal spaces i:A→ B 
  is a flat cofibrationsubjectflat cofibration!of orthogonal spaces 
  if the latching morphism
  
    ν_m i =  i(^m)∪ν_m^B : A(^m)∪_L_m AL_m B → B(^m)

  is an O(m)-cofibration  for all m≥ 0.
  An orthogonal space B is 
  flatsubjectflat!orthogonal spacesubjectorthogonal space!flat 
  if the unique morphism from the empty orthogonal space
  to B is a flat cofibration.
  Equivalently, for every m≥ 0 the latching map ν_m:L_m B → B(^m)
  is an O(m)-cofibration.



We are ready to establish the strong level model structure.



    The strong level equivalences, strong level fibrations 
    and flat cofibrations form a topological cofibrantly generated model structure,
    the strong level model structure,subjectstrong level model structure!for orthogonal spacessubjectmodel structure!strong level|seestrong level model structure
    on the category of orthogonal spaces.


We apply Proposition <ref> as follows.
We let (m) be the projective
model structure on the category of O(m)-spaces
(with respect to the set of all closed subgroups of O(m)),
compare Proposition <ref>.
The classes of level equivalences, level fibrations and cofibrations
in the sense of Proposition <ref> 
then precisely become the strong level equivalences,
strong level fibrations and flat cofibrations.

In this situation the consistency condition 
(see Definition <ref>)
is a consequence of a stronger property, namely that the functor

    (^m,^m+n)×_O(m) -  :  O(m) → O(m+n)

takes acyclic cofibrations to acyclic cofibrations 
(in the two relevant projective model structures).
Since the functor is a left adjoint, it suffices to prove the claim for the generating
acyclic cofibrations, i.e., the maps

    O(m)/H × j_k

for all k≥ 0 and all closed subgroups H of O(m),
where j_k:D^k×{0}→ D^k× [0,1] is the inclusion.
The functor under consideration takes this generator to the map
(^m,^m+n)/ H× j_k,
which is an acyclic O(m+n)-cofibration because 
(^m,^m+n)/H
is cofibrant as an O(m+n)-space, by Proposition <ref> (iii). 

We describe explicit sets of generating cofibrations 
and generating acyclic cofibrations.
We let I^ be the set of all morphisms G_m i for m≥ 0
and for i in the set of generating cofibrations for the projective model
structure on the category of O(m)-spaces specified 
in (<ref>). 
Then the set I^ detects the acyclic fibrations 
in the strong level model structure 
by Proposition <ref> (iii). 
Similarly, we let J^ be the set of all morphisms G_m j 
for m≥ 0 and for j in the set
of generating acyclic  cofibrations for the projective model
structure on the category of O(m)-spaces specified 
in (<ref>).
Again by Proposition <ref> (iii),
J^ detects the fibrations in the strong level model structure. 

The model structure is topological by Proposition <ref>,
where we take  as the set of orthogonal spaces L_H,^m
for all m≥ 0 and all closed subgroups H of O(m), and we take =∅.



For easier reference we make the generating (acyclic) cofibrations
of the strong level model structure even more explicit. Using the isomorphism  

    G_m ( O(m)/H )  =  (^m,-)×_O(m) ( O(m)/H )   (^m,-)/H   = _H,^m ,

we can identify I^ with the set of all morphisms

    _H,^m× i_k  :  _H,^m×∂ D^k 
     → _H,^m× D^k

for all k,m≥ 0 and all closed subgroups H of O(m).
The tautological action of H on ^m is faithful; 
conversely every pair (G,V) consisting of a compact Lie group and a faithful
representation is isomorphic to a pair (H,^m) 
for some closed subgroup H of ^m.
We conclude that I^ is a set of representatives of the 
isomorphism classes of morphisms

    _G,V× i_k  :  _G,V×∂ D^k
       → _G,V× D^k

for G a compact Lie group, V a faithful G-representation and k≥ 0.
Similarly, J^ is a set of representatives of the 
isomorphism classes of morphisms

    _G,V×  j_k :  _G,V× D^k×{0} → _G,V× D^k× [0,1]

for G a compact Lie group, V a faithful G-representation and k≥ 0.



  Let K be a compact Lie group and φ:W→ U a linear isometric embedding 
  of K-representations, where W is finite-dimensional, and U is finite-dimensional
  or countably infinite dimensional.
  
  
  * For every flat cofibration of orthogonal spaces i:A→ B 
    the maps
    
    i(U)    :  A(U) → B(U)    and  
    
        i(U)∪ B(φ)    :  A(U)∪_A(W) B(W) → B(U)

    are K-cofibrations of K-spaces.
  
  * For every flat orthogonal space B the map 
    B(φ): B(W)→ B(U) is a K-cofibration of K-spaces and the 
    K-space B(U) is K-cofibrant. 
  
  * Every flat orthogonal space is closed.subjectorthogonal space!closed
  


(i) The class of those morphisms of orthogonal spaces i
such that the map i(U)∪ B(φ)
is a K-cofibration of K-spaces is
closed under coproducts, cobase change, composition and
retracts. Similarly, the class of those morphisms of orthogonal spaces i
such that the map i(U) is a K-cofibration of K-spaces is
closed under coproducts, cobase change, composition and
retracts. 
So it suffices to show each of the two claims for a set of generating cofibrations.
We do this for the  morphisms _G,V× i_k
for all k≥ 0, all compact Lie groups G and all G-representations V,
where i_k:∂ D^k→ D^k is the inclusion.
In this case the first map specializes to (V,U)/G× i_k.
The map i_k is a cofibration and (V,U)/G is cofibrant as
a K-space by Proposition <ref> (ii).
So (V,U)/G× i_k is a K-cofibration of K-spaces.

The second map in question becomes the pushout product of
the sphere inclusion i_k with the map

    (V,φ)/G  :  (V,W)/G → (V,U)/G .

The map i_k is a cofibration and (V,φ)/G is a K-cofibration
by Proposition <ref> (i).
So their pushout product is again a K-cofibration.

Part (ii) is the special case of part (i) where A=∅ 
is the empty orthogonal space.
Part (iii) is the special case of (ii) where K
is a trivial group, using that cofibrations of spaces are in particular 
h-cofibrations (Corollary <ref>) 
and hence closed embeddings (Proposition <ref>).




Now we proceed towards the global model structure 
on the category of orthogonal spaces,
see Theorem <ref>.
The weak equivalences in this model structure are the global equivalences
and the cofibrations are the flat cofibrations. The fibrations in the
global model structure are defined as follows.

subjectglobal fibration!of orthogonal spacessubjectfibration!global|seeglobal fibration
    A morphism f:X→ Y of orthogonal spaces is a global fibration
    if it is a strong level fibration
    and for every compact Lie group G,
    every faithful G-representation V 
    and every equivariant linear isometric embedding φ:V→ W 
    of G-representations, the map
    
    (f(V)^G, X(φ)^G) :  X(V)^G  → 
        Y(V)^G ×_Y(W)^G   X(W)^G

    is a weak equivalence.
    
    An orthogonal space X is staticsubjectstatic!orthogonal space
    if for every compact Lie group G, every faithful G-representation V,
    and every G-equivariant linear isometric embedding φ:V→ W 
    the structure map 
    
    X(φ) :  X(V) → X(W)

    is a G-weak equivalence.



Equivalently, a morphism f is a global fibration if and only if f
is a strong level fibration and for every compact Lie group G,
every faithful G-representa­tion V and 
equivariant linear isometric embedding φ:V→ W 
the square of G-fixed point spaces
    
    @C=18mm X(V)^G [d]_f(V)^G[r]^-X(φ)^G    
               X(W)^G [d]^f(W)^G
    
              Y(V)^G [r]_-Y(φ)^G    Y(W)^G

    is homotopy cartesian.  


Clearly, an orthogonal space X is static if and only if the unique morphism
to a terminal orthogonal space is a global fibration;
the static orthogonal spaces will thus turn out to be the fibrant objects in the
global model structure.
The static orthogonal spaces are those which,
roughly speaking, don't change the equivariant homotopy type
once a faithful representation has been reached. 

 

 
  
   
  * Every global equivalence that is also a global fibration
    is a strong level equivalence.
  
  * Every global equivalence between static orthogonal spaces
    is a strong level equivalence.



(i) We let f:X→ Y be a morphism of orthogonal spaces 
that is both a global fibration 
and a global equivalence. We consider a compact Lie group G, 
a faithful G-representation V, a finite G-CW-pair (B,A) and a commutative square:

    A[r]^-α[d]_incl    X(V) [d]^f(V)
    
    B[r]_-β    Y(V)

We will exhibit a continuous G-map μ:B→ X(V) such that
μ|_A=α and such that f(V)∘μ
is homotopic, relative A, to β. 
This shows that the map f(V) is a G-weak equivalence,
so f is a strong level equivalence.

Since f is a global equivalence, there is 
a G-equivariant linear isometric embedding φ:V→ W 
and a continuous G-map λ:B→ X(W)
such that λ|_A=X(φ)∘α:A→ X(W) and
such that f(W)∘λ:B→ Y(W)
is G-homotopic, relative to A, to Y(φ)∘β.
Since f is a strong level fibration, 
we can improve λ into a continuous G-map λ':B→ X(W)
such that λ'|_A=λ|_A=X(φ)∘α
and such that f(W)∘λ' is equal to Y(φ)∘β.

Since f is a global fibration the G-map

    (f(V), X(φ)) :  X(V)  → 
              Y(V) ×_Y(W)  X(W)

is a G-weak equivalence.
So we can find a continuous G-map μ:B→ X(V) such that
μ|_A=α and (f(V), X(φ))∘μ
is G-homotopic, relative A, to 
(β,λ'):B→ Y(V) ×_Y(W)   X(W):

    @C=15mm
    A[r]^-α[d]_incl    X(V)  [d]^(f(V), X(φ))
    
    B[r]_-(β,λ')@–>[ur]^-(.4)μ      
    Y(V) ×_Y(W)   X(W)

This is the desired map.

(ii) We let f:X→ Y be a global equivalence between static orthogonal spaces.
We let G be a compact Lie group, V a faithful G-representation,
(B,A) a finite G-CW-pair 
and α:A→ X(V) and β:B→ Y(V) continuous G-maps 
such that f(V)∘α=β|_A.
Since f is a global equivalence, there is 
a G-equivariant linear isometric embedding φ:V→ W and
a continuous G-map λ:B→ X(W) such that
λ|_A=X(φ)∘α and f(W)∘λ is G-homotopic to
Y(φ)∘β relative A. 
Since X is static, the map X(φ):X(V)→ X(W)
is a G-weak equivalence, so there is a continuous G-map λ̅:B→ X(V)
such that λ̅|_A=α and X(φ)∘λ̅ 
is G-homotopic to λ relative A.
The two G-maps f(V)∘λ̅ and β:B→ Y(V)
then agree on A and become G-homotopic, relative A,
after composition with Y(φ):Y(V)→ Y(W).
Since Y is static, the map Y(φ) is a G-weak equivalence,
so f(V)∘λ̅ and β:B→ Y(V)
are already G-homotopic relative A.
This shows that f(V):X(V)→ Y(V) is a G-weak equivalence,
and hence f is a strong level equivalence.








We let j:A→ B be a morphism in a topological model category.
We factor j through the mapping cylinder as the composite

    A  c(j)  Z(j) =
    (A× [0,1])∪_j B   r(j)   B  ,

where c(j) is the `front' mapping  cylinder inclusion and r(j) 
is the projection, which is a homotopy equivalence.
In our applications we will assume that both A and B are cofibrant,
and then the morphism c(j) is a cofibration by the pushout product property.
We then define (j)symbol(j) - set of pushout products of cylinder inclusions
 as the set of all pushout product maps

    c(j) i_k  :    A× D^k ∪_A×∂ D^k Z(j)×∂ D^k
     → Z(j)× D^k

for k≥ 0, where i_k:∂ D^k→ D^k is the inclusion.




 
Let  be a topological model category, j:A→ B
a morphism between cofibrant objects and f:X→ Y a fibration.
Then the following two conditions are equivalent:


  * The square of spaces

    @C=15mm(B,X)[r]^-(j,X)[d]_(B,f)   (A,X) [d]^(A,f)
    (B,Y)[r]_-(j,Y)   (A,Y)

is homotopy cartesian.

  * The morphism f has the right lifting property with respect to the
set (j).



The square (<ref>) maps to the square

    @C=20mm(Z(j),X)[r]^-(c(j),X)[d]_(Z(j),f)   (A,X) [d]^(A,f)
    (Z(j),Y)[r]_-(c(j),Y)   (A,Y)

via the map induced by r(j):Z(j)→ B on the left part
and the identity on the right part. Since r(j) is a homotopy equivalence,
the map of squares is a weak equivalence at all four corners.
So the square (<ref>) is homotopy cartesian if and only if
the square (<ref>) is homotopy cartesian.

Since A is cofibrant and f a fibration, 
(A,f) is a Serre fibration. So the 
square (<ref>) is homotopy cartesian
if and only if the map

    ((Z(j),f),(c(j),X))    :  
    (Z(j),X) →    (Z(j),Y) ×_(A,Y)(A,X)

is a weak equivalence. Since c(j) is a cofibration and f is a fibration,
the map (<ref>) is always a Serre fibration.
So (<ref>) is a weak equivalence
if and only if it is an acyclic fibration,
which is equivalent to the right lifting property for
the inclusions i_k:∂ D^k→ D^k for all k≥ 0.
By adjointness, the map (<ref>) 
has the right lifting property with respect to the maps i_k
if and only if the morphism f has the right lifting property with respect to the
set (j).




The set J^ was defined in 
the proof of Proposition <ref> as the set of morphisms G_m j 
for m≥ 0 and for j in the set of generating acyclic cofibrations 
for the projective model structure on the category of O(m)-spaces specified 
in (<ref>).
The set J^ detects the fibrations 
in the strong level model structure. We add another set of morphisms K 
that detects when the squares (<ref>)
are homotopy cartesian.
Given any compact Lie group G and G-representations V and W,
the restriction morphism

    ρ_G,V,W =  ρ_V,W/G  :  _G,V⊕ W → _G,V

restricts (the G-orbit of) a linear isometric embedding
from V⊕ W to V.
If the representation V is faithful, 
then this morphism is a global equivalence 
by Proposition <ref> (ii).
We set

    K  =  ⋃_G,V,W(ρ_G,V,W)  ,

the set of all pushout products of boundary inclusions ∂ D^k→ D^k
with the mapping cylinder inclusions of the morphisms ρ_G,V,W;
here the union is over a set of representatives
of the isomorphism classes of triples (G,V,W) consisting of
a compact Lie group G, a faithful G-representation V and
an arbitrary G-representation W.
The morphism ρ_G,V,W represents the map of G-fixed point spaces
X(i_V,W)^G:X(V)^G→ X(V⊕ W)^G;
every G-equivariant linear isometric embedding is isomorphic
to a direct summand inclusion i_V,W,
so by Proposition <ref>, the right lifting property 
with respect to the union J^∪ K 
characterizes the global fibrations, i.e., we have shown:



A morphism of orthogonal spaces is a global fibration if and only if it has
the right lifting property with respect to the set J^∪ K.  



Now we are ready for the main result of this section.

 
    The global equivalences, global fibrations and flat cofibrations 
    form a model structure,
    the global model structuresubjectglobal model structure!for orthogonal spacessubjectmodel structure!global|seeglobal model structure
    on the category of orthogonal spaces. 
    The fibrant objects in the global model structure are the static orthogonal spaces.
    The global model structure is proper, topological and cofibrantly generated.


We number the model category axioms as in <cit.>. 
The category of orthogonal spaces is complete and cocomplete,
so axiom MC1 holds.
Global equivalences satisfy the 2-out-of-6 property
by Proposition <ref> (iii),
so they also satisfy the 2-out-of-3 property MC2.
Global equivalences are closed under retracts
by Proposition <ref> (iv);
it is straightforward that cofibrations and global fibrations 
are closed under retracts, so axiom MC3 holds.

The strong level model structure shows that every morphism of orthogonal spaces
can be factored as f∘ i for a flat cofibration i
followed by a strong level equivalence f that is also a strong level fibration.
For every G-equivariant linear isometric embedding φ:V→ W
between faithful G-representations, 
 both vertical maps in the commutative square 
of fixed point spaces (<ref>)
are then weak equivalences, so the square is homotopy cartesian.
The morphism f is thus a global fibration and a global equivalence,
so this provides one of the factorizations as required by MC5.
For the other half of the factorization axiom MC5
we apply the small object argumentsubjectsmall object argument 
(see for example <cit.> or <cit.>)
to the set J^∪ K.
All morphisms in J^ are flat cofibrations and strong level equivalences.
Since _G,V⊕ W and _G,V are flat, the morphisms in K
are also flat cofibrations, and they are global equivalences
because the morphisms ρ_G,V,W are 
(Proposition <ref> (ii)).
The small object argument provides a functorial factorization
of every morphism X→ Y of orthogonal spaces as a composite

    X   i   W   q   Y

where i is a sequential composition of cobase changes of coproducts
of morphisms in J^∪ K, and q has the right lifting property with respect 
to J^∪ K. 
Since all morphisms in J^∪ K are flat cofibrations
and global equivalences, the morphism i is a
flat cofibration and a global equivalence by the closure properties
of Proposition <ref>.
Moreover, q is a global fibration by Proposition <ref>.

Now we show the lifting properties MC4. 
By Proposition <ref> (i)
a morphism that is both a global equivalence and a global fibration
is a strong level equivalence, and hence an acyclic fibration
in the strong level model structure. So every morphism that is
simultaneously a global equivalence and  a global fibration has the
right lifting property with respect to flat cofibrations.
Now we let j:A→ B be a flat cofibration that is also a global equivalence and 
we show that it has the left lifting property with respect to all global fibrations.
We factor j=q∘ i, 
via the small object argument for J^∪ K,
where i:A→ W is a (J^∪ K)-cell complex and q:W→ B a global fibration.
Then q is a global equivalence since j and i are,
and hence an acyclic fibration in the strong level model structure,
again by Proposition <ref> (i).
Since j is a flat cofibration, a lifting in

    A [r]^-i [d]_j     W [d]^q_(.6)∼
    
    B @=[r] @..>[ur]     B

exists. Thus j is a retract of the morphism i that has the left lifting property
with respect to global fibrations. But then j itself has this lifting property.
This finishes the verification of the model category axioms.
Alongside we have also specified sets of generating flat cofibrations I^
and generating acyclic cofibrations J^∪ K.
Sources and targets of all morphisms in these sets are small with
respect to sequential colimits of flat cofibrations. So the global
model structure is cofibrantly generated.

Left properness of the global model structure follows from
Proposition <ref> (xi) and the fact that
flat cofibrations are h-cofibrations 
(Corollary <ref> (iii)).
Right properness follows from
Proposition <ref> (xii) because
global fibrations are in particular strong level fibrations.

The global model structure is topological by 
Proposition <ref>,
with  the set of semifree orthogonal spaces _G,V
indexed by a set of representatives (G,V) of the isomorphism classes
of pairs consisting of a compact Lie group G and a faithful G-representation V,
and with  the set of mapping cylinder inclusions c(ρ_G,V,W)
of the morphisms ρ_G,V,W.



The global model structure of orthogonal spaces is also monoidal,
even with respect to two different monoidal structures.
Indeed, the categorical product of orthogonal spaces has the pushout product
property for flat cofibrations, 
by Proposition <ref> below.
Moreover,  Proposition <ref> (iii)
(for the global family of all compact Lie groups)
shows that  global model structure of orthogonal spaces 
satisfies the pushout product property
with respect to the box product of orthogonal spaces.




We also introduce a `positive' version of the global model structure for
orthogonal spaces; our main use of this variation is for the
global model structure of ultra-commutative monoids 
in Section <ref>.
As is well known from similar contexts (for example, the stable model
structure for commutative orthogonal ring spectra),
model structures cannot usually be lifted naively to 
multiplicative objects with strictly commutative products.
The solution is to lift a `positive'
version of the global model structure in which the values at the
trivial inner product space
are homotopically meaningless and where the fibrant objects
are the `positively static' orthogonal spaces.


 A morphism f:A→ B of orthogonal spaces is a positive cofibrationsubjectcofibration!positivesubjectpositive cofibration!of orthogonal spaces
if it is a flat cofibration and the map f(0):A(0)→ B(0) is a homeomorphism.  
An orthogonal space Y 
is positively staticsubjectpositively staticsubjectstatic!positively
if for every compact Lie group G, every faithful G-representation V
with V 0 and every G-equivariant linear isometric embedding φ:V→ W  
the structure map 

    Y(φ) :  Y(V) → Y(W)

is a G-weak equivalence.


If G is a non-trivial compact Lie group, then any faithful G-representation
is automatically non-trivial. So a positively static orthogonal space
is static (in the absolute sense) if the structure map Y(0)→ Y()
is a non-equivariant weak equivalence.



 
  The global equivalences and positive cofibrations are part of 
  a cofibrantly generated, proper, topological model structure,
  the positive global model structuresubjectpositive global model structure!for orthogonal spacessubjectglobal model structure!positive
  on the category of orthogonal spaces. 
  A morphism f:X→ Y of orthogonal spaces is a fibration 
  in the positive global model structure if and only if
  for every compact Lie group G,
  every faithful G-representation V with V 0 
  and every equivariant linear isometric embedding φ:V→ W  
  the map f(V)^G:X(V)^G→ Y(V)^G is a Serre fibration and
  the square of G-fixed point spaces
    
    @C=18mm X(V)^G [d]_f(V)^G[r]^-X(φ)^G    
               X(W)^G [d]^f(W)^G
    
              Y(V)^G [r]_-Y(φ)^G    Y(W)^G

    is homotopy cartesian.  
    The fibrant objects in the positive global model structure 
    are the positively static orthogonal spaces.


  We start by establishing a positive strong level model structure.
  We call a morphism f:X→ Y of orthogonal spaces a
  positive strong level equivalencesubjectlevel equivalence!positive strong!of orthogonal spaces 
  (respectively positive strong level fibration)subjectlevel fibration!positive strong!of orthogonal spaces
  if for every inner product space V with V 0
  the map f(V):X(V)→ Y(V)  is an O(V)-weak equivalence
  (respectively an O(V)-fibration).
  Then we claim that the positive strong level equivalences, 
  positive strong level fibrations and positive cofibrations 
  form a topological model structure on the category of orthogonal spaces.

  The proof is another application of the general construction method
  for level model structures in 
  Proposition <ref>.
  Indeed, we let (0) be the degenerate model structure on 
  the category  of spaces in which every morphism
  is a weak equivalence and a fibration, but only the isomorphisms are cofibrations. 
  For m≥ 1 we let (m) be the projective
  model structure (for the set of all closed subgroups)
  on the category of O(m)-spaces,
  compare Proposition <ref>.
  With respect to these choices of model structures,
  the classes of level equivalences, level fibrations and cofibrations
  in the sense of Proposition <ref> 
  become the positive strong level equivalences,
  positive strong level fibrations and positive cofibrations.
  The consistency condition 
  (Definition <ref>)
  is now strictly weaker than for the strong level model structure,
  so it holds.
  
  We obtain the positive global model structure
  for orthogonal spaces by `mixing' the 
  positive strong level model structure 
  with the global model structure of Theorem <ref>.
  Every positive strong level equivalence is a global equivalence and every 
  positive cofibration is a flat cofibration.
  The global equivalences and the positive cofibrations
  are part of a model structure by Cole's mixing theorem <cit.>,
  which is our first claim. 
  By <cit.> (or rather its dual formulation),
  an orthogonal space is fibrant in the positive global model structure 
  if and only if it is weakly equivalent in the positive strong level model structure to 
  a static orthogonal space; this is equivalent to
  being positively static.

  Cofibrant generation, properness and topologicalness 
  of the positive global model structure are proved in much
  the same way as for the absolute global model structure
  in Theorem <ref>.




We can relate the unstable global homotopy theory
of orthogonal spaces to the homotopy theory of
G-spaces for a fixed compact Lie group G. 
Then evaluation at a faithful G-representation V 
and the semifree functor at (G,V) are a pair of adjoint functors

    _G,V :  G @<.4ex>[r]      :  _G,V@<.4ex>[l]

between the categories of G-spaces and orthogonal spaces.
This adjoint pair is a Quillen pair
with respect to the global model structure of orthogonal spaces and
the `genuine' model structure of G-spaces
(i.e., the projective model structure with respect to the family
of all subgroups, 
compare Proposition <ref>).
The adjoint total derived functors

    L(_G,V)  :  ( G )  @<.4ex>[r]       ( ) :  R(_G,V)  @<.4ex>[l]
 
are independent of the faithful representation V 
up to preferred natural isomorphism, by 
Proposition <ref> (ii).

Every G-space is G-weakly equivalent to a G-CW-complex,
and these are built from the orbits G/H.
So the derived left adjoint L(_G,V):G→ 
is essentially determined by its values on the coset spaces G/H.
Since _G,V(G/H) is isomorphic to _H,V=B_H, 
the derived left adjoint takes the homogeneous space G/H
to a global classifying space of H.

The derived right adjoint also has a more explicit description,
at least for closed orthogonal spaces Y, as the underlying G-space Y(_G).
Indeed, we can choose a fibrant replacement of Y 
in the global model structure, i.e., a flat cofibration j:Y→ Z
that is also a global equivalence, and such that Z is
globally fibrant (i.e., static).
Then Z is also closed, and so the induced map
j(_G):Y(_G)→ Z(_G) is a G-weak equivalence by
Proposition <ref>.
We may assume that V is a subrepresentation of _G;
we choose a nested sequence 

    V = V_1  ⊂ V_2  ⊂ … ⊂ V_n  ⊂ …

of finite-dimensional G-subrepresentations that exhaust _G.
Since V is faithful and Z is closed and static, the induced maps

    Z(V) = Z(V_1) → Z(V_2)  → … → Z(V_n)  → …

are all closed embeddings and G-weak equivalences.
So the canonical map

    Z(V)  → _n≥ 1  Z(V_n)  =  Z(_G)

is also a G-weak equivalence. Since Z is a globally fibrant
replacement of Y, the G-space Z(V) calculates the right derived functor
of _G,V at Y. This exhibits a chain of two G-weak equivalences

    R(_G,V)(Y) =  Z(V)  ≃  Z(_G) 
      ≃  Y(_G)  .




[Cofree orthogonal spaces]
subjectcofree orthogonal spacesubjectorthogonal space!cofree|seecofree orthogonal space
For every compact Lie group K, we will now define
a right adjoint to the functor that takes an orthogonal space Y
to the underlying K-space Y(_K).
We refer to the right adjoint R_K as the cofree functor.symbolR_K(A) - cofree orthogonal space of a K-space A
We consider the continuous functor

    (-,_K)  :   → (K)^ ,    V  ⟼ (V,_K) ,

with functoriality by precomposition with linear isometric embeddings.
The group K acts on the values of this functor through the action on
the complete universe _K.
The cofree orthogonal space R_K(A) associated to a K-space A is then the composite

    (-,_K) K^ ^K(-,A)  .

The unit of the adjunction is the morphism

    η_Y  :  Y  → R_K (Y(_K))

whose value at an inner product space V is the adjoint of the action map

    (V,_K)×  Y(V)  → Y(_K)  ,   
    (φ,y) ⟼ Y(φ)(y) .

The counit of the adjunction is the continuous K-map

    ϵ_A  :  R_K(A)(_K)    → A

assembled from the compatible K-maps

    R_K(A)(V) =  ^K((V,_K),A) → A  ,  
    f⟼ f(i_V) ,

for V∈ s(_K), where i_V:V→_K is the inclusion.
This data makes the functors

    (-)(_K) :    @<.4ex>[r]        K :  R_K @<.4ex>[l]

into an adjoint pair.





Let K be a compact Lie group.  


  * The adjoint functor pair ((-)(_K),R_K) is a Quillen pair for
the global model structure of orthogonal spaces and the projective model
structure of K-spaces.

  * For every K-space A the orthogonal space R_K(A) is static.

  * For every closed orthogonal space Y the map

    (η_Y)^*∘ R_K :  (K)(Y(_K),A)  → ()(Y, R_K(A))

is bijective.



(i)
We let f:X→ Y be a fibration of K-spaces.
We let G be another compact Lie group and V a faithful G-representation.
Then the K-space (V,_K)/G is K-cofibrant
by Proposition <ref> (ii).
The projective model structure on K-spaces is topological, 
so ^K((V,_K)/G,-) takes fibrations of K-spaces
to fibrations of spaces. Because

    ^K((V,_K)/G,X)  =  (R_K(X)(V))^G ,

this means that R_K takes fibrations of K-spaces to strong level fibrations
of orthogonal spaces. By the same argument, 
R_K takes acyclic fibrations of K-spaces to acyclic fibrations
in the strong level model structure, which coincide with the
acyclic fibrations in the global model structure of orthogonal spaces.

Now we let φ:V→ W be a G-equivariant linear isometric embedding.
Then the map

    ρ_V,W(_K)/G :  (V⊕ W,_K)/G  → (V,_K)/G

is a K-homotopy equivalence by Proposition <ref> (ii).
So the induced map 

    (R_K(X)(φ))^G :  (R_K(X)(V))^G → (R_K(X)(V⊕ W))^G

is a homotopy equivalence. So in the commutative square

    @C=20mm 
    (R_K(X)(V))^G[r]^-R_K(X)(φ)^G[d]_(R_K(f)(V))^G    
    (R_K(X)(V⊕ W))^G [d]^(R_K(f)(V⊕ W))^G
    
    (R_K(Y)(V))^G[r]_-R_K(Y)(φ)^G    (R_K(Y)(V⊕ W))^G

both vertical maps are Serre fibrations and both horizontal maps
are weak equivalences. 
The square is then homotopy cartesian, and so the morphism
R_K(f):R_K(X)→ R_K(Y) is a global fibration of orthogonal spaces.
Altogether this shows that the right adjoint R_K preserves fibrations and 
acyclic fibrations, so ((-)(_K),R_K) is a Quillen pair.

(ii) Every K-space A is fibrant in the projective model structure.
So R_K(A) is fibrant in the global model structure of orthogonal spaces;
by Theorem <ref> these fibrant object
are precisely the static orthogonal spaces.

(iii)
We choose a global equivalence f:Y^c→ Y with flat source.
Then Y^c and Y are both closed, the former by Proposition <ref> (iii). 
So the map f(_K):Y^c(_K)→ Y(_K)
is a K-weak equivalence by Proposition <ref>.
So the morphism f induces bijections on both sides of the map in question,
and it suffices to prove the claim for Y^c instead of Y.
But Y^c is cofibrant and A is fibrant, so in this case the claim
is just the derived adjunction isomorphism.





For K=e the trivial group we drop the subscript
and abbreviate the cofree functor R_e to R. 


An orthogonal space Y is cofreesubjectcofree orthogonal space 
if it is globally equivalent to an orthogonal space of the form R A
for some space A.


We will now develop criteria for detecting cofree orthogonal spaces,
and then recall some non-tautological examples. 
One criterion involves the unit of the adjunction, 
the special case 

    η_Y  :  Y  → R (Y(^∞))

of (<ref>) for _K=^∞. The next proposition shows that the 
morphism η_Y is always a non-equivariant weak equivalence,
provided Y is closed.



For every closed orthogonal space Y the morphism   
η_Y :Y → R (Y(^∞)) induces a weak equivalence

    η_Y(^∞)  : Y(^∞)  →  R (Y(^∞))(^∞)

on underlying non-equivariant spaces.


We start with a general observation about cofree orthogonal spaces.
Since R A is static (Proposition <ref> (ii)) and closed,
the map (R A)(φ):(R A)(V)→ (R A)(W) 
induced by any linear isometric embedding φ:V→ W
is a weak equivalence and a closed embedding.
So the canonical map

    A   ((0,^∞),A) =     (R A)(0) 
    →    _W∈ s(^∞)  (R A)(W)  =  (R A)(^∞)

is a weak equivalence as well.
The adjunction counit ϵ_A : (R A)(^∞) →  A
is a retraction to the map (<ref>), 
so ϵ_A is also a weak equivalence. 

Now we turn to the proof of the proposition.
Even though the map η_Y(^∞) under consideration is not the same 
as the canonical map (<ref>) for A=Y(^∞),
the counit ϵ_Y(^∞):R(Y(^∞))(^∞)→ Y(^∞) 
is also a retraction to η_Y(^∞). 
Since ϵ_Y(^∞) is a weak equivalence, so is η_Y(^∞). 


While the morphism η_Y:Y→ R(Y(^∞)) 
tends to be a non-equivariant equivalence,
it is typically not a global equivalence. 
We will now see that for a closed orthogonal space Y 
the morphism η_Y is a global equivalence if and only if Y is cofree.

We recall that a universal free K-space,
for a compact Lie group K, is a K-cofibrant free K-space 
whose underlying space in non-equivariantly contractible.
Any two universal free K-spaces are K-homotopy equivalent,
see Proposition <ref>.
We call a K-space A cofreesubjectcofree equivariant space 
if for some (hence any) universal free K-space E K the map

    const :   A → (E K, A)

that sends a point to the corresponding constant map is a K-weak equivalence.



For a closed orthogonal space Y the following three conditions are equivalent.


  * The orthogonal space Y is cofree.

  * For every compact Lie group K the K-space
Y(_K) is cofree.

  * The adjunction unit η_Y:Y→ R(Y(^∞)) is a global equivalence.



In a first step we show that for every space A and every compact Lie group K,
the K-space (R A)(_K) is cofree.
We choose a faithful K-representa­tion W.
Then (W,^∞) is a universal 
free K-space by Proposition <ref> (i).
So the projection from E K×(W,^∞) to the second factor
is a K-weak equivalence between cofibrant K-spaces, hence
a K-homotopy equivalence. So the induced map

    const :  (R A)(W)  =     ((W,^∞), A) 
    →    (E K×(W,^∞), A)   (E K,  (R A)(W))

is a K-homotopy equivalence. Hence the K-space (R A)(W)
is cofree as soon as K acts faithfully on W.
Since  R A is static (by Proposition <ref> (ii))
and closed, the canonical map

    (R A)(W)  → (R A)(_K)

is a K-weak equivalence. So (R A)(_K) is K-cofree.
Now we prove the equivalence of conditions (i), (ii) and (iii).

(i)⟹(ii)
The global equivalences are part of the global model structure on
the category of orthogonal spaces, compare Theorem <ref>. 
Moreover, the orthogonal space R A is static, hence fibrant in the global model
structure.
So if Y is globally equivalent to R A, then for some (hence any)
global equivalence p:Y^c→ Y with cofibrant (i.e., flat)
source, there is a global equivalence f:Y^c→ R A.

Now we let K be any compact Lie group.
The orthogonal space Y^c is closed by 
Proposition <ref> (iii). 
Since Y and R A are also closed, the global equivalences induce
K-weak equivalences

    Y(_K) [∼] p(_K) Y^c(_K) [∼] f(_K) (R A)(_K)

by Proposition <ref>.
Since (R A)(_K) is K-cofree by the introductory remark, so is Y(_K).

(ii)⟹(iii)
We start with a preliminary observation.
We let Y and Z be two closed orthogonal spaces
such that the K-spaces Y(_K) and Z(_K) are cofree
for all compact Lie groups K. We claim that every morphism f:Y→ Z
of orthogonal spaces such that f(^∞):Y(^∞)→ Z(^∞)
is a non-equivariant weak equivalence is already a global equivalence.
Indeed, for every compact Lie group K the two vertical maps in 
the commutative square of K-spaces

    @C=25mm 
    Y(_K)[d]_const[r]^-f(_K)   
    Z(_K)[d]^const
    (E K, Y(_K))[r]_-(E K, f(_K))^-≃   (E K, Z(_K))

are K-weak equivalences by hypothesis.
Since _K is non-equivariantly isometrically isomorphic to ^∞,
the K-map f(_K):Y(_K)→ Z(_K) is a non-equivariant weak equivalence
by hypothesis. So the lower horizontal map is a K-weak equivalence.
We conclude that the upper  horizontal map is a K-weak equivalence.
Since Y and Z are closed, 
the criterion of Proposition <ref> 
shows that f is a global equivalence.

Now we apply the criterion to the morphism η_Y:Y→ R(Y(^∞)). 
The map η_Y(^∞) is a weak equivalence
by Proposition <ref>. 
Moreover, for every compact Lie group K
the space Y(_K) is K-cofree by hypothesis (ii),
and R(Y(^∞))(_K) is K-cofree by the introductory remark.
The criterion of the previous paragraph thus applies and shows that the morphism
η_Y:Y→ R(Y(^∞)) is a global equivalence.

Condition (i) is a special case of (iii).



We recall now that the global classifying spaces of certain compact Lie groups
are cofree, namely of those with abelian identity path component.
Said differently, the group must be an extension of a finite group by a torus.
The following theorem is a reinterpretation of the main result
of Rezk's paper <cit.>,
who calls these groups `1-truncated' 
because the homotopy groups of the underlying spaces
vanish in dimensions larger than 1.
The special case of abelian compact Lie groups was proved
earlier by Lashof, May and Segal <cit.>.
The case of finite groups seems to be folklore,
going all the way back to Hurewicz <cit.> who proved 
that for finite groups G, homotopy classes of continuous maps B K→ B G 
biject with conjugacy classes of continuous group homomorphisms
from K to G.


 
Let G be a compact Lie group whose identity path component is abelian.
Then the global classifying space B_ G is cofree.subjectglobal classifying space!of an abelian compact Lie group


We let V be any faithful G-representation, so that B_G=_G,V. 
Then _G,V(_K)=(V,_K)/G is a classifying space
for principal (K,G)-bundles, by Proposition <ref>.
Since G has abelian identity component, it is 1-truncated
in the sense of Rezk, and so the K-space _G,V(_K) is cofree by 
<cit.>.
So criterion (ii) of Proposition <ref> is satisfied;
since the orthogonal space _G,V is closed, we have thus shown that it is cofree.



subjectglobal classifying space
The global classifying space B_ G is not cofree in general,
i.e., when the identity component of G is not abelian.
Indeed, for another compact Lie group K, the homotopy set
π_0^K(B_ G) (to be introduced in Definition <ref> below)
bijects with conjugacy classes of continuous homomorphisms from K to G, 
by Proposition <ref> (ii).
On the other hand, the set π_0^K(R(B G)) bijects 
with homotopy classes of continuous maps from B K to B G.
However, there are continuous maps B K→ B G that are not homotopic to Bα
for any continuous homomorphism α:K→ G.
Whenever this happens, the adjunction unit η_B_ G:B_ G→ R(B G) 
is not surjective on π_0^K, and hence not a global equivalence 
(by Corollary <ref> below).
The first examples of such `exotic' maps between classifying spaces
of compact Lie groups were constructed by Sullivan
and appeared in his widely circulated and highly influential
MIT lecture notes;
an edited version of Sullivan's notes was eventually published
in <cit.>.
Indeed, Corollary 5.11 of <cit.>
constructs `unstable Adams operations' ψ^p:B U(n)→ B U(n)
for a prime p and all n<p;
for n>1 these maps are not induced by any continuous homomorphism.




[Realization of simplicial objects]subjectrealization!of simplicial objects
We will occasionally want to realize simplicial objects, so we quickly
recall the necessary background. We let  be a cocomplete category
tensored over the category  of spaces. 
We let symbol - simplicial index category denote
the simplicial indexing category, with objects the finite
totally ordered sets [n]={0≤ 1≤…≤ n}
for n≥ 0. Morphisms in  are all weakly monotone maps.
We letsymbolΔ^n - topological n-simplex

    Δ^n  =  { (t_1,…,t_n)∈ [0,1]^n  | t_1≤ t_2≤…≤ t_n }

be the topological n-simplex.
As n varies, these topological simplices assemble into a covariant functor

    → ,    [n] ⟼ Δ^n ;

the coface maps are given by

    (d_i)_*(t_1,…,t_n)  = {[         (0,t_1,…,t_n)                      ; (t_1,…,t_i,t_i,…,t_n)                      ;         (t_1,…,t_n,1)                       ].

For 0≤ i ≤ n-1, the codegeneracy map (s_i)_*:Δ^n→Δ^n-1
drops the entry t_i+1.

A simplicial objectsubjectsimplicial object
in  is functor X:^→, i.e., a contravariant functor from .
We use the customary notation X_n=X([n]) for the value of a simplicial object at [n]. 
The realization of X is the coend

    |X| =  ∫^[n]∈Δ X_n×Δ^n

of the functor

    ^× →  ,    
    ([m],[n]) ⟼ X_m×Δ^n  .

We also need to recall the latching objects of a simplicial object.
We let (n) denote category with objects the weakly 
monotone surjections σ:[n]→ [k];
a morphism from σ:[n]→ [k] to σ':[n]→ [k']
is a morphism α:[k]→ [k'] in  
(necessarily surjective as well)
with α∘σ=σ'.
We let (n)_∘ denote the full subcategory of (n)
with all objects except the identity of [n].
A simplicial object X can be restricted along the forgetful functor

    (n)^_∘  u   ^ ,   
    (σ:[n]→ [k]) ⟼ [k] .

The n-th latching object of X is the colimit over (n)^_∘
of the restricted functor:

    L_n^(X)  =  _(n)^_∘  (X∘ u) .

The morphisms

    σ^*  :  X_k =  (X∘ u)(σ:[n]→ [k]) → X_n

assemble into a 
latching morphismsubjectlatching morphism!of a simplicial object

    l_n :  L_n^(X)  → X_n

from the n-th latching object of X to the value at [n].
For example, the category (0)_∘ is empty, so L_0^(X)
is an initial object of . 
The category (1)_∘ has a unique object s_0:[1]→ [0],
so L_1^(X) = X_0 and the latching morphism is given by s_0^*:X_0→ X_1.
The category (2)_∘ has three objects and two non-identity morphisms,
and L_2^(X) is a pushout of the diagram

    X_1   s_0^*   X_0   s_0^*  X_1  .



We specialize the above to realizations of simplicial orthogonal spaces,
i.e., simplicial objects in the category of orthogonal spaces.
For orthogonal spaces, coends and product with Δ^n 
are objectwise, and hence 

    |X|(V)   =  |X (V)| ,

i.e., the value of |X| at an inner product space V is 
the realization of the simplicial space [n]↦ X_n(V),
as discussed in Construction <ref>.
By Proposition <ref>, the realization
can be formed in the ambient category of all topological spaces,
and the result is automatically compactly generated.


A simplicial orthogonal space X is Reedy flatsubjectReedy flat!orthogonal space
if the latching morphism l_n:L_n^(X)→ X_n is a flat cofibration
of orthogonal spaces for every n≥ 0.


The terminology `Reedy flat' stems from Reedy's theorem <cit.>
that the simplicial objects in any model category admit a certain model
structure, nowadays called the `Reedy model structure', 
in which the equivalences are the level equivalences of simplicial objects. 
Reedy's paper – albeit highly influential – remains unpublished, but an account of 
the Reedy model structure can 
for example be found in <cit.>.
If we form the Reedy model structure starting
with the global model structure of orthogonal spaces,
then the cofibrant objects are precisely the 
Reedy flat simplicial orthogonal spaces.

The most important consequence of Reedy flatness for our purposes is that for these
simplicial orthogonal spaces, realization is homotopical.
Indeed, the global model structure of orthogonal spaces is
a topological model structure by Theorem <ref>.
We can turn this into a simplicial model structure by defining
the tensor of an orthogonal space X with a simplicial set A as

    X A  =  X× |A|  ,

the objectwise product with the geometric realization of A.
The following proposition is then a special case of the fact
that realization is a left Quillen functor for the Reedy model
structure on simplicial orthogonal spaces, see <cit.>.

subjectrealization!of simplicial orthogonal spaces
  
  
  * The realization of every Reedy flat simplicial orthogonal space is flat.
  
  * Let f:X→ Y be a morphism of Reedy flat
    simplicial orthogonal spaces. If f_n:X_n→ Y_n is a global equivalence
    for every n≥ 0, then the morphism of realizations
    |f|:|X|→ |Y| is a global equivalence.
  






§ MONOIDAL STRUCTURES



This section is devoted to monoidal products on the category of orthogonal spaces, 
with emphasis on global homotopical features.
Our main focus is the box product of orthogonal spaces,
a special case of a Day type convolution product,
and the `good' monoidal structure for orthogonal spaces.
We prove in Theorem <ref> that  the box product 
is fully homotopical with respect to global equivalences.
While the box product is the most useful monoidal
structure for orthogonal spaces, the cartesian product
is also relevant for our purposes.
We show in Proposition <ref>
that the categorical product, too, satisfies the pushout product property
for flat cofibrations.
In particular, the product of two flat orthogonal spaces is again flat.

The last part of this section introduces complex analogues ^_G,W 
of the semi­free orthogonal spaces,
indexed by unitary G-representations W,
see Construction <ref>.
While these complex versions are not (semi)free in any categorical
sense, they are similar to the semifree orthogonal spaces
in many ways; for example, the orthogonal spaces ^_G,W 
are flat (Proposition <ref> (ii)) 
and behave well multiplicatively under box and cartesian product
(Proposition <ref>).





We define a bimorphism 
b:(X,Y)→ Zsubjectbimorphism!of orthogonal spaces
from a pair of orthogonal spaces (X,Y)
to another orthogonal space Z as a collection of continuous maps 

    b_V,W :  X(V) ×  Y(W)  → Z(V⊕ W),

for all inner product spaces V and W, such that for all
linear isometric embeddings φ:V→ V' and ψ:W→ W' the following
square commutes:

    @C=20mm X(V) × Y(W) [r]^-b_V,W[d]_X(φ)× Y(ψ)   
    Z(V⊕ W) [d]^Z(φ⊕ψ)
    
    X(V') × Y(W') [r]_-b_V',W'     Z(V'⊕ W')

We define a box productsubjectbox product!of orthogonal spaces|( 
of X and Y as a universal example
of an orthogonal space with a bimorphism from X and Y.
More precisely, a box product is a pair (X⊠ Y,i) 
consisting of an orthogonal space X⊠ Y
and a universal bimorphism i:(X,Y)→ X⊠ Y,
i.e., such that for every orthogonal space Z the mapsymbol⊠ - box product of orthogonal spaces

    (X⊠ Y,Z)  → Bimor((X,Y),Z)  ,   
    f⟼ f i = {f(V⊕ W)∘ i_V,W}_V,W

is bijective. 
We will often refer to this bijection as the
universal property of the box product of orthogonal spaces.
subjectuniversal property!of the box product
Very often only the object X⊠ Y will be referred to 
as the box product, but one should keep in mind that it comes equipped
with a specific, universal bimorphism. 


The existence of a universal bimorphism out of any
pair of orthogonal spaces X and Y, and thus of a box product X⊠ Y, 
is a special case of the existence of Day type convolution products on
certain functor categories; the construction is an enriched
Kan extension of the `pointwise' cartesian product of X and Y
along the direct sum functor ⊕:×→
(see Proposition <ref>), or more explicitly an enriched
coend (see Remark <ref>).

Also by the general theory of convolution products, 
the box product X⊠ Y is a functor in both variables
(Construction <ref>) and it supports a preferred
symmetric monoidal structure (see Theorem <ref>);
so there are specific natural associativity and symmetry isomorphisms

    (X⊠ Y)⊠ Z  → X⊠(Y⊠ Z)
         respectively      
     X⊠ Y  → Y⊠ X

and a strict unit, the terminal orthogonal space 1,
i.e., such that 1⊠ X= X= X⊠1.
The upshot is that the associativity and symmetry isomorphisms 
make the box product
of orthogonal spaces into a symmetric monoidal product with 
the terminal orthogonal space as unit object. 
The box product of orthogonal spaces is closed symmetric monoidal
in the sense that the box product is adjoint to an
internal Hom orthogonal space. We won't use the internal function
object, so we do not elaborate on it.



The next result proves a key feature, namely that up to global equivalence,
the box product of orthogonal spaces coincides with the categorical product.
Given two orthogonal spaces X and Y, the maps

    X(V)× Y(W)  X(i_1)× Y(i_2)X(V⊕ W)× Y(V⊕ W)  =  (X× Y)(V⊕ W)

form a bimorphism (X,Y)→ X× Y as V and W vary through 
all inner product spaces; here i_1:V→ V⊕ W and
i_2:W→ V⊕ W are the two direct summand embeddings. 
This bimorphism is represented by a morphism

    ρ_X,Y :  X⊠ Y → X× Y

of orthogonal spaces that is natural in both variables.


Let X and Y be orthogonal spaces.


  * The morphism 
ρ_X,Y:X⊠ Y→ X× Y is a global equivalence. 

  * The functor X⊠ - preserves global equivalences.



(i) For an orthogonal space Z we denote by Z the orthogonal space defined by

    ( Z)(V)  =  Z( V⊕ V)      and    
    ( Z)(φ) =  Z(φ⊕φ) ;

thus Z is isomorphic to ^^2_(Z), 
the multiplicative shiftsubjectshift!of an orthogonal space!multiplicative
of Z by ^2 as defined 
in Example <ref>.
We define a morphism of orthogonal spaces

    λ :  X× Y  → (X⊠ Y)

at an inner product space V as the composite

    X(V)× Y(V) i_V,V 
    (X⊠ Y)(V⊕ V)  =  ( (X⊠ Y))(V)  .

Now we consider the two composites λ∘ρ_X,Y 
and (ρ_X,Y)∘λ:

    @C=12mm  
    X⊠ Y[r]^-ρ_X,Y      X× Y [r]^-λ   (X⊠ Y) [r]^-(ρ_X,Y)   ( X× Y)

We claim that the composite λ∘ρ_X,Y:X⊠ Y→( X⊠ Y)
is homotopic to the morphism (X⊠ Y)∘ i_1, where
i_1 is the natural linear isometric embedding V→ V⊕ V
as the first summand.
Indeed, for every t∈ [0,1] we define a natural linear isometric embedding

    j_t  :  V⊕ W  → V⊕ W⊕ V⊕ W   by  
    j_t(v,w) =  (v, t · w, 0, √(1-t^2)·  w) .

Then the maps

    X(V)× Y(W) i_V,W    (X⊠ Y)(V⊕ W)
    (X⊠ Y)(j_t)    (X⊠ Y)(V⊕ W⊕ V⊕ W) 
     =  ( (X⊠ Y))(V⊕ W)

form a bimorphism as V and W vary; 
the universal property of the box product turns this into a morphism

    f_t  :  X⊠ Y  →  (X⊠ Y)

of orthogonal spaces.
The linear isometric embeddings j_t vary continuously
with t, hence so do the morphisms f_t.
Moreover, f_0=λ∘ρ_X,Y and f_1=(X⊠ Y)∘ i_1, so this is
the desired homotopy.
The morphism (X⊠ Y)∘ i_1 is a global equivalence by
Theorem <ref>, hence so is the morphism λ∘ρ_X,Y.


The shift functor preserves products, and under the canonical isomorphism
(X× Y) ( X)× ( Y) the morphism
(ρ_X,Y)∘λ becomes the product of the two morphisms

    X∘ i_1 :  X →  X      and    
    Y∘ i_2 :  Y →  Y  .

The morphisms X∘ i_1 and Y∘ i_2 are global equivalences by
Theorem <ref>, hence so is their product
(by Proposition <ref> (vi)).
The global equivalences satisfy the 2-out-of-6 property
by Proposition <ref> (iii);
since λ∘ρ_X,Y and (ρ_X,Y)∘λ are
global equivalences, so is the morphism ρ_X,Y.

(ii) The cartesian product X×- preserves global equivalences 
by Proposition <ref> (vi).
Together with part (i) this implies part (ii).



[Box product of semifree orthogonal spaces]
We show that the box product of two semifree orthogonal spaces is another semifree
orthogonal space. This can be deduced from the general fact that a convolution
product of two representable functors is again representable 
(see Remark <ref>); however, the argument
is simple enough that we make it explicit for orthogonal spaces.

We consider two compact Lie groups G and K,
representations V and W of G respectively K, a G-space A and
a K-space B.
Then V⊕ W is a (G × K)-representation via 

    (g,k)· (v,w) =  (g v,  k w) ,

and A× B is a (G × K)-space in much the same way.
The map

    A× B   (_V· G,-)× (_W· K,- )    ( _G,V A)(V)× (_K,W B) (W)
       i_V,W       ( ( _G,V A)⊠ (_K,W B) )(V⊕ W)

is (G× K)-equivariant, so it extends freely to a morphism of orthogonal spaces

    _G× K,V⊕ W(A× B)  → (_G,V A)⊠ (_K,W B)  .

The maps

    ((V,U)×_G A) × ((W,U')×_K B)     → (V⊕ W,U⊕ U')×_G× K(A ×  B) 
    
    ( [φ,a],  [ψ,b])            ⟼     
     [φ⊕ψ,(a,b)]

form a bimorphism from (_G,VA,_K,WB) to _G× K,V⊕ W(A× B)
as the inner product spaces U and U' vary. The universal property of the box
product translates this into a morphism

    ( _G,V A)⊠ (_K,W B) → _G× K,V⊕ W(A× B) .

These two morphisms are mutually inverse isomorphisms, i.e., the box product
(_G,V A)⊠ (_K,W B) is isomorphic to _G× K,V× W(A× B).
A special case of this shows that the box product of two global classifying
spaces is another global classifying space.subjectglobal classifying space
Indeed, if G acts faithfully on V and K acts faithfully on W, then
the (G× K)-action on V⊕ W is also faithful, and hence

    ( B_G) ⊠ (B_ K)  =  _G,V⊠_K,W  _G× K,V⊕ W =  B_(G× K) .

If we compose the inverse of the isomorphism (<ref>)
with the global equivalence ρ__G,V, _K, W
from (<ref>), we obtain a global 
equivalence of orthogonal spaces

    B_(G× K)  ∼  ( B_G) × (B_ K)  .


subjectbox product!of orthogonal spaces|) 


Now we show that the categorical product of orthogonal spaces has the pushout product
property for flat cofibrations,
see Proposition <ref> below.
In particular, the product of two flat orthogonal spaces
is again flat, which is not completely obvious from the outset.
To this end we establish a useful sufficient condition for flatness;
the criterion is inspired by a flatness criterion for -spaces
proved by Sagave and Schlichtkrull in <cit.>.
The conditions may seem technical at first sight,
but we give two examples where they are easily verified,
see Propositions <ref> 
and <ref>.
The linear isometries category  does not have very many limits;
however, for property (b) of the next proposition we note that it does have
pullbacks. 


Let Y be an orthogonal space.


  * Suppose that Y satisfies the following conditions.


  * For every inner product space V, the space Y(V) is compact. 

  * As a functor from the category  to sets, Y preserves
pullbacks.

Then for all m≥ 0 the canonical morphism  i_m:^m Y→ Y 
from the m-skeleton is objectwise a closed embedding.

  * Let G be a compact Lie group acting continuously on Y
through automorphisms of orthogonal spaces. 
Suppose that in addition to the conditions (a) and (b),
the following also hold for all inner product spaces V:



  * The (G× O(V))-space Y(V) admits a (G× O(V))-CW-structure.

  * If an element of Y(V) is fixed by a reflection in O(V), then
it is in the image of Y(U) for some proper subspace U of V.

Then the orthogonal orbit space Y/G is flat.



We let V be any inner product space and  consider the composite

    ∐_0≤ i ≤  m(^i,V)× Y(^i)  →(^m  Y)(V) i_m(V) Y(V)   ,

where the first map is from the definition of ^m Y
as a coequalizer (<ref>).
The first map in (<ref>)
is a quotient map, and its source is compact by hypothesis (a)
and because the spaces (^i,V) are all compact.
So the space (^m Y)(V) is quasi-compact. 
Since Y(V) is Hausdorff, the map i_m(V) is a closed map.
So all that remains is to show that i_m(V) is injective.

We consider two pairs

    (φ, x)  ∈ (^i,V)× Y(^i)         and     
    (φ', x') ∈ (^j,V)× Y(^j)

with i, j≤ m, such that

    Y(φ)(x)  =  Y(φ')(x')   ∈ Y(V)  .

We show that the two pairs represent the same element in the
coequalizer (^m Y)(V).
We choose a pullback square in the category :

    ^k [r]^-h'[d]_h    ^j[d]^φ'
    ^i[r]_-φ    V

Condition (b) provides
an element z ∈ Y(^k) such that Y(h)(z)=x and Y(h')(z)=x'.
Then

    (φ,x)    = (φ, Y(h)(z)) ∼ (φ h, z) =  (φ' h' ,z) ∼
     (φ', Y(h')(z)) = (φ', x') .

So the pairs (φ,x) and (φ',x') 
describe the same element in (^m Y)(V).

Now we let G act on Y and also assume conditions (c) and (d).
By the first part, the latching morphism

    ν_m^Y =  i_m-1(^m) :  L_m(Y) =  (^m-1Y)(^m) → Y(^m)

is a closed embedding for every m≥ 0,
and the space Y(^m) admits a (G× O(m))-CW-structure by hypothesis (c).
We claim that ν_m^Y is a (G× O(m))-cofibration;
to this end we characterize its image by a stabilizer condition.
An element of y∈ Y(^m) is in the image of the latching map 
if an only if it is in the image of the map Y(U)→ Y(^m)
for some (m-1)-dimensional subspace U of ^m. 
Then the orthogonal reflection in the hyperplane U fixes y.
Conversely, if y is fixed by a reflection,
then it is in the image of Y(U) for some proper subspace U, by hypothesis (d).
So the image of the latching morphism coincides with the subspace 
of all elements of Y(^m)
whose stabilizer group contains the reflection in some hyperplane of ^m.

If K is any compact Lie group and X a K-CW-complex, then
the elements of X whose stabilizer group contains a conjugate
of some fixed subgroup is automatically a K-CW-subcomplex.
In the situation at hand this means that the image of
the latching map ν_m^Y is a (G× O(m))-subcomplex
in any (G× O(m))-CW-structure on Y(^m).
So the latching map is a (G× O(m))-cofibration.

The latching space construction commutes with colimits,
so the canonical map

    (L_m Y)/G →  L_m(Y/G)

is a homeomorphism, and the latching map for Y/G
is obtained from the latching map for Y by passage to G-orbits.
Since the latching map for Y 
is a (G× O(m))-cofibration, 
the induced map on G-orbits is then an O(m)-cofibration by  
Proposition <ref> (iii).
This shows that the orthogonal space Y/G is flat.


We apply the flatness criterion to show that the
product of two semifree orthogonal spaces is flat.


Let G and K be compact Lie groups and let V and W 
be faithful representations of G respectively K. Then the
orthogonal space _G,V×_K,W is flat.


We verify the conditions (a)-(d) of Proposition <ref>
for the orthogonal space _V×_W with the action of (G× K)
by precomposition on linear isometric embeddings.
The space (V,U) is compact and a Stiefel manifold,
so it comes with a `standard' smooth structure; 
moreover, the action of G× O(U) by pre- respectively postcomposition is smooth. 
By the same argument, (W,U) admits the structure of
a smooth closed (K× O(U))-manifold.
Hence the product (V,U)×(W,U)
underlies a closed smooth (G× K× O(U))-manifold.
So Illman's theorem <cit.> provides
a finite (G× K× O(U))-CW-structure.
This verifies conditions (a) and (c).

Conditions (b) is straightforward. Finally, if a pair

    (φ,ψ )  ∈ (V,U)×(W,U)

is fixed by the reflection in some hyperplane U' of U,
then the images of φ and ψ are contained in U', 
and hence (φ,ψ) lies in the image of
(V,U')×(W,U'). This verifies condition (d).
Proposition <ref> thus applies, and
shows that the orthogonal space (_V×_W)/(G× K) is flat.
So the isomorphic orthogonal space _G,V×_K,W is flat as well.


Since the semifree orthogonal spaces _G,V `generate' the flat cofibrations,
the previous Proposition <ref>
is the key input for the pushout product property for
the categorical product of orthogonal spaces.

 
Let f:A→ B and g:X→ Y be flat cofibrations of orthogonal spaces.
Then the pushout product morphism

    f g =  (f× Y)∪(B× g)  :  
    A× Y∪_A× XB× X  → B× Y

is a flat cofibration.
In particular, the product of two flat orthogonal spaces is again flat.

 
Since the cartesian product preserves colimits in both variables,
it suffices to show the claim for two generating flat cofibrations
  
    _G,V× i_k :  _G,V×∂ D^k 
       → _G,V× D^k

  and _K,W× i_m, where G and K are compact Lie groups,
  V and W are faithful representations of G respectively K,
  and k,m≥ 0.
  The pushout product (_G,V× i_k)(_K,W× i_m) 
  of two such generators is isomorphic to the morphism
  
    _G,V×_K,W× i_k+m .

  This morphism is a flat cofibration 
  since the strong level model structure of orthogonal spaces is topological
  and _G,V×_K,W is flat (Proposition <ref>).



In Definition <ref>
we introduced global classifying spaces as the semifree orthogonal spaces
defined from faithful orthogonal representations of compact Lie groups. 
As we shall explain in Proposition <ref> below, 
we can also use faithful unitary representations instead. 
This extra flexibility will come in handy
when we study global objects with an intrinsic complex flavor,
such as global classifying spaces of unitary groups,
complex Grassmannians, or complex Bott periodicity.
The next construction introduces the orthogonal space
^_G,W for a unitary G-representation W.
In contrast to their orthogonal cousins _G,V,
the unitary analogs are not representable nor semifree in any sense.
However, the unitary versions also enjoy various useful properties,
for example that they are flat 
(see Proposition <ref> (ii))
and behave well 
under box product (see Proposition <ref>).



To define the orthogonal space ^_G,W for a unitary G-representation W,
we introduce notation for going back and forth 
between euclidean inner product spaces over  and hermitian inner
product spaces over .
Throughout, we shall denote euclidean inner products on -vector spaces
by pointy brackets -,-, and
hermitian inner products on -vector spaces
by round parentheses (-,-).
For a euclidean inner product space V we let 

    V_ =  _ V

be the complexification;
symbolV_ - complexification of the inner product space V
the euclidean inner product -,- on V induces a hermitian
inner product (-,-) on the complexification V_, defined
as the unique sesquilinear form that satisfies

    (1 v, 1 v')  =  v,v'

for all v, v'∈ V.
For a hermitian inner product space W, we let u W 
denote the underlying -vector space,
equipped with the euclidean inner product

    w, w' =  Re ( w ,w' )  ,

the real part of the given hermitian inner product. 
Every -linear isometric embedding is in particular
an -linear isometric embedding of underlying euclidean vector spaces;
so U(W)⊆ O(u W), i.e., the unitary group of W
is a subgroup of the orthogonal group of u W. We thus view a unitary
representation on W as an orthogonal representation on u W.
If V and W are two finite-dimensional -vector spaces equipped
with hermitian inner products, we denote by ^(V,W) 
the space of -linear isometric embeddings.symbol^(V,W) - space of -linear isometric embeddings  
We topologize this as a complex Stiefel manifold, i.e., homeomorphic 
to the space of complex _(V)-frames in W.

Now we can define complex analogs of semifree orthogonal spaces.
We let G be a compact Lie group and W a finite-dimensional 
unitary G-representation. We define the orthogonal space ^_G,W by

    ^_G,W(V)  =   ^(W,V_) / G .

We define a morphism of orthogonal spaces

    f_G,W :  _G, u W → ^_G, W

as follows.
The map

    j_W  :  W  → _ W  =  ( u W)_ ,    
    j_W(w)  =  1/√(2)· ( 1 w - i(i w)  )

is a G-equivariant -linear isometric embedding. 
At a real inner product space V, we can thus define

    f_G,W(V) :  (u W,V)/ G   → ^(W,V_)/G    by  
    f_G,W(V)(φ G) =  ( φ_∘ j_W) G  .





Let G be a compact Lie group and W a faithful unitary G-representation.


  * The morphism f_G,W:_G, u W→^_G, W is a global equivalence.

  * The orthogonal space ^_G,W is flat.  



(i) Both source and target of f are closed orthogonal spaces; 
so by Proposition <ref> 
we may show that for every compact Lie group K the map

    f_G,W(_K) :  (u W,_K)/ G   → ^(W,__K) / G

is a K-weak equivalence.
We consider the (K× G)-equivariant continuous map

    f̃ :  (u W,_K)   → ^(W ,__K)
     ,   φ ⟼ φ_∘ j_W

that `covers' f_G,W(_K).
The source of f̃ is a universal (K× G)-space
for the family (K;G) of graph subgroups, 
by Proposition <ref> (i).
Since __K is a complete complex G-universe, the complex analog
of Proposition <ref> (i), proved in much the same way,
shows that the target of f̃ is also such a universal space for the
same family of subgroups of K× G.
So f̃ is a (K× G)-equivariant homotopy equivalence,
compare Proposition <ref>.
The map f_G,W(_K)=f̃/G induced on G-orbit spaces
is thus a K-equivariant homotopy equivalence.

(ii)
We verify conditions (a)-(d) of Proposition <ref>
for the orthogonal G-space ^_W.
The space ^(W,V_) is compact and a complex Stiefel manifold,
so it comes with a `standard' smooth structure; 
moreover, the action of G× O(V) by pre- respectively postcomposition is smooth. 
So Illman's theorem <cit.> provides 
a (G× O(V))-CW-structure on ^_W(V).
This verifies conditions (a) and (c).

For condition (b) we observe that complexification
takes pullback squares in  to pullback squares of hermitian 
inner product spaces and complex linear isometric embeddings.
So the functor ^(W,(-)_) preserves pullbacks.
Finally, if φ∈^(W,V_)
is fixed by the reflection in some hyperplane U of V,
then the image of φ is contained in U_, and hence φ
lies in the image of ^(W,U_). This verifies condition (d).
Proposition <ref> thus applies, and
shows that the orthogonal space
^_W/G = ^_G,W is flat.



Now we explain in which way the orthogonal spaces ^_G,W
are multiplicative in the pair (G,W).
We let K be another compact Lie group and U a unitary K-representation.
The maps

    ^(W,V_)/G ×^(U,V'_)/K     → ^(W⊕ U,(V⊕ V')_)/ ( G× K )
    
    ( φ· G, ψ· K )          ⟼     
     (φ⊕ψ)·(G× K)

form a bimorphism from (^_G,W,^_K,U) to ^_G× K,W⊕ U
as the inner product spaces V and V' vary. The universal property of the box
product translates this into a morphism of orthogonal spaces

    ζ_G,K;W,U :  ^_G,W⊠^_K,U → ^_G× K,W⊕ U .

The analogous morphism for semifree orthogonal spaces
(i.e., for orthogonal representations and without the superscript ^)
is an isomorphism, see Example <ref>.
For the complex analogs, an isomorphism would be too much to hope for, 
but the next best thing is true:


Let G and K be compact Lie groups, W a unitary G-representation
and U a unitary K-representation.  
Then the morphism ζ_G,K;W,U is a global equivalence.


The global equivalences discussed 
in Proposition <ref> (i) 
make the following square commute:

    @C=20mm_G, u W⊠_K,u U[r]^-_-(<ref>)[d]_f_G,W⊠ f_K, U^≃   _G× K, u( W⊕ U)[d]_≃^f_G× K, W⊕ U
    ^_G, W⊠^_K,U[r]_-ζ_G,K;W,U   ^_G× K, W⊕ U

The left vertical morphism is a global equivalence because these
are stable under box product (Theorem <ref> (ii)).
Since the vertical morphisms are global equivalences and the upper horizontal
one is an isomorphism, the lower horizontal morphism ζ_G,K;W,U
is also a global equivalence.






§ GLOBAL FAMILIES



In this section we explain a variant of unstable global homotopy
theory based on a global family, i.e., a class  of 
compact Lie groups with certain closure properties. 
We introduce -equivalences, a relative version of global equivalences,
and establish an -relative version of the global model
structure in Theorem <ref>. 
We also discuss the compatibility of the
-global model structure with the box product of
orthogonal spaces, see Proposition <ref>.
Finally, we record that for multiplicative global families,
the -global model structure lifts to category of modules and
algebras, see Corollary <ref>.

 
A global familysubjectglobal familysymbol - global family 
is a non-empty class of compact Lie groups
that is closed under isomorphisms, closed subgroups and quotient groups.


Some relevant examples of global families are: 
all compact Lie groups;
all finite groups; 
all abelian compact Lie groups; 
all finite abelian groups;
all finite cyclic groups; 
all finite p-groups.
Another example is the global family G generated by a compact Lie group G,
i.e., the class of all compact Lie groups isomorphic to a quotient 
of a closed subgroup of G.
A degenerate case is the global family esymbole - global family of trivial groups of all trivial groups.
In this case our theory specializes to the non-equivariant
homotopy theory of orthogonal spaces.
 
For a global family  and a compact Lie group G we write ∩ Gsymbol∩ G - subgroups of G belonging to 
for the family of those closed subgroups of G
that belong to .  We also write (m) for ∩ O(m),
the family of closed subgroups of O(m) that belong to .symbol(m) - subgroups of O(m) belonging to 
We recall that an equivariant continuous map of O(m)-spaces
is an (m)-cofibration if it has 
the right lifting property with respect to all morphisms q:A→ B
of O(m)-spaces such that the map q^H:A^H→ B^H is a weak equivalence
and Serre fibration for all H∈(m).

The following definitions of -level equivalences,
-level fibrations and -cofibrations 
are direct relativizations of the corresponding concepts in the
strong level model structure of orthogonal spaces.





 Let  be a global family.
  A morphism f:X→ Y of orthogonal spaces is
  
  
  * an -level equivalencesubjectF-level equivalence@-level equivalence!of orthogonal spaces 
    if for every compact Lie group G in  and every G-representation V the map
    f(V)^G:X(V)^G→ Y(V)^G is a weak equivalence;
  
  * an -level fibrationsubjectF-level fibration@-level fibration!of orthogonal spaces 
    if for every compact Lie group G in  and every G-representation V the map
    f(V)^G:X(V)^G→ Y(V)^G is a Serre fibration;
  
  * an -cofibration if the latching morphism
    ν_m f:X(^m)∪_L_m XL_m Y→ Y(^m) is an (m)-cofibration 
    for all m≥ 0.



Every inner product space V is isometrically isomorphic to ^m
with the standard scalar product, where m is the dimension of V.
So a morphism f:X→ Y of orthogonal spaces is an
-level equivalence (respectively -level fibration)
precisely if for every m≥ 0
the map f(^m):X(^m)→ Y(^m)  is an (m)-equivalence
(respectively (m)-projective fibration).
The formal argument is analogous to Lemmas <ref> and <ref> 
which treat the case =.
Clearly, the classes of -level equivalences,
-level fibrations and -cofibrations are closed under composition, retracts 
and coproducts.


Now we discuss the  -level model structures on orthogonal spaces.
When = is the global family of all compact Lie groups, 
then (m) is the family of all closed subgroups of O(m).
For this maximal global family,
an -level equivalence is just a strong level equivalence
in the sense of Definition <ref>.
Moreover, the -level fibrations coincide with the strong level fibrations.
The -cofibrations coincide with the flat cofibrations. 
So for the global family of all compact Lie groups the  
-level model structure on orthogonal spaces is the strong level model structure 
of Proposition <ref>.


  Let  be a global family.
  The -level equivalences, -level fibrations 
  and -cofibrations form a topological and cofibrantly generated model structure,
  the -level model structure,subjectF-level model structure@-level model structure!for orthogonal spaces
  on the category of orthogonal spaces.


We specialize Proposition <ref> 
by letting (m) be the (m)-projective
model structure on the category of O(m)-spaces,
compare Proposition <ref>.
With respect to these choices of model structures (m),
the classes of level equivalences, level fibrations and cofibrations
in the sense of Proposition <ref> 
precisely become the -level equivalences,
 -fibrations and -cofibrations.
Every acyclic cofibration in the (m)-projective model structure of O(m)-spaces
is also an acyclic cofibration in the -projective model structure of O(m)-spaces.
So the consistency condition (see Definition <ref>)
in the present situation is a special case of the consistency condition
for the strong level model structure that we established in the proof of
Proposition <ref>.

We describe explicit sets of generating cofibrations 
and generating acyclic cofibrations for the  -level model structure.
We let I_ be the set of all morphisms G_mi for m≥ 0
and for i in the set
of generating cofibrations for the (m)-projective model
structure on the category of O(m)-spaces specified in (<ref>).
Then the set I_ detects the acyclic fibrations 
in the  -level model structure 
by Proposition <ref> (iii). 
Similarly, we let J_ be the set of all morphisms G_m j 
for m≥ 0 and for j in the set
of generating acyclic  cofibrations for the (m)-projective model
structure on the category of O(m)-spaces specified 
in (<ref>).
Again by Proposition <ref> (iii),
J_ detects the fibrations in the  -level model structure. 
The -level model structure is topological 
by Proposition <ref>,
where we take  as the set of orthogonal spaces L_H,^m
for all m≥ 0 and all H∈(m).



Now we proceed towards the construction of the 
-global model structure,
see Theorem <ref> below.
The weak equivalences in this model structures
are the -equivalences of the following definition,
the direct generalization of global equivalences in the presence of a global family.



  Let  be a global family. A morphism f:X→ Y of orthogonal spaces
  is an -equivalencesubjectF-equivalence@-equivalence!of orthogonal spaces
  if the following condition holds: for every compact Lie group G in ,
  every G-representation V, every k≥ 0 
  and all maps α:∂ D^k→ X(V)^G
  and β:D^k→ Y(V)^G such that f(V)^G∘α=β|_∂ D^k
  there is a G-representation W,
  a G-equivariant linear isometric embedding φ:V→ W
  and a continuous map λ:D^k→ X(W)^G
  such that λ|_∂ D^k=X(φ)^G∘α and
  such that f(W)^G∘λ
  is homotopic, relative to ∂ D^k, to Y(φ)^G∘β.



When = is the maximal global family of all compact Lie groups,
then -equivalences are precisely the global equivalences.
The following proposition generalizes Proposition <ref>,
and it is proved in much the same way. 


 
Let  be a global family.
For every morphism of orthogonal spaces f:X→ Y, the following
three conditions are equivalent.


  * The morphism f is an -equivalence.

  * Let G be a compact Lie group,
V a G-representation and (B,A) a finite G-CW-pair 
all of whose isotropy groups belong to .
Then for all continuous G-maps α:A→ X(V)
and β:B→ Y(V) such that β|_A=f(V)∘α,
there is a G-representation W, a G-equivariant linear
isometric embedding φ:V→ W and a continuous G-map λ:B→ X(W)
such that λ|_A=X(φ)∘α and
such that f(W)∘λ
is G-homotopic, relative to A, to Y(φ)∘β.

  * For every compact Lie group G in the family 
and every exhaustive sequence {V_i}_i≥ 1
of G-represen­tations the induced map

    _i f(V_i) :  _i X(V_i) → _i Y(V_i)

is a G-weak equivalence.





subjectF-global fibration@-global fibration!of orthogonal spaces
    A morphism f:X→ Y of orthogonal spaces is an -global fibration
    if it is an -level fibration
    and for every compact Lie group G in the family ,
    every faithful G-representation V 
    and every equivariant linear isometric embedding φ:V→ W 
    of G-representations, the map
    
    (f(V)^G, X(φ)^G) :  X(V)^G  → 
        Y(V)^G ×_Y(W)^G   X(W)^G

    is a weak equivalence.


The next proposition contains various properties of -equivalences
that generalize Proposition <ref>
and Proposition <ref> (i). 




 
  Let  be a global family.
  
  
  * Every -level equivalence is an -equivalence.
  
  * The composite of two -equivalences is an -equivalence.
  
  * If f, g and h are composable morphisms of orthogonal spaces 
    such that g f and h g are -equivalences, then f, g, h and h g f
    are also -equivalences.
  
  * Every retract of an -equivalence is an -equivalence.
  
  * A coproduct  of any set of -equivalences is an -equivalence.
  
  * A finite product of -equivalences is an -equivalence.
  
  * For every space K the functor -× K preserves -equivalences
    of orthogonal spaces.
  
  * Let e_n:X_n→ X_n+1 and f_n:Y_n→ Y_n+1 be morphisms 
    of orthogonal spaces that are objectwise closed embeddings, for n≥ 0. 
    Let ψ_n:X_n→ Y_n be -equivalences of orthogonal spaces
    that satisfy ψ_n+1∘ e_n=f_n∘ψ_n for all n≥ 0.
    Then the induced morphism ψ_∞:X_∞→ Y_∞ 
    between the colimits of the sequences is an -equivalence.
  
  * Let f_n:Y_n→ Y_n+1 be an -equivalence
   and a closed embedding of orthogonal spaces, for n≥ 0. 
   Then the canonical morphism 
   f_∞:Y_0→ Y_∞ to the colimit of the sequence {f_n}_n≥ 0
   is an -equivalence.

  * Let 
    
    C  [d]_γ    A [l]_-g [d]^α[r]^-f     B [d]^β
    
          C'      A' [l]^-g'[r]_-f'    B'

    be a commutative diagram of orthogonal spaces such that f and f' are
    h-cofibrations.
    If the morphisms α,β and γ are -equivalences,
    then so is the induced morphism of pushouts
    
    γ∪β :  C∪_A B  → C'∪_A' B' .

  
  * Let 
    
    A [r]^-f [d]_g     B [d]^h
    
          C [r]_-k     D

    be a pushout square of orthogonal spaces such that f is an -equivalence.
    If in addition f or g is an h-cofibration, then the morphism 
    k is an -equivalence.
 
  * Let 
   
    P[r]^-k [d]_g     X [d]^f 
     Z [r]_-h     Y

   be a pullback square of orthogonal spaces in which f is an -equivalence.
   If in addition one of the morphisms f or h is
   an -level fibration, then the morphism g is also an -equivalence.
   
  * Every -equivalence that is also an -global fibration 
    is an -level equivalence.
  
  * The box product of two -equivalences is an -equivalence.



The proofs of (i) through (xii)
are almost verbatim the same as the corresponding parts
of Proposition <ref>, and we omit them.
Part (xiii) is proved in the same way as 
Proposition <ref> (i). 

(xiv)
The product of orthogonal spaces preserves 
-equivalences in both variables by part (vi).
The morphism ρ_X,Y:X⊠ Y→ X× Y is a global equivalence,
hence an -equivalence, for all orthogonal spaces X and Y,
by Theorem <ref> (i); this implies the claim.




Now we establish the  -global model structures on the
category of orthogonal spaces.
We spell out sets of generating cofibrations and
generating acyclic cofibrations for the -global model structures.
In Proposition <ref>
we introduced I_ as the set of all morphisms G_m i for m≥ 0
and for i in the set of generating cofibrations for the (m)-projective model
structure on the category of O(m)-spaces specified in (<ref>).
The set I_ detects the acyclic fibrations 
in the  -level model structure, 
which coincide with the acyclic fibrations 
in the  -global model structure.

Also in Proposition <ref>
we defined  J_ as the set of all morphisms G_m j 
for m≥ 0 and for j in the set
of generating acyclic  cofibrations for the (m)-projective model
structure on the category of O(m)-spaces specified 
in (<ref>).
The set J_ detects the fibrations in the  -level model structure. 
We add another set of morphisms K_ 
that detects when the squares (<ref>)
are homotopy cartesian for G∈.
We set

    K_ =  ⋃_G,V,W :  G∈(ρ_G,V,W)  ,

the set of all pushout products of sphere inclusions i_k:∂ D^k→ D^k
with the mapping cylinder inclusions of the morphisms ρ_G,V,W,
compare Construction <ref>;
here the union is over a set of representatives
of the isomorphism classes of triples (G,V,W) consisting of
a compact Lie group G in , a faithful G-representation V and
an arbitrary G-representation W.
By Proposition <ref>, the right lifting property 
with respect to the union J_∪ K_ 
characterizes the  -global fibrations.


The proof of the following theorem
proceeds by mimicking the proof in the special case =,
and all arguments in the proof of Theorem <ref> 
go through almost verbatim. Whenever the small object argument is used,
it now has to be taken with respect to the set
J_∪ K_ (as opposed to the set J^∪ K).




 
  Let  be a global family.
  
  
  * The -equivalences, -global fibrations and -cofibrations 
    form a model structure,
    the -global model structure,subjectF-global model structure@-global model structure!for orthogonal spacessubjectglobal model structure!-
    on the category of orthogonal spaces. 
  
  * The fibrant objects in the  -global model structure 
    are the -static orthogonal spaces,subjectF-static@-static
    i.e., those orthogonal spaces X such that for every compact Lie group G in ,
    every faithful G-representation V
    and every G-equivariant linear isometric embedding φ:V→ W
    the map of G-fixed point spaces X(φ)^G: X(V)^G→ X(W)^G
    is a weak equivalence.
  
  * A morphism of orthogonal spaces is:
    
    
  * an acyclic fibration in the  -global model structure
      if and only if it has the right lifting property 
      with respect to the set I_; 
    
  * a fibration in the  -global model structure
      if and only if it has the right lifting property 
      with respect to the set J_∪ K_.  
    
  
  * The  -global model structure is cofibrantly generated, 
    proper and topological. 
  




In the case =e of the minimal global family of trivial groups,
the e-global homotopy theory of orthogonal spaces
just another model for the (non-equivariant) homotopy theory of spaces.
Indeed, the evaluation functor _0:→ is a right Quillen
equivalence with respect to the  e-global model structure.
So the total derived functor

    (_0)  : ^e()  →()

is an equivalence of homotopy categories.

In fact, for the global family =e, most of what we do here has already been
studied before: The  e-global model structure 
and the fact that it is Quillen equivalent to the model category of spaces 
were established by Lind <cit.>;
in <cit.>, orthogonal spaces are called `ℐ-spaces' and
e-global equivalences 
are called `weak homotopy equivalences' and 
are defined as those morphisms that induce weak equivalences on homotopy colimits.





  Let f:A→ B be a morphism of orthogonal spaces
  and  a global family.
  Then the following conditions are equivalent.
  
  
  * The morphism f is an -equivalence.subjectF-equivalence@-equivalence!of orthogonal spaces
  
  * The morphism can be written as f=w_2∘ w_1 for an
    -level equivalence w_2 and a global equivalence w_1.
  
  * For some (hence any) -cofibrant approximation
    f^c:A^c→ B^c in the  -level model structure
    and every -static orthogonal space Y 
    the induced map
    
    [f^c,Y]  :  [B^c,Y]  → [A^c,Y]

    on homotopy classes of morphisms is bijective.



(i)⟺(ii) 
The -equivalences contain the global equivalences by definition,
and the -level equivalences by 
Proposition <ref> (i), and are closed under composition by
Proposition <ref> (ii);
so all composites w_2∘ w_1 as in the claim are -equivalences. 
On the other hand, every -equivalence f can be factored 
in the global model structure of Theorem <ref> as f=q j
where j is a global equivalence and q is a global fibration.
Since f and j are -equivalences, so is q by 
Proposition <ref> (iii);
so q is an -equivalence and a global fibration, 
hence an -level equivalence by
Proposition <ref> (xiii).

(i)⟺(iii) The morphism f is an -equivalence
if and only if the -co­fibrant approximation f^c:A^c→ B^c is
an -equivalence.
Since A^c and B^c are -cofibrant, they are
cofibrant in the  -global model structure.
So by general model model category theory
(see for example <cit.>),
f^c is an -equivalence if and only if the induced map [f^c,X] 
is bijective for every fibrant object in the
 -global model structure.
By Theorem <ref> (ii) 
these fibrant objects are precisely the -static orthogonal spaces.






[Mixed global model structures]
Cole's `mixing theorem' for model structures <cit.>  
allows to construct many more `mixed' -global model structures 
on the category of orthogonal spaces.
We consider two global families such that ⊆. 
Then every -equivalence is an -equivalence and every 
fibration in the  -global model structure
is a fibration in the  -global model structure.
By Cole's theorem <cit.> 
the -equivalences and the fibrations of the
 -global model structure are part of a model structure,
the -mixed -global model structuresubjectmixed global model structuresubjectglobal model structure!mixed
on the category of orthogonal spaces.
By <cit.> 
the cofibrations in the -mixed -global model structure 
are precisely the retracts of all composites
h∘ g in which g is an -cofibration and h is simultaneously
an -equivalence and an -cofibration.
In particular, an orthogonal space is cofibrant in the
-mixed -global model structure if it is -cofibrant and
-equivalent to an -cofibrant 
orthogonal space <cit.>.
The -mixed -global model structure is again proper 
(Propositions 4.1 and 4.2 of <cit.>).

When =e is the minimal family of trivial groups,
this provides infinitely many -mixed model structures on
the category of orthogonal spaces that are all Quillen equivalent
to the model category of (non-equivariant) spaces, with respect to
weak equivalences.







The next topic is the compatibility of the -global model structure 
with the box product of orthogonal spaces.
Given two morphisms f:A→ B and g:X→ Y of orthogonal spaces
we denote by 

    f g = (f⊠ Y)∪(B⊠ g)  :  
    A⊠ Y∪_A⊠ XB⊠ X  → B⊠ Y

the pushout product morphism.subjectpushout product
We recall that a  model structure on a symmetric monoidal category
satisfies the pushout product property
if the following two conditions hold:


  * for every pair of cofibrations f and g the
  pushout product morphism f g is also a cofibration;

  * if in addition f or g is a weak equivalence, then so is the
  pushout product morphism  f g.

We let  and  be two global families. We denote by
× the smallest global family that contains all
groups of the form G× K for G∈ and K∈.
So a compact Lie group H belongs to × if and
only if H is isomorphic to a closed subgroup of a 
group of the form (G× K)/N for some groups G∈ and K∈, 
and some closed normal subgroup N of G× K.

 
  Let  and  be two global families.
  
  
  * The pushout product of an -cofibration with an -cofibration
    is an (×)-cofibration.  
  
  * The pushout product of a flat cofibration that is also
    an -equivalence with any morphism of orthogonal spaces is an -equivalence.
  
  * Let  be a multiplicative global family, i.e., × =.
    Then the  -global model structure  
    satisfies the pushout product property
    with respect to the box product of orthogonal spaces.
  
  * The positive global model structure satisfies the pushout product property
    with respect to the box product of orthogonal spaces.
  

 
  (i) It suffices to show the claim for sets of generating cofibrations. 
  The -cofibrations are generated by the morphisms
  
    _G,V× i_k  :  _G,V×∂ D^k
       → _G,V× D^k

  for G∈, V a G-representation and k≥ 0.
  Similarly, the -cofibrations are generated by the morphisms
  _K,W× i_m for K∈, W a K-representation and m≥ 0.
  The pushout product of two such generators is isomorphic to the morphism
  
    _G× K,V⊕ W× i_k+m :  _G× K,V⊕ W×∂ D^k+m → _G× K,V⊕ W× D^k+m ,

  compare Example <ref>.
  Since G× K belongs to the family ×,
  this pushout product morphism is an (×)-cofibration.

  (ii) We let i:A→ B and j:K→ L be morphisms of orthogonal spaces
  such that i is a flat cofibration and an -equivalence.
  Then i⊠ K and i⊠ L are -equivalences
  by Proposition <ref> (xiv).
  Moreover, i is an h-cofibration 
  by Corollary <ref> (iii),
  hence so is i⊠ K:A⊠ K→ B⊠ K.
  Thus its cobase change, the canonical morphism
  
    A⊠ L  → A⊠ L ∪_A⊠ K B⊠ K

  is an -equivalence by Proposition <ref> (xi).
  Since i⊠ L:A⊠ L→ B⊠ L is also an -equivalence,
  so is the pushout product map, by 2-out-of-6 
  (Proposition <ref> (iii)).

  (iii) The part of the pushout product property that refers only to cofibrations
  is true by part (i) with = and the hypothesis that ×=. 
  Every cofibration in the  -global model structure is
  in particular a flat cofibration,
  so the part of the pushout product property that also refers to 
  acyclic cofibrations in the  -global model structure
  is a special case of (ii).

  Part (iv) is proved in the essentially the same way as (iii), 
  for the global family of all compact Lie groups.



Finally, we will discuss another important relationship between
the -global model structures and the box product, 
namely the monoid axiom <cit.>.
We only discuss a slightly weaker form of the monoid axiom 
in the sense that we only cover sequential (as opposed to more general transfinite)
compositions.

subjectmonoid axiom!for the box product of orthogonal spaces 

We let  be a global family.
For every flat cofibration i:A→ B that is also an -equivalence
and every orthogonal space Y the morphism

    i⊠ Y  :  A⊠ Y  → B⊠ Y

is an h-cofibration and an -equivalence.subjecth-cofibration
Moreover, the class of h-cofibrations that are also -equivalences
is closed under cobase change, coproducts and sequential compositions.


Every flat cofibration is an h-cofibration 
(Corollary <ref> (iii)
applied to the strong level model structure),
and h-cofibrations are closed under box product with any orthogonal space
(Corollary <ref> (ii)),
so i⊠ Y is an h-cofibration.
Since i is an -equivalence, so is i⊠ Y 
by Proposition <ref> (xiv).

Proposition <ref> shows that the class of h-cofibrations 
that are also -equivalences is closed under cobase change, 
coproducts and sequential compositions.



We let  be a multiplicative global family, i.e., ×=.
The constant one-point orthogonal space 1 
is the unit object for the box product 
of orthogonal spaces, and it is `free', i.e., e-cofibrant.
So 1 is cofibrant in the -global model structure. 
So with respect to the box product, the -global model structure is a
symmetric monoidal model category
in the sense of <cit.>.
A corollary is that the unstable -global homotopy category, i.e., 
the localization of the category
of orthogonal spaces at the class of -equivalences,
inherits a closed symmetric monoidal structure, 
compare <cit.>.
This `derived box product'subjectbox product!of orthogonal spaces 
is nothing new, though: 
since the morphism ρ_X,Y:X⊠ Y→ X× Y
is a global equivalence for all orthogonal spaces X and Y, 
the derived box product is just a categorical product in ^().





subjectorthogonal monoid space 
subjectmonoid space!orthogonal|seeorthogonal monoid space 
An orthogonal monoid space
is an orthogonal space R equipped with unit morphism η:1→ R
and a multiplication morphism μ:R⊠ R→ R
that is unital and associative in the sense that the diagram

    @C=15mm
    (R⊠ R)⊠ R [r]_-^-associativity[d]_μ⊠ R   
    R⊠(R⊠ R)[r]^-R⊠μ   
    R⊠ R [d]^μ
    
    R⊠ R [rr]_-μ       R

commutes.
An orthogonal monoid space R is 
commutativesubjectorthogonal monoid space!commutative if moreover
μ∘τ_R,R=μ, where τ_R,R:R⊠ R→ R⊠ R
is the symmetry isomorphism of the box product.
A morphism of orthogonal monoid spaces is a morphism of orthogonal spaces
f:R→ S such that f∘μ^R= μ^S∘(f⊠ f) and
f∘η_R=η_S.



One can expand the data of an orthogonal monoid space into an `external' form
as follows. The unit morphism η:1→ R is determined by
a unit element 0∈ R(0), the image of the map η(0):1(0)→ R(0).
The multiplication map corresponds to continuous maps
μ_V,W:R(V) × R(W) → R(V⊕ W) for all inner product spaces
V and W that form a bimorphism as (V,W) varies and such that

    μ_V,0(x,0) =  x      and    μ_0,W(0,y) =  y .

Put another way, the data of an orthogonal monoid space in external form
is that of a lax monoidal functor.
The commutativity condition can be expressed in terms of the external multiplication
as the commutativity of the diagrams
 
    @C=15mm  R(V) × R(W) [r]^-μ_V,W[d]_twist   
    R(V⊕ W) [d]^R(τ_V,W)
    
    R(W) × R(V) [r]_-μ_W,V    R(W⊕ V)

where τ_V,W:V⊕ W→ W⊕ V interchanges the summands.
So commutative orthogonal monoid spaces
in external form are lax symmetric monoidal functors.
We will later refer to commutative orthogonal monoid spaces as
ultra-commutative monoids.subjectultra-commutative monoid

Every -cofibration is in particular a flat cofibration,
so the monoid axiom in the -global model structure holds.
If the global family  is closed under products,
Theorem 4.1 of <cit.> applies
to the  -global model structure 
of Theorem <ref> and shows:


 
Let R be an orthogonal monoid space and  a multiplicative 
  global family.
  
  
  * The category of R-modules admits the -global model structuresubjectF-global model structure@-global model structure!for R-modules  
    in which a morphism is an equivalence (respectively fibration)
    if the underlying morphism of orthogonal spaces is an -equivalence
    (respectively -global fibration).
    This model structure is cofibrantly generated.
    Every cofibration in this -global model structure 
    is an h-cofibration of underlying orthogonal spaces. 
    If the underlying orthogonal space of R is -cofibrant,
    then every cofibration of R-modules is an -cofibration of underlying
    orthogonal spaces.
  
  * If R is commutative, then with respect to ⊠_R 
    the -global model structure of R-modules is a monoidal model category 
    that satisfies the monoid axiom.
  
  * If R is commutative, then the category of R-algebras 
    admits the -global model structuresubjectF-global model structure@-global model structure!for R-algebras  
    in which a morphism is an equivalence (respectively fibration)
    if the underlying morphism of orthogonal spaces is an -equivalence
    (respectively -global fibration).
    Every cofibrant R-algebra is also cofibrant as an R-module.
  


Almost of the statements are in Theorem 4.1 of <cit.>.
The only additional claims that require justification are the
two statements in part (i) that concern the behavior of
the forgetful functor on the cofibrations in the -global model structure.

Since the forgetful functor from R-modules to orthogonal spaces preserves
all colimits and the classes of h-cofibrations and of -cofibrations
of orthogonal spaces are both closed under coproducts, cobase change, 
sequential colimits and retracts, it suffices to show each claim
for the generating cofibrations in the -global model structure
on the category of R-modules. These are of the form 

    ( R⊠_H,^m) × i_k

for some k,m≥ 0 and H a closed subgroup of O(m) that belongs to the
global family ; as usual i_k:∂ D^k→ D^k is the inclusion.
Since i_k is an h-cofibration of spaces, 
the morphisms (R⊠_H,^m)× i_k
are h-cofibrations of orthogonal spaces.
This concludes the proof that every cofibration of R-modules
is an h-cofibration of underlying orthogonal spaces.

Now we suppose that the underlying orthogonal space of R is -cofibrant.
Because H belongs to , the orthogonal space _H,^m
is -cofibrant. Hence the orthogonal space R⊠_H,^m
is -cofibrant by Proposition <ref> (iii).
So (R⊠_H,^m)× i_k is an -cofibration
of orthogonal spaces.
This concludes the proof that every cofibration of R-modules
is an -cofibration of underlying orthogonal spaces.


Strictly speaking, Theorem 4.1 of <cit.>
does not apply verbatim to the -global model structure
because the hypothesis that every object is small (with respect to some regular cardinal)
is not satisfied, and our version of the monoid axiom 
in Proposition <ref>
is weaker than Theorem 3.3 of <cit.>
in that we do not close under transfinite compositions.
However, in our situation the sources of the generating cofibrations
and generating acyclic cofibrations are small with respect to sequential
compositions of flat cofibrations, and this suffices to
run the countable small object argument 
(compare also Remark 2.4 of <cit.>).

 
Let R be an orthogonal monoid space and N a right R-module
that is cofibrant in the -global model structure 
of Corollary <ref> (i). 
Then for every global family ,
the functor N⊠_R - takes -equivalences of left R-modules
to -equivalences of orthogonal spaces.  


For the course of this proof we call an R-module N homotopical
if the functor N⊠_R - takes -equivalences of left R-modules
to -equivalences of orthogonal spaces.  
Since the -global model structure on the category of right R-modules
is obtained by lifting the global model structure of orthogonal spaces
along the free and forgetful adjoint functor pair, every cofibrant right R-module
is a retract of an R-module that arises as the colimit of a sequence

    ∅ = M_0  → M_1  →… → M_k  → …

in which each M_k is obtained from M_k-1 as a pushout

    @C=15mm 
     A_k⊠ R [r]^-f_k⊠ R[d]     
     B_k ⊠ R [d]    
    
    M_k-1[r]     M_k

for some flat cofibration f_k:A_k→ B_k of orthogonal spaces.
For example, f_k can be chosen as a disjoint union of morphisms 
in the set I^ of generating flat cofibrations.
We show by induction on k that each module M_k is homotopical.
The induction starts with the empty R-module, where there is nothing to show.
Now we suppose that M_k-1 is homotopical, and we claim that then M_k
is homotopical as well.
To see this we consider an -equivalence of left R-modules
φ:X→ Y. Then M_k⊠_R φ
is obtained by passing to horizontal pushouts in the following commutative diagram
of orthogonal spaces:

    @C=15mm 
    M_k-1⊠_R X [d]_M_k-1⊠_R φ    
     A_k ⊠ X[d]^A_k⊠φ[l][r]^-f_k⊠ X    B_k ⊠ X[d]^B_k⊠φ
    
    M_k-1⊠_R Y      A_k ⊠ Y[l][r]_-f_k⊠ Y    B_k ⊠ Y

Here we exploit that (A_k⊠ R)⊠_R X
is naturally isomorphic to A_k⊠ X.
In the diagram, the left vertical morphism is an -equivalence by hypothesis. 
The middle and right vertical morphisms are 
-equivalences because box product is homotopical for -equivalences
(Proposition <ref> (xiv)).
Moreover, since the morphism f_k is a flat cofibration, it is an h-cofibration
(by Corollary <ref> (iii)),
and so the morphisms f_k⊠ X and f_k⊠ Y are h-cofibrations.
Proposition <ref> (x)
then shows that the induced morphism on horizontal
pushouts M_k⊠_R φ is again an -equivalence.

Now we let M be a colimit of the sequence (<ref>).
Then M⊠_R X is a colimit of the sequence M_k ⊠_R X.
Moreover, since f_k:A_k→ B_k is an h-cofibration, so is the 
morphism f_k⊠ R, and hence also its cobase change M_k-1→ M_k.
So the sequence whose colimit is M⊠_R X consists of h-cofibrations,
which are objectwise closed embeddings 
by Proposition <ref>.
The same is true for M⊠_R Y.
Since each M_k is homotopical and colimits of orthogonal spaces
along closed embeddings are homotopical 
(by Proposition <ref> (viii)), 
we conclude that the morphism M⊠_Rφ:M⊠_R X→ M⊠_R Y is an -equivalence, so that M is homotopical.
Since -equivalences are closed under retracts, 
the class of homotopical R-modules is closed under retracts, and so every
cofibrant right R-module is homotopical.





§ EQUIVARIANT HOMOTOPY SETS



In this section we define the equivariant homotopy sets π_0^G(Y)
of orthogonal spaces and relate them by restriction maps defined from 
continuous homomorphisms between compact Lie groups.
As the Lie groups vary, the resulting structure is a `Rep-functor' _0(Y), 
i.e., a contravariant functor
from the category of compact Lie groups and conjugacy classes of 
continuous homomorphisms.
The Rep-functor _0(B_G) associated to a global classifying space
is the Rep-functor represented by G, 
by Proposition <ref>. 
We identify the category of all natural operations 
with the category Rep of conjugacy classes of continuous homomorphisms,
compare Corollary <ref>.
Construction <ref>
introduces a pairing of equivariant homotopy sets
×:π_0^G(X)×π_0^G(Y)→π_0^G(X⊠ Y)
for any pair of orthogonal spaces, and
Proposition <ref> summarizes its main properties.




We recall that _G is a chosen complete G-universe and s(_G)
denotes the poset, under inclusion,
of finite-dimensional G-subrepresentations of _G.


  Let Y be an orthogonal space, G be a compact Lie group and A a G-space.
  We define 
  
    [A,Y]^G  =  _V∈ s(_G)  [A, Y(V)]^G  ,

  the colimit over the poset s(_G) of the sets of G-homotopy 
  classes of G-maps from A to Y(V).



The canonical G-maps Y(V)→ Y(_G)  
induce maps from [A, Y(V)]^G to [A, Y(_G)]^G and hence a canonical map

    [A,Y]^G → [A, Y(_G)]^G .

In general there is no reason for this map to be injective or surjective.
If Y is closed and A is compact, the situation improves:

 
Let G be a compact Lie group.


  * For every closed orthogonal space Y and every compact G-space A
the canonical map (<ref>) is bijective.

  * Let  be a global family and f:X→ Y an -equivalence of orthogonal spaces.
Then for every finite G-CW-complex A 
all of whose isotropy groups belong to , the induced map

    [A,f]^G :  [A,X]^G → [A,Y]^G

is bijective.

  * For every pair of orthogonal spaces X and Y and
every G-space A, the canonical map

    ([A,p_X]^G,[A,p_Y]^G) :  [A,X× Y]^G  → [A,X]^G ×[A, Y]^G

is bijective, where p_X and p_Y are the projections.



(i) Since the poset s(_G) contains a cofinal subsequence, 
Y(_G) is a sequential colimit of values of Y along closed embeddings. 
By Proposition <ref> (i),
every continuous G-map A→ Y(_G) thus factors through Y(V)
for some finite-dimensional V∈ s(_G),
which shows surjectivity. Injectivity follows by the same argument
applied to the compact G-space A×[0,1].

(ii) We let β:A→ Y(V) be a continuous G-map, for some V∈ s(_G),
that represents an element of [A,Y]^G. 
Together with the unique map from the empty space
this specifies an equivariant lifting problem  on the left:

    @C=15mm∅[r] [d]     X(V) [d]^f(V)   ∅[r][d]     X(V) [r]^-X(φ)     X(W) [d]^f(W)
    
    A[r]_-β    Y(V)     
    A[r]_-β@–>[urr]^-(.4)λ    Y(V) [r]_-Y(φ)    Y(W)

Since (A,∅) is a finite G-CW-pair with isotropy in 
and f an -equivalence, Proposition <ref> (ii) 
provides a G-equivariant linear isometric embedding φ:V→ W 
and a continuous G-map λ on the right hand side such that f(W)∘λ 
is G-homotopic to Y(φ)∘β.
We choose a G-equivariant linear isometric embedding j:W→_G
extending the inclusion of V. Then the class in [A,X]^G
represented by the G-map

    X(j)∘λ :  A → X(j(W))

is taken to [β] by the map [A,f]^G.
This shows that [A,f]^G is surjective.

For injectivity we consider two G-maps g,g':A→ X(V), for some V∈ s(_G), 
such that [A,f]^G[g]=[A,f]^G[g'].
By enlarging V, if necessary, we can assume that the two composites
f(V)∘ g and f(V)∘ g' are G-homotopic.
A choice of such a homotopy specifies an equivariant lifting problem on the left:

    @C=12mm
    A×{0,1}[r]^-g,g'[d]     X(V) [d]^f(V)   
    A×{0,1}[r]^-g,g'[d]     X(V) [r]^-X(φ)    
    X(W) [d]^f(W)
    
    A×[0,1][r]_-β    Y(V)     
    A×[0,1][r]_-β@–>[urr]^-(.4)λ    
    Y(V) [r]_-Y(φ)    Y(W)

Proposition <ref> (ii) 
provides a G-equivariant linear isometric embedding φ:V→ W 
and a lift λ on the right hand side such that λ(-,0)=g, λ(-,1)=g'
and f(W)∘λ is G-homotopic, relative A×{0,1}, 
to Y(φ)∘β.
As in the first part, we use a G-equivariant 
linear isometric embedding j:W→_G,
extending the inclusion of V, to transform λ into the G-homotopy

    X(j)∘λ :  A×[0,1]  → X(j(W))

that connects the images of g and g' in X(j(W)).
This shows that [g]=[g'] in [A,X]^G, so [A,f]^G is also injective.

(iii) Products of orthogonal spaces are formed objectwise, so the canonical map

    [A, (X× Y)(V)]^G  → [A,X(V)]^G ×[A, Y(Y)]^G

is bijective for every G-representation V.
Filtered colimits commute with finite products, so the claim follows
by passage to colimits over the poset s(_G).




subjectglobal classifying space
We specialize to the case where Y=B_G is the global classifying space
of a compact Lie group G. Proposition <ref> 
above already gave an explanation for the name `global classifying space'
by exhibiting (B_ G)(_K) as a classifying space for 
principal (K,G)-bundles over paracompact K-spaces.
We now reinterpret this result as follows.

We choose a faithful G-representation V, so that B_G=_G,V.
We let A be a compact K-space and consider the composite

    [A, B_G]^K     
    [A, (B_G)(_K) ]^K     _(K,G)(A) ,

where the first map is the bijection of Proposition <ref> (i),
exploiting that the orthogonal space _G,V is closed.
The second map is the bijection provided 
by Proposition <ref>. 
The composite bijection

    [A,  B_ G]^K   _(K,G)(A)

sends the class represented by a continuous K-equivariant map f:A→(V,W)/G
to the class of the pullback principal (K,G)-bundle f^*q over A,
where q:(V,W)→(V,W)/G is the projection.




Now we specialize the equivariant homotopy sets [A,Y]^G to
the case A={∗} of a one-point G-space, and then give it
a new name.


Let G be a compact Lie group.
The G-equivariant homotopy set of an orthogonal 
space Ysubjectequivariant homotopy set!of an orthogonal space 
is the set

    π_0^G(Y)   =  _V∈ s(_G) π_0( Y(V)^G)   .
  



Specializing Proposition <ref> 
to a one-point G-space yields:


 
Let G be a compact Lie group.


  * For every closed orthogonal space Y the canonical map

    π_0^G(Y) → π_0 ( Y(_G)^G )

is bijective.

  * Let  be a global family and f:X→ Y an -equivalence of orthogonal spaces.
Then for every compact Lie group G in  the induced map

    π_0^G(f) :  π_0^G(X) → π_0^G(Y)

of equivariant homotopy sets is bijective.





As the group varies, the homotopy sets π_0^G(Y) 
have contravariant functoriality in G:
every continuous group homomorphism α:K→ G
between compact Lie groups 
induces a restriction map α^*:π_0^G(Y)→π_0^K(Y),
as we shall now explain.
We denote by α^* the
restriction functor from G-spaces to K-spaces 
(or from G-representations to K-representations)
along α, i.e., α^* Z (respectively α^* V)
is the same topological space as Z 
(respectively the same inner product space as V) endowed with
K-action via

    k· z  =  α(k)· z  .

Given an orthogonal space Y,
we note that for every G-representation V, the K-spaces 
α^*(Y(V)) and Y(α^*V) are equal (not just isomorphic).

The restriction α^*(_G) is a K-universe, 
but if α has a non-trivial kernel, then this K-universe is 
not complete. When α is injective,
then α^*(_G) is a complete K-universe
(by Remark <ref>),  but typically
different from the chosen complete K-universe _K. 
To deal with this we explain how a G-fixed point y∈ Y(U)^G,
for an arbitrary G-representation U,
gives rise to an unambiguously defined element y in π_0^G(Y).
The point here is that U need not be a subrepresentation
of the chosen universe _G
and the resulting class does not depend on any additional choices.

To construct y we choose a G-equivariant linear isometry j:U→ V
onto a G-subrepresentation V of _G.
Then Y(j)(y) is a G-fixed point of Y(V), so we obtain an element 

    y =  [Y(j)(y)]  ∈ π_0^G (Y) .

It is crucial, but not completely obvious, 
that f does not depend on the choice of isometry j.



Let Y be an orthogonal space, G a compact Lie group,
U a G-representation and y∈ Y(U)^G a G-fixed point.


  * The class y in π_0^G(Y) is independent of the choice 
of linear isometry from U to a subrepresentation of _G.

  * For every G-equivariant linear isometric embedding 
φ:U→ W the relation

    Y(φ)(y) =  y     holds in  π_0^G(Y).




(i) We let j:U→ V and j':U→ V' be two G-equivariant linear
isometries, with V,V'∈ s(_G).
We choose a third G-equivariant linear isometry j”:U→ V”
such that V”∈ s(_G) and V” is orthogonal to both V and V'.
We let W be the span of V, V' and V” inside _G. 
We can then view j, j' and j” as equivariant linear isometric embeddings
from U to W. 

Since the images of j and j” are orthogonal,
they are homotopic through G-equivariant linear isometric embeddings into W, 
via the homotopy H:U× [0,1]→ W given by

    H(u,t) = √(1-t^2) · j(u)   +  t· j”(u)  .

By the same argument, j' and j” 
are homotopic through G-equivariant linear isometric embeddings.
In particular, j and j' are homotopic to each other. 
If H(-,t):U→ W  is a continuous 1-parameter family of 
G-equivariant linear isometric embeddings from j to j', then 

    t ⟼ Y(H(-,t))(y)

is a path in Y(W)^G from Y(j)(y) to Y(j')(y), so
[Y(j)(y)]=[Y(j')(y)] in π_0^G(Y).

(ii) If j:W→ V is an equivariant linear isometry with V∈ s(_G),
we define V̅=j(φ(U)) and we let k:U→V̅ be the equivariant
linear isometry that is defined by k(u)=j(φ(u))
(i.e., k is essentially j∘φ, but with range V̅ instead of V).
Then

    Y(φ)(y) =  [Y(j)(Y(φ)(y))]  =  [Y(jφ)(y)] 
     =  [Y(k)(y)]  =  y .






We can now define the restriction mapsubjectrestriction map!of equivariant homotopy sets associated to
a continuous group homomorphism α:K→ G by

    α^*  :  π^G_0(Y)  → π^K_0(Y)  ,  
    [y] ⟼ y .

This makes sense because every G-fixed point of Y(V) is also
a K-fixed point of α^*(Y(V))=Y(α^*V).
For a second continuous group homomorphism β:L→ K we have

    β^*∘α^*  =  (αβ)^*  :  π_0^G (Y)  → π_0^L(Y)  .

Clearly, restriction along the identity homomorphism is the identity,
so we have made the collection of equivariant homotopy sets π_0^G(Y)
into a contravariant functor in the group variable.


An important special case of the restriction homomorphisms
are conjugation maps.
Here we consider a closed subgroup H of G, an element g∈ G and denote by 

    c_g  :  H  →  H^g ,    c_g(h) = g^-1h g
 
the conjugation homomorphism, where 
H^g={g^-1 h g |  h∈ H} is the conjugate subgroup.
As any group homomorphism, 
c_g induces a restriction mapsymbolc_g - conjugation by g

    g_⋆ =  (c_g)^*  :  π^H^g_0 (Y)  → π^H_0(Y)

of equivariant homotopy sets.
For g,g̅∈ G we have c_gg̅=c_g̅∘ c_g:H→ H^gg̅
and thus 

    (gg̅)_⋆ =   (c_gg̅)^* =  (c_g̅∘ c_g)^* = (c_g)^*∘ (c_g̅)^* =  g_⋆∘g̅_⋆ .

A key fact is that inner automorphisms act trivially, 
i.e., the restriction map g_⋆ is the identity on π_0^G (Y).
So the action, by the restriction maps, 
of the automorphism group of G on π_0^G(Y) 
factors through the outer automorphism group.


For every orthogonal space Y, every compact Lie group G, 
and every g∈ G, the conjugation mapsubjectconjugation map!on equivariant homotopy sets
g_⋆:π_0^G(Y) →π_0^G(Y) is the identity.

 
We consider a finite-dimensional G-subrepresentation V 
of _G and a G-fixed point y∈ Y(V)^G that represents an element in π_0^G(Y). 
Then the map l_g:c_g^*(V)→ V given by left multiplication by g
is a G-equivariant linear isometry.
So 

    g_⋆[y] =  (c_g)^*[y] =  [Y(l_g)(y)] =  [g· y] =  [y]  ,

by the very definition of the restriction map. The third equation is the definition
of the G-action on Y(V) through the G-action on V.
The fourth equation is the hypothesis that y is G-fixed.



We note that because inner automorphisms act as the identity,
the restriction map α^* only depends on the homotopy class of α.
More precisely, suppose that α,α':K→ G 
are homotopic through continuous group homomorphisms. 
Then α and α' belong to the same path component of the space (K,G)
of continuous homomorphisms, and so they are conjugate by an element of G,
compare Proposition <ref>.



We denote by Repsymbol   - category of compact Lie groups and conjugacy classes of homomorphisms 
the category whose objects are the compact Lie groups
and whose morphisms are conjugacy classes of continuous group homomorphisms.
We can summarize the discussion thus far by saying that for every
orthogonal space Y the restriction maps
make the equivariant homotopy sets
{π_0^G(Y)} into a functor

    _0(Y)  :  ^ →(sets) .

We will refer to such a functor as a Rep-functor.subjectRep-functor



Our next aim is to show that the homotopy -functor
_0(B_G) of a global classifying space is represented by G.
For every G-representation V 
we define the tautological classsubjecttautological class!unstable|seeunstable tautological classsubjectunstable tautological classsymbolu_G,V - tautological class in π_0^G(_G,V)subjectglobal classifying space

    u_G,V ∈ π_0^G(_G,V)

as the path component of the G-fixed point

    _V· G  ∈ ((V,V)/G)^G = (_G,V(V))^G ,

the G-orbit of the identity of V.




 
Let G and K be compact Lie groups and V a faithful G-representation.


  * The K-fixed point space  (_G,V(_K))^K is a 
disjoint union, indexed by conjugacy classes
of continuous group homomorphisms α:K→ G,
of classifying spaces of the centralizer  of the image of α.

  * The map

    (K,G)  → π_0^K(_G,V) ,   
    [α:K→ G]  ⟼ α^*(u_G,V)

is bijective. 



Part (i) works for any universal (K× G)-space E
for the family (K;G) of graph subgroups, for example for E=(V,_K). 
The argument can be found in Theorem 2.17 of <cit.>
or Proposition 5 of <cit.>.subjectuniversal space!for the family of graph subgroups
We repeat the proof for the convenience of the reader.
For a continuous homomorphism α:K→ G, we let C(α)
denote the centralizer, in G, of the image of α, and we set

    E^α =  {x∈ E | (k,α(k))· x = x for all k∈ K} ,

the space of fixed points of the graph of α.
Since the G-action on the universal space E is free,
Proposition <ref> provides
a homeomorphism

    ∐α^♭ :  ∐_α   E^α/C(α) →  (E/G)^K  ,

where the disjoint union is indexed by conjugacy classes of continuous homomorphisms.
The graph of α belongs to the family (K;G),
so E^α is a contractible space. 
The action of C(α) on E^α is a restriction of the G-action on E,
hence free. Since E is (K× G)-cofibrant, 
the fixed point space E^α is cofibrant for the action
of the normalizer (inside K× G) of the graph of α,
by Proposition <ref>.
Hence E^α is also cofibrant as a C(α)-space, by 
Proposition <ref> (i). 
So for every homomorphism α the space 
E^α/C(α) is a classifying space for the group C(α).
This shows part (i).

(ii) Since the classifying space of a topological group is connected,
part (i) identifies the path components of (_G,V(_K))^K
with the conjugacy classes of continuous homomorphisms α:K→ G.
The bijection sends the class of α to α^* (u_G,V).
The claim then follows by applying Corollary <ref> (i).





Now we show that the restriction maps along continuous group
homomorphisms give all natural operations between equivariant homotopy sets
of orthogonal spaces. 
We will want to do similar calculations several other times
in this book, so we state the argument 
in the more general situation of a category  related to 
the category of orthogonal spaces by an adjoint functor pair.
Our present context is the degenerate case = and the identity functors.
Later we will also consider the cases of the categories
of ultra-commutative monoids, of orthogonal spectra and 
of ultra-commutative ring spectra.

We recall that the restriction morphism

    ρ_G,V,W =  ρ_V,W/G  :  _G,V⊕ W =  (V⊕ W,- )/G → (W,-)/G  =  _G,W

restricts the orbit of a linear isometric embedding from V⊕ W
to the second summand W.
This morphism is a global equivalence of orthogonal spaces by 
Proposition <ref> (ii),
as long as G acts faithfully on W.


 
Let  be a category and

    Λ :   @<.4ex>[r]      :  U @<.4ex>[l]

an adjoint functor pair such that the composite functor
UΛ:→ is continuous.
Suppose in addition that
for every compact Lie group G, all G-representations V 
and all non-zero faithful G-representations W, the morphism of -functors

    _0( UΛ(ρ_G,V,W))  :  _0( UΛ(_G,V⊕ W))  → _0( UΛ(_G,W))

is an isomorphism.
Let W be a non-zero faithful G-representation, and set

    u_G,W^ =  η_*(u_G,W) ∈ π_0^G( UΛ(_G,W)) ,

where η:_G,W→ UΛ(_G,W) is the adjunction unit.
Then for every compact Lie group K,
evaluation at the class u^_G,W is a bijection

    _→(sets)(π_0^G∘ U, π_0^K∘ U) 
     → π_0^K( UΛ( _G,W ) )  ,   τ ⟼ τ(u^_G,W)

between the set of natural transformations 
from the functor π_0^G∘ U to π_0^K∘ U, 
and the set π_0^K( UΛ(_G,W)).


To show that the evaluation map is injective 
we show that every natural transformation τ:π_0^G∘ U→π_0^K∘ U 
is determined by the element τ(u^_G,W).
We let X be any object of  and [x]∈π_0^G(U X)
a G-equivariant homotopy class, represented by a G-fixed point

    x  ∈ (U X)(V)^G  ,

for some G-representation V. We can stabilize with the representation W
and obtain another representative

    (U X)(i)(x)  ∈ (U X)(V⊕ W)^G  ,

where i:V→ V⊕ W is the embedding of the first summand. 
This G-fixed point is adjoint to a morphism of orthogonal spaces

    x̂ :  _G,V⊕ W → U X

and hence adjoint to a -morphism

    x^♭ :  Λ(_G,V⊕ W)  → X

that satisfies

    π_0^G(U x^♭)(u^_G,V⊕ W)  =  [x]   in  π_0^G(U X)  .

The restriction morphism of orthogonal spaces 
ρ_G,V,W :_G,V⊕ W→_G,W sends u_G,V⊕ W to u_G,W.
The morphism of orthogonal spaces

    UΛ(ρ_G,V,W) :  UΛ( _G,V⊕ W) → UΛ( _G,W )

then sends u^_G,V⊕ W to u^_G,W.
The diagram

    @C=25mmπ_0^G(UΛ( _G,W))[d]_τ   π_0^G(UΛ(_G,V⊕ W))[d]_τ[l]_-π_0^G(UΛ(ρ_G,V,W))^-[r]^-π_0^G(U x^♭)   π_0^G(U X)[d]^τ
    π_0^K(UΛ(_G,W))    π_0^K(UΛ( _G,V⊕ W)) [l]^-π_0^K(UΛ(ρ_G,V,W))_-[r]_-π_0^K(U x^♭)   π_0^K(U X)

commutes and the two left horizontal maps are bijective by hypothesis.
Since 

    [x]  =  π_0^G(U x^♭)(π_0^G(UΛ(ρ_G,V,W))^-1(u^_G,W))  ,

naturality yields that

    τ[x]     =  τ(π_0^G(U x^♭)(π_0^G(UΛ(ρ_G,V,W))^-1(u^_G,W)))
       =  π_0^K(U x^♭)(π_0^K(UΛ(ρ_G,V,W))^-1(τ(u^_G,W))) .

This shows that the transformation τ is determined by the value
τ(u^_G,W).

It remains to construct, for every element 
z∈π_0^K(UΛ(_G,W)), a natural transformation
τ:π_0^G∘ U→π_0^K∘ U with τ(u^_G,W)=z.
The previous paragraph dictates what to do: 
we represent a given class in π_0^G(U X) by
a G-fixed point x∈(U X)(V⊕ W)^G and set

    τ[x]  =  π_0^K(U x^♭)(π_0^K(UΛ(ρ_G,V,W))^-1(z))  .

We must verify that the element τ[x] 
is independent of the representative.
If y∈(U X)(V⊕ W)^G is in the same path component as x,
then any path adjoins to a homotopy of morphisms of orthogonal spaces

    H  :  _G,V⊕ W× [0,1] → U X

from x̂ to ŷ.
Since the functor UΛ is continuous, the composite morphisms

    UΛ(_G,V⊕ W) UΛ(H(-,t)) UΛ(U X) U(ϵ_X) U X

form a continuous 1-parameter family of morphisms for t∈[0,1], 
where ϵ_X:Λ(U X)→ X is the counit of the adjunction.
This witnesses that the morphism U x^♭ is homotopic to the morphism U y^♭.
So π_0^K(U x^♭)=π_0^K(U y^♭), and the class τ[x]
is independent of the representative in the given path component of (U X)(V⊕ W)^G.

Now we let V' be another G-representation and φ:V→ V' 
a G-equivariant linear isometric embedding. 
Then

    y  =  (U X)(φ⊕ W)(x)     in   (U X)(V'⊕ W)^G

is another representative of the class [x].
The restriction morphism 

    φ^♯ =  (φ⊕ W,-)/G :  _G,V'⊕ W → _G,V⊕ W

makes the following diagram of orthogonal spaces commute:

    @C=15mm_G,W   _G,V'⊕ W[d]_φ^♯[r]^-ŷ[l]_-ρ_G,V',W    U X
       _G,V⊕ W[ur]_x̂[ul]^ρ_G,V,W

Passing to adjoints and applying U yields another commutative diagram:

    @C=15mm 
    UΛ(_G,W)     
    UΛ(_G,V'⊕ W) [d]_UΛ(φ^♯)[r]^-U y^♭[l]_-UΛ(ρ_G,V',W)    U X
       UΛ(_G,V⊕ W)[ur]_U x^♭[ul]^UΛ(ρ_G,V,W)

So

    π_0^K(U x^♭)∘π_0^K(UΛ(ρ_G,V,W))^-1    = π_0^K(U x^♭)∘π_0^K(UΛ(φ^♯))∘π_0^K(UΛ(ρ_G,V',W))^-1
       =  π_0^K(U y^♭)∘π_0^K(UΛ(ρ_G,V',W))^-1 ,

and hence the class τ[x] remains unchanged upon stabilization of x
along φ.
Altogether this shows that τ[x] is well-defined.

Naturality of τ is straightforward: if ψ:X→ Y is a -morphism
and x∈(U X)(V⊕ W)^G is as above,
then (U ψ)(V⊕ W)(x)∈ (U Y)(V⊕ W)^G 
represents the class π_0^G(Uψ)[x].
Moreover, the adjoint of (Uψ)(V⊕ W)(x) coincides with the composite

    Λ(_G,V⊕ W)   x^♭  X   ψ  Y .

So naturality follows:

    τ(π_0^G(Uψ)[x])    =  π_0^K(Uψ∘ U x^♭)(π_0^K(UΛ(ρ_G,V,W))^-1(z)) 
       =  π_0^K(Uψ)( π_0^K(U x^♭)(π_0^K(UΛ(ρ_G,V,W))^-1(z))) =  π_0^K(Uψ)(τ[x])  .

Finally, the class u^_G,W 
is represented by the G-fixed point 

    η__G,W(_W· G) ∈ (UΛ(_G,W))(W)^G ,

which is adjoint to the identity of Λ(_G,W). Hence
τ(u_G,W^)= π_0^K()(z) =z.




Every natural transformation π_0^G →π_0^K
of set valued functors on the category of orthogonal spaces is of the form
α^*
for a unique conjugacy class of continuous group homomorphism α:K→ G.  


We let W be any non-zero faithful G-representation.
The composite

    (K,G)  [α]↦α^* (π_0^G,π_0^K) 
        π_0^K(_G,W)

is bijective by Proposition <ref> (ii), 
where the second map is evaluation at the tautological class u_G,W.
The evaluation map is bijective by 
Proposition <ref>, applied to =
and the identity adjoint functor pair.
So the first map is bijective as well.






Given two orthogonal spaces X and Y, we endow the equivariant homotopy sets
with a pairingsubjectproduct!on equivariant homotopy sets

    × : π_0^G(X)  × π_0^G(Y)  → π_0^G(X⊠ Y)  ,

where G is any compact Lie group.
We suppose that V and W are G-represen­tations
and x∈ X(V)^G and y∈ Y(W)^G are fixed points that
represent classes in π^G_0(X) respectively π^G_0(Y).
We denote by x× y the image of the G-fixed point (x,y)
under the G-map

    i_V,W :    X(V)× Y(W) → (X⊠ Y)(V⊕ W)

that is part of the universal bimorphism.
If φ:V→ V' and ψ:W→ W' are equivariant linear isometric embeddings,
then 

    X(φ)(x)× Y(ψ)(y)     =   i_V',W'(X(φ)(x),Y(ψ)(y))
       =   (X⊠ Y)(φ⊕ψ)(i_V,W(x,y)) 
     =   (X⊠ Y)(φ⊕ψ)(x× y)  .

So by Proposition <ref> (ii)
the classes x× y and X(φ)(x)× Y(ψ)(y)
coincide in π_0^G(X⊠ Y). The upshot is that the assignment

    [x]× [y] =  x× y ∈ π_0^G(X⊠ Y)

is well-defined.


The pairings of equivariant homotopy sets have several expected properties
that we summarize in the next proposition.

 
  Let G be a compact Lie group and X, Y and Z orthogonal spaces.
  
  
  * (Unitality)
    Let 1∈π_0^G(1) be the unique element.
    Then 1× x=x=x× 1 for all x∈π_0^G(X).
  
  * (Associativity) For all classes x∈π_0^G ( X ), y∈π_0^G( Y )
    and z∈π_0^G( Z ) the relation
    
    a_*( (x× y)× z)  =  x× (y× z)

    holds in π_0^G(X⊠ (Y⊠ Z)),
    where a:(X⊠ Y)⊠ Z X⊠ (Y⊠ Z) 
    is the associativity isomorphism.
  
  * (Commutativity) For all classes x∈π_0^G ( X ) and y∈π_0^G( Y )
    the relation
    
    τ_*( x× y )  =   y× x

    holds in π_0^G( Y⊠ X),
    where τ:X⊠ Y→ Y⊠ X is the symmetry isomorphism.
  
  * (Restriction) For all classes x∈π_0^G ( X ) and y∈π_0^G( Y )
    and all continuous homomorphisms α:K→ G the relation
    
    α^*(x)×α^*(y) =  α^*(x× y)

    holds in π_0^K(X⊠ Y).
      


The unitality property (i), the associativity property (ii) 
and compatibility with restriction (iv)
are straightforward from the definitions.
Part (iii) exploits that the square

    @C=18mm 
    X(V)× Y(W) [r]^-i_V,W[d]_twist   
    (X⊠ Y)(V⊕ W)[d]^τ(χ_V,W)
    
    Y(W)× X(V) [r]_i_W,V   
    (Y⊠ X)(W⊕ V)

commutes. The image of (x,y) under the upper right composite represents
τ_*(x× y), whereas the image of (y,x) 
under the lower left composite represents y× x, so
τ_*( x× y ) =  y× x.






[External versus internal products]
If G and K are two compact Lie groups, we can define an `external' form
of the pairing (<ref>) as the composite

    π_0^G(X)×π_0^K(Y)   p_1^*× p_2^* π_0^G× K(X)×π_0^G× K(Y)  × π_0^G× K(X⊠ Y) ,

where p_1:G× K→ G and p_2:G× K→ K are the two projections.
These external pairings also satisfy various naturality, unitality, associativity and
commutativity properties that we do not spell out.
On the other hand, the internal pairing (<ref>)
can be recovered from the external products (<ref>)
by taking G=K and restricting along the diagonal
embedding Δ_G:G→ G× G.
Indeed, the p_1∘Δ_G=p_2∘Δ_G=_G, and hence

    Δ_G^* (p_1^*(x)× p_2^*(y)) =  Δ_G^*( p_1^*(x))×Δ_G^*(p_2^*(y))  =  x× y .




Theorem <ref> (i) and the fact that
the functor π_0^G commutes with finite products 
(Proposition <ref> (iii) for A=∗)
imply:


For every compact Lie group G and all orthogonal spaces X and Y the three maps

    π_0^G(X)×π_0^G(Y)  ×  π_0^G( X⊠ Y)  (ρ_X,Y)_* π_0^G( X× Y)  ((p_X)_*,(p_Y)_*) π_0^G(X)×π_0^G(Y)

are bijective, where p_X:X× Y→ X and p_Y:X× Y→ Y
are the projections. Moreover, the composite is the identity.


[Infinite box products]
We end this section with a generalization of the previous corollary to
`infinite box products' of based orthogonal spaces, but we first have
to clarify what we mean by that. We let I be an indexing set and {X_i}_i∈ I
a family of based orthogonal spaces, i.e., each equipped with a 
distinguished point x_i∈ X_i(0).
If K⊂ J are two nested, finite subsets of I, then
the basepoints of X_k for k∈ J-K provide a morphism

    ⊠_k∈ K X_k  → ⊠_j∈ J X_j  .

In terms of the universal property of the box product, this morphism arises
from the maps

    ∏_k∈ K  X_k(V_k)  → ∏_k∈ K  X_k(V_k)    ×∏_j∈ J-K  X_j(0) → (⊠_j∈ J  X_j)(⊕_k∈ K  V_k)  ,

where the second map is part of the universal multi-morphism.
We can thus define the infinite box productsubjectbox product!of orthogonal spaces!infinite
as the colimit of the finite box products over the filtered poset of
finite subsets of I:

    ⊠'_i∈ I X_i =  _J⊂ I, |J|<∞ ⊠_j∈ J X_j  .

If I happens to be finite, then this recovers the iterated box product.


The distinguished basepoint of X_i represents a distinguished basepoint
in the equivariant homotopy set π_0^G(X_i) for every compact Lie group G.
In fact, these points all arise from the basepoint in π_0^e(X_i)
by restriction along the unique homomorphism G→ e.
The weak product ∏'_i∈ Iπ_0^G(X_i)
is the subset of the product consisting of all tuples (y_i)_i∈ I
with the property that almost all y_i are the distinguished basepoint.
Equivalently, the weak product is the filtered colimit,
over the poset of finite subsets of I, of the finite products.

If we iterate the pairing (<ref>), it provides a multi-pairing

    ∏_j∈ J π_0^G(X_j) → π_0^G(⊠_j∈ J  X_j)

for every finite set J. Passing to colimits over finite subsets
of I on both sides yields a map

    ∏'_i∈ I π_0^G(X_i) → π_0^G(⊠'_i∈ I X_i) .




Let I be a set and {X_i}_i∈ I a family of based orthogonal spaces.
Then for every compact Lie group G the map (<ref>)
is bijective.  


For every  k∈ I we define a `projection'

    Π_k  :  ⊠'_i∈ I X_i  → X_k

as follows. Since the infinite box product is defined as a colimit,
we must specify the `restriction' of Π_k to ⊠_j∈ JX_j
for every finite subset J of I, compatibly as J increases.
For k∉J we define this restriction as the constant morphism
factoring through the basepoint of X_k.
For k∈ J we define the restriction

    ⊠_j∈ J X_j  → X_k

as the morphism corresponding, under the universal property of the box product,
to the multi-morphism with components

    ∏_j∈ J   X_j(V_j)  proj_k  X_k(V_k) X_k(incl) X_k(⊕_j∈ J  V_j)   .

Then the composite

    ∏'_i∈ I π_0^G(X_i) (<ref>) π_0^G(⊠'_i∈ I X_i) π_0^G(Π_k) π_0^G(X_k)

is the projection onto the k-th factor.
So if two tuples in the weak product have the same image
under the map (<ref>), they coincide.
This shows injectivity.

Now we show surjectivity. Every element of π_0^G(⊠'_i∈ I X_i) 
is represented by a G-fixed point of (⊠'_i∈ I X_i)(V)
for some G-representation V.
Colimits of orthogonal spaces are formed objectwise, so
(⊠'_i∈ I X_i)(V) is a colimit
of the spaces (⊠_j∈ J X_j)(V),
formed over the filtered poset of finite subsets J of I.
For every nested pair of finite subsets K⊂ J
of I the morphism (<ref>) has a retraction, by `projection'.
So at every inner product space V, the map

    (⊠_k∈ K X_k)(V)  → (⊠_j∈ J X_j)(V)

is a closed embedding by Proposition <ref>.
For fixed V, the colimit (⊠'_i∈ I X_i)(V)
in the category  of compactly generated spaces
can thus be calculated in the ambient category of all topological spaces,
by Proposition <ref> (ii).
In particular, every G-fixed point of (⊠'_i∈ I X_i)(V)
arises from a G-fixed point of (⊠_j∈ J X_j)(V)
for some finite subset J of I.
In other words, the canonical map

    _J⊂ I,|J|<∞π_0^G(⊠_j∈ J X_j)  → π_0^G(⊠'_i∈ I X_i)

is surjective. 
For finite sets J the map ∏_j∈ Jπ_0^G(X_j)→π_0^G(⊠_j∈ J X_j)
is bijective by Corollary <ref>,
so this shows surjectivity.







CHAPTER: ULTRA-COMMUTATIVE MONOIDS





Orthogonal monoid spaces are the lax monoidal continuous functors
from the linear isometries category  to the category of spaces.
Orthogonal monoid spaces with strictly commutative multiplication
(i.e., the lax symmetric monoi­dal functors)
play a special role, and we honor this
by special terminology, referring to them 
as ultra-commutative monoids.
This chapter is devoted to the study of ultra-commutative monoids,
including a global model structure, an algebraic study of their
homotopy operations, and many examples.

I want to motivate the adjective `ultra-commutative'.
In various contexts of homotopy theory, 
highly structured multiplications that are
only associative or commutative up to higher coherence homotopies
can in fact be rigidified to multiplications that are strictly associative
or possibly strictly commutative.
One example is the fact that E_∞-spaces can be rigidified to 
strictly commutative ℐ-space 
monoids <cit.>;
another example is the fact that E_∞-ring objects internal
to symmetric spectra can be rigidified to strictly commutative
symmetric ring spectra, see for example 
<cit.>
and the paragraph immediately following thereafter.
More to the point of our present discussion, 
in <cit.> Lind establishes
a Quillen equivalence between the non-equivariant homotopy theory of E_∞-spaces
(i.e., spaces with an action of the linear isometries operad)
and the non-equivariant homotopy theory of commutative
orthogonal monoid spaces (there called `commutative ℐ-FCPs').

Our use of the word `ultra-commutative' is intended as a reminder
that the slogan `E_∞=commutative' is no longer true in equivariant
or global contexts. 
More specifically, one can consider orthogonal monoid spaces
with an action of an E_∞-operad;
up to non-equivariant equivalence, these objects model E_∞-spaces,
and they can be replaced by equivalent strictly commutative orthogonal monoid spaces.
The analogous statement for global equivalences is false, i.e.,
E_∞-orthogonal spaces cannot in general be replaced by
globally equivalent ultra-commutative monoids.
In fact, the definition of power operations and transfer maps requires
strict commutativity, 
and Remark <ref> illustrates 
how the lack of transfers obstructs ultra-commutativity.

The study of orthogonal monoid spaces goes back to 
Boardman and Vogt <cit.>,
who introduce them as a delooping machine in a non-equivariant context. 
More precisely, they show that for every ultra-commutative monoid R
the space R(^∞) has the structure of an `E-space'
(nowadays called an E_∞-space), and if in addition π_0(R(^∞))
is a group, then  R(^∞) is an infinite loop space.
Ultra-commutative monoids also appear, 
with an extra pointset topological hypothesis and
under the name ℐ_*-prefunctor, in <cit.>;
in <cit.>, they are studied under the name
`commutative ℐ-FCPs'.

In Section <ref>
we formally define ultra-commutative monoids and establish the global model structure.
Section <ref> is devoted to
the algebraic structure on the homotopy -functor _0(R)
of an ultra-commutative monoid. 
We refer to this structure as a `global power monoid';
it consist of an abelian monoid structure
on the set π_0^G(R)  for every compact Lie group G, 
natural for restriction along continuous homomorphisms,
and additional structure that can equivalently be encoded
as power operations (see Definition <ref>)
or as transfer maps (see Construction <ref>).
In this section we also show that these operations are the entire natural structure
(see Theorem <ref>).
Section <ref> collects various examples of ultra-commutative monoids:
among these are examples made from the infinite families of classical Lie groups
(orthogonal, special orthogonal, unitary, special unitary, symplectic,
spin and spin^c); examples consisting of Grassmannians under direct sum
of subspaces (in real, oriented, complex and quaternionic flavors);
examples made from Grassmannians under tensor product
of subspaces (in a real and complex version);
and ultra-commutative multiplicative models for global classifying spaces
of abelian compact Lie groups.


Section <ref> is a case study of how
non-equivariant homotopy types can `fold up' into many different global
homotopy types. We define, discuss and compare different ultra-commutative
and E_∞-orthogonal monoid spaces whose
underlying non-equivariant homotopy type is B O, 
a classifying space for the infinite orthogonal group; 
in all examples we also identify
the associated global power monoids and fixed point spaces.
Section <ref>
discusses `units' and `group completion' of ultra-commutative monoids.
The two constructions are dual to each other, and they are
homotopically right adjoint respectively left adjoint to
the inclusion of group-like ultra-commutative monoids.
On the algebraic level of global power monoids, the topological constructions
pick out the invertible elements and perform the algebraic group completion 
respectively. A naturally occurring example of a global group
completion is the morphism from the additive Grassmannians to
the periodic global version of B O.
As an application we end the section 
with a global, highly structured version of Bott periodicity:
Theorem <ref>
shows that  is globally equivalent, as an
ultra-commutative monoid, to Ω.






§ GLOBAL MODEL STRUCTURE




In this section we formally define ultra-commutative monoids
and establish various formal properties of the category umon of
ultra-commutative monoids.
We introduce free ultra-commutative monoids in Example <ref>.  
The main result is the model structure with global equivalences as the weak equivalences,
see Theorem <ref>.





An ultra-commutative monoid is a commutative orthogonal monoid space.  
We write umon for the category of ultra-commutative monoids.symbol  umon - category of ultra-commutative monoids

subjectmonoid!ultra-commutative|seeultra-commutative monoidsubjectultra-commutative monoid|(
subjectultra-commutative monoid


As we explained after Definition <ref>,
the data of an ultra-commutative monoid is the same as that
of a lax symmetric monoidal continuous functor 
from the linear isometries category  (under orthogonal direct sum)
to the category  of spaces (under cartesian product).




One can think of an ultra-commutative monoid 
as encoding a collection of E_∞-G-spaces,
one for every compact Lie group G, compatible under restriction.
If R is a closed orthogonal space and G a compact Lie group,
then the G-equivariant homotopy type encoded in R
can be accessed as the underlying G-spacesubjectunderlying G-space!of an orthogonal space 

    R(_G)  =  _V∈ s(_G)  R(V)  .

The additional structure of an ultra-commutative monoid 
gives rise to an action of a specific E_∞-G-operad on this G-space,
namely the linear isometries operad of the complete G-universe _G.
The n-th space of this operad is the space
(_G^n,_G)subjectlinear isometries operad
of linear isometric embeddings (not necessarily equivariant)
of _G^n into _G. 
The group G acts on (_G^n,_G) by conjugation,
and the operad structure is by direct sum and composition of linear isometric embeddings.
The symmetric group Σ_n permutes the summands in the source.
The space (_G^n,_G) 
is G-equivariantly contractible by <cit.>,
and the Σ_n-action is free; in fact,
(_G^n,_G) even has the (G×Σ_n)-equivariant
homotopy type of a universal space for (G,Σ_n)-bundles.

By simultaneous passage to colimit over s(_G) in all n variables,
the iterated multiplication maps 

    R(V_1)×…× R(V_n)  → R(V_1⊕…⊕ V_n)

give rise to a map μ_(n):R(_G)^n→ R(_G^n).
A linear isometric embedding ψ:→' between 
countably infinite dimensional inner product spaces induces a map
R(ψ) :  R() → R('); the resulting `action map'

    (,')× R() → R(')  ,    
    (ψ,y) ⟼ R(ψ)(y)

is continuous.
The operadic action map is then simply the composite

    (_G^n,_G) × R(_G)^n (_G^n,_G) ×μ_(n) (_G^n,_G) × R(_G^n)   act    
     R(_G)  .







Now we work towards the main result of this section,
the global model structure for ultra-commutative monoids.
Before we start with the homotopical considerations, we get some
of the necessary category theory out of the way.
For a moment we consider more generally 
any symmetric monoidal category  with monoidal
product ⊠ and unit object I. We can then study operads
in  and algebras over a fixed operad.
The following (co-)completeness and preservation results
can be found in <cit.> 
or <cit.>.

 
Let (,⊠,I) be a complete and cocomplete symmetric monoidal category 
such that the monoidal product preserves colimits in each variable.
Let  be an operad in . 
Then the category of -algebras is complete and cocomplete.
Moreover, the forgetful functor from the category of -algebras
to the underlying category  creates all limits, 
all filtered colimits and those coequalizers that are reflexive in
the underlying category .


We let  denote the incarnation of the commutative operad
internal to the category of orthogonal spaces, under box product.
So for every n≥ 0 the orthogonal space (n)
of n-ary operations is constant with value a one-point space.
Equivalently,  is a terminal operad in orthogonal spaces.
Endowing an orthogonal space with an ultra-commutative multiplication
is the same as giving it an algebra structure over the commutative operad . 
More formally, the category of ultra-commutative monoids is
isomorphic to the category of -algebras.
So Proposition <ref> has the following special case:

  
The category of ultra-commutative monoids is complete and cocomplete.
The forgetful functor from the category of ultra-commutative mo­noids
to the category of orthogonal spaces creates all limits, 
all filtered colimits and those coequalizers that are reflexive in
the category of orthogonal spaces.


[Free ultra-commutative monoids] 
We quickly recall that ultra-commutative monoids are monadic over the
category of orthogonal spa­ces; this is not particular to our context,
and the analogous fact holds for 
commutative monoids in any cocomplete symmetric monoidal category.
For every orthogonal space Y and m≥ 0 we denote by

    ^m(Y) =   Y^⊠ m/Σ_m

the m-symmetric power, with respect to the box product, of Y.
In particular, ^0(Y) is the terminal, constant one-point orthogonal space,
and ^1(Y)=Y.
Then the orthogonal space

    (Y) =  ∐_m≥ 0 ^m(Y)  =  ∐_m≥ 0   Y^⊠ m/Σ_m

is an ultra-commutative monoid under the concatenation product, 
and it is in fact the free ultra-commutative monoid generated by Y.symbol(Y) - free ultra-commutative monoid generated by Ysubjectultra-commutative monoid!free|seefree ultra-commutative monoidsubjectfree ultra-commutative monoid
More precisely, the functor

    :   → umon

becomes a left adjoint to the forgetful functor with respect to
the morphism η_Y:Y =^1(Y) → Y, the inclusion of the `linear' summand,
as the adjunction unit. 
In other words, the following composite is bijective
for every ultra-commutative monoid R:

    umon( Y, R) forget ( Y, R)  η_Y^*  (Y, R) .

Moreover, this adjunction is monadic, i.e., the category of ultra-commutative monoids
is isomorphic to the category of algebras over the monad .




We will also exploit that the category of ultra-commu­tative monoids is
tensored and cotensored over the category  of spaces,
so we spend a few words explaining this enrichment.
In fact, the constructions work more generally for algebras over 
continuous monads on any category enriched in spaces,
see for example <cit.>.  
We only spell out the case of ultra-commutative monoids,
which are the algebras over the free ultra-commutative monoid monad :→.

The mapping space of morphisms between two orthogonal spaces X and Y is
defined as follows. Since every inner product space is isometrically isomorphic
to ^n for some n, the map

    (X,Y) → ∏_n≥ 0 (X(^n),Y(^n))  ,   
    f ⟼ { f(^n)}_n≥ 0

is injective with closed image.
So we endow (X,Y) with the subspace topology of the product 
(which is taken internal to the category  of
compactly generated spaces, i.e., it is the Kelleyfied product topology).

If R and S are ultra-commutative monoids, then the set
umon(R,S) of morphisms of ultra-commutative monoids is a closed subset
of the space (R,S), and we give it the subspace topology.
We omit the verification that composition is continuous in this topology,
so we have indeed defined an enrichment of the category of
ultra-commutative monoids in spaces.

The cotensors of ultra-commutative monoids are defined `pointwise'.
In more detail, we consider an ultra-commutative monoid R and a space A.
Then the orthogonal space (A,R) inherits an ultra-commutative multiplication

    (A,R)⊠(A,R) →(A,R)

from the bimorphism with (V,W)-component

    (A,R(V))×(A,R(W)) → (A,R(V⊕ W))  , (f,g) ⟼ μ_V,W∘(f,g) .

The multiplicative unit is the constant map A→ R(0)
with value the unit of R.

The tensor of an ultra-commutative monoid R with a space A is, 
however, not pointwise. To avoid confusion with
the objectwise product we denote this tensor by R A;
its defining property is that it represents the functor

    (A,umon(R,-)) :  umon  → (sets) .

So R A comes equipped with a continuous map
i:A→ umon(R,R A) such that the map

    umon(R A,S) → (A,umon(R,S))  ,   
    f  ⟼ umon(R,f)∘ i

is bijective.
One way to construct a tensor R A is as a coequalizer,
in the category of ultra-commutative monoids, 
of the two morphisms:

    @C=18mm( ( R)× A)  @<.4ex>[r]^-(α× A)@<-.4ex>[r]_-ν     (R× A)

Here α: R→ R is the structure morphism
(i.e., the counit of the free-forgetful adjunction) and ν
is adjoint to the morphism of orthogonal spaces

    ( R)× A  → (R× A)

that in turn is adjoint to the composite

    A  a↦ (-,a) (R, R× A)     ( R, (R× A))  .

The above coequalizer defining R A is reflexive in the underlying category
of orthogonal spaces, so it can be calculated in the underlying category,
by Proposition <ref>. 



In our discussion of global group completions 
in Section <ref>
we will want to realize simplicial ultra-commutative monoids.
We refer to Construction <ref> for generalities
about realization of simplicial objects.subjectrealization!of simplicial ultra-commutative monoids|(
For a simplicial ultra-commutative monoid B:^→ umon, 
the term `geometric realization' actually has two potentially different interpretations,
and we take some time to clarify this issue.
On the one hand we can form the geometric realization |B|_un 
in the underlying category of orthogonal spaces; this is, by definition, a coend, in the
category of orthogonal spaces, of the functor

    ^× →  ,    
    ([m],[n]) ⟼ B_m×Δ^n  .

We call this the underlying realization of B.
Coends of orthogonal spaces are calculated objectwise, so |B|_un(V)
is a realization of the simplicial space [m]↦ B_m(V).
It is not a priori obvious, however, whether this realization inherits any multiplication.

On the other hand, we explained in Construction <ref>
that the category of ultra-commutative monoids 
is tensored over the category  of spaces.
We continue to write R A for the tensor of 
an ultra-commutative monoid R with a space A, in order to distinguish it
from the (objectwise) product of the underlying orthogonal space of R with A. 
We can also consider the realization |B|_in
internal to ultra-commutative monoids,
i.e., a coend, in the category of ultra-commutative monoids, of the functor

    ^× → umon  ,    
    ([m],[n]) ⟼ B_mΔ^n  .

We call this the internal realization.
The internal realization is, by definition, an ultra-commutative monoid, but it is not 
immediately clear how it relates to the underlying realization of |B|_un
as an orthogonal space.
As we shall now show, the forgetful functor from a category of
ultra-commutative monoids to orthogonal spaces commutes with
realization of simplicial objects.
We do not claim any originality here, 
and many results of this kind can be found in the literature, see for example
<cit.>,
<cit.>,
<cit.>,
<cit.>
or <cit.>.


 
Let B be a simplicial object in the category of ultra-commu­ta­tive monoids. 
Then the canonical morphism |B|_ un→ |B|_ in
from the underlying realization to the internal realization 
is an isomorphism of orthogonal spaces. 


We adapt an argument given by Mandell
in an unpublished preprint <cit.>.
We start by considering two simplicial orthogonal spaces
X, Y:^→.
We denote by X⊠ Y the diagonal of the external box product, 
i.e., the composite simplicial orthogonal space 

    ^ diagonal ^×^ X× Y ×  ⊠   .

For every n≥ 0 we consider the composite

    (X_n⊠ Y_n)×Δ^n  ×diag    (X_n⊠ Y_n)× (Δ^n×Δ^n)
     shuffle     (X_n×Δ^n)⊠(Y_n×Δ^n)
     → |X|⊠|Y|  ,

where the last morphism is the box product of the two canonical morphisms
X_n×Δ^n→ |X| and Y_n×Δ^n→|Y|.
These composites are compatible with the coend relations,
so they assemble into a morphism of orthogonal spaces

    | X⊠ Y|  →  |X|⊠ |Y|  .

We claim that this morphism is an isomorphism. 
Indeed, since ⊠ preserves colimits
in each variable, the right hand side is a coend of the functor

    (^2)^×^2  →  ,    
    ([k],[l],[m],[n]) ⟼ (X_k⊠ Y_l) ×Δ^m×Δ^n .

Coends of orthogonal spaces are calculated objectwise, and
for bisimplicial spaces the bi-realization is homeomorphic to the
realization of the diagonal 
(see <cit.>
or Proposition <ref> (iii)).

By iterating, we obtain a Σ_m-equivariant isomorphism of orthogonal spaces

    |X^⊠ m|    |X|^⊠ m

for every m≥ 0. Since coends commute with colimits, we can pass to
Σ_m-orbits and take coproduct over m≥ 0, resulting in an isomorphism 

    |  (X) |_un =   | ⨿_m≥ 0 (X^⊠ m)/Σ_m |   ⨿_m≥ 0  |X|^⊠ m/Σ_m   =   |X| .

On the other hand, the ultra-commutative monoid 
|X| has the universal property of the internal realization
of the simplicial ultra-commutative monoid ∘ X. 
This shows the claim in the special case where B is
freely generated by a simplicial orthogonal space.

Now we treat the general case. The diagram

    ( B) @<.4ex>[r]^-α@<-.4ex>[r]_-μ    B[r]^-α    B

is a coequalizer diagram of simplicial ultra-commutative monoids.
Here α: R→ R is the adjunction counit
and μ is the monad structure of the free functor.
Moreover, the coequalizer is reflexive in the underlying category of orthogonal spaces,
by the morphisms

    ( B)     B[l]_-η_ B    B [l]_-η_B

where η:R→ R is the unit of the free-forget adjunction,
i.e., the inclusion as the `linear' summand ^1(R).

Applying the two functors under consideration
gives a commutative diagram of orthogonal spaces

    |( B)|_un@<.4ex>[r]@<-.4ex>[r] [d] 
        | B|_un[r] [d]     |B|_un[d] 
    
    |( B)|_in@<.4ex>[r]@<-.4ex>[r]     
    | B|_in[r]     |B|_in

We claim that both rows are coequalizer diagrams of orthogonal spaces.
For the upper row we argue as follows. For every n≥ 0 the diagram

    ( B_n) @<.4ex>[r]@<-.4ex>[r]     B_n[r]     B_n
 
is a coequalizer in the category of ultra-commutative monoids.
Since the diagram splits in the underlying category
of orthogonal spaces, it is also a coequalizers diagram there,
by Proposition <ref>  or <cit.>.
So the diagram

    ( B) @<.4ex>[r]@<-.4ex>[r]     B [r]     B
 
is also a coequalizer diagram of simplicial orthogonal spaces.
Since product with Δ^n and coends commute with colimits
in the category of orthogonal spaces, the diagram stays
a coequalizer after (underlying) geometric realization.
Since coends commute with all colimits,
the bottom row of (<ref>)
is a coequalizer diagram of ultra-commutative monoids.
Again the diagram splits in the underlying category of orthogonal spaces,
so the lower diagram is also a coequalizer diagram of orthogonal spaces.
Since the two left vertical morphisms in (<ref>)
are isomorphisms of orthogonal spaces by the special case above, 
this proves that the morphism 
|B|_un→ |B|_in is an isomorphism as well.

subjectrealization!of simplicial ultra-commutative monoids|)

Ultra-commutative monoids form a pointed category: 
the constant one-point orthogonal monoid space is a zero object.
The enrichment, tensors and cotensors over spaces extend to
enrichment, tensors and cotensors over the category of based topological spaces. 
We shall write R A for the tensor of an ultra-commutative monoid R
with a based space (A,a_0), in order to distinguish it
from the (objectwise) smash product of 
the underlying based orthogonal space of R with A.
Thus R A is a pushout, in the category of ultra-commutative monoids,
of the diagram

    ∗       R{a_0} Rincl R A .

As may be familiar from similar contexts,
the bar construction B(R) of an ultra-commutative monoid R
can be interpreted as R S^1, the based tensor of R with the based space S^1,
see (<ref>) below.
Another way to say this is that 
the bar construction is the internal suspension, 
in the category of an ultra-commutative monoids.
We show a more general statement
and consider a based simplicial set A.
We define a simplicial object of ultra-commutative monoids by

    B_m(R,A) =  R A_m    ,

with simplicial structure induced by that of A. Since A_m is a based set,
R A_m is in fact a categorical coproduct (i.e., box product) 
of copies of R, indexed by the non-basepoint elements of A_m.

The next proposition constructs an isomorphism 
of ultra-commutative mo­noids between R |A|
and |B_∙(R,A) |_in, the internal geometric realization.
By Proposition <ref>,
we can (and will) confuse the internal realization with 
the underlying realization of B_∙(R,A) 
in the category of orthogonal spaces.
Variations of the following proposition appear in various places in the literature,
and they go back, at least, to the interpretation,
by McClure, Schwänzl and Vogt <cit.>,
of topological Hochschild homology of a commutative ring spectrum 
as the tensor with S^1.

 
Let R be an ultra-commutative monoid and A a based simplicial set.
Then R |A| is an internal realization of the simplicial ultra-commutative monoid
B_∙(R,A).


The geometric realization |A| is a coend of the functor

    ^× → _*  ,    
    ([m],[n]) ⟼ A_mΔ^n_+   .

Since the functor R - preserves colimits, R |A| is a coend, 
in the category of ultra-commutative monoids, of the functor

    ^× → umon  ,  
    ([m],[n]) ⟼ R ( A_mΔ^n_+)  .

The isomorphisms

    R ( A_mΔ^n_+)   (R A_m)Δ^n_+  (R A_m)Δ^n =  B_m(A,R)Δ^n

are natural in ([m],[n])∈^×,
and they show that R|A| is an internal realization
of the simplicial ultra-commutative monoid B_∙(R,A).



Now we approach the global model structure on the category 
of ultra-commu­ta­tive monoids.
We will establish this model structure as a special case of a lifting theorem
for model structures to categories of commutative monoids that 
was formulated by White <cit.>.
Like its predecessor for 
associative monoids <cit.>,
the input is a cofibrantly generated symmetric monoidal model category
that satisfies the monoid axiom.
However, lifting a model structure to commutative monoids is more subtle 
and needs extra hypotheses; the essence of the additional condition is
that, loosely speaking, symmetric powers must be `sufficiently homotopy invariant'.
Before White, Gorchinskiy and Guletskii <cit.>
also studied symmetric power constructions in a symmetric monoidal model category,
and there is a substantial overlap in the arguments of <cit.>
and <cit.>. 



We let  be a symmetric monoidal category with monoidal product ⊠.
To simplify the exposition we follow the common abuse
to suppress the associativity and unit isomorphisms from the notation,
i.e., we pretend that the underlying monoidal structure is strict
(i.e., a permutative structure).
We let i:A→ B be a -morphism and arrange the n-fold ⊠-power
of i into an n-dimensional cube K^n(i) in , i.e., a functor

    K^n(i) : (n)  →

from the poset category of subsets of {1,2,…,n} and
inclusions to . 
More explicitly, if S⊆{1,2,…,n} is a subset, 
then the vertex of the cube at S is 

    K^n(i)(S)  = C_1 ⊠ C_2 ⊠…⊠ C_n      with    
     C_j  = {[      A   j ∉S;      B j ∈ S. ].

All morphisms in the cube K^n(i) are ⊠-products of identities and
copies of the morphism i:A→ B.
The initial vertex of the cube is K^n(i)(∅)=A^⊠ n
and the terminal vertex is K^n(i)({1,…,n})=B^⊠ n.

We denote by Q^n(i) the colimit of the punctured cube,
i.e., the cube K^n(i) with the terminal vertex removed, 
and by i^ n:Q^n(i)→ K^n(i)({1,…,n})=B^⊠ n 
the canonical morphism, an iterated  pushout product morphism.
Indeed, for n=2 the cube K^2(i) is a square and looks like

    A⊠ A[r]^-A⊠ i[d]_i⊠ A    A⊠ B[d]^i⊠ B
     
    B⊠ A[r]_-B⊠ i    B⊠ B

Hence 

    i^ 2 = i i  =  (B⊠ i)∪(i⊠ B) :  
    B⊠ A∪_A⊠ A A⊠ B  → B⊠ B .

Similarly, i^ 3 is the morphism from the colimit of the punctured
cube to the terminal vertex of the following cube:

    @C=8mm@R=8mm A⊠ A⊠ A [rr]^A ⊠ A ⊠ i[dr]_A ⊠ i⊠ A[dd]_i⊠ A ⊠ A        
    A⊠ A⊠ B [dr]^A ⊠ i⊠ B'[d][dd]^i⊠ A⊠ B
        A⊠ B⊠ A [rr]_(.35)A ⊠ B⊠ i[dd]^(.7)i⊠ B⊠ A        A⊠ B⊠ B [dd]^i⊠ B⊠ B
    
    B⊠ A⊠ A [dr]_B ⊠ i⊠ A'[r]^(.6)B ⊠ A⊠ i[rr]        B⊠ A⊠ B [dr]^B ⊠ i⊠ B
        B⊠ B⊠ A [rr]_-B⊠ B⊠ i       B⊠ B⊠ B





We observe that the symmetric group Σ_n acts on 
Q^n(i) and B^⊠ n by permuting the factors,
and the iterated pushout product morphism
i^ n:Q^n(i)→ B^⊠ n is Σ_n-equivariant.
We recall from <cit.> 
the notions of symmetrizable cofibration
and symmetrizable acyclic cofibration. 



<cit.>  Let  be a symmetric monoidal model category. 
A morphism i:A→ B is a  symmetrizable cofibrationsubjectsymmetrizable cofibrationsubjectsymmetrizable acyclic cofibration
(respectively a symmetrizable acyclic cofibration) if the morphism

    i^ n/Σ_n  : Q^n(i)/Σ_n  → B^⊠ n/Σ_n = ^n(B)

is a cofibration (respectively an acyclic cofibration) for every n≥ 1.  


Since the morphism i^ 1/Σ_1 is the original morphism i, 
every symmetrizable cofibration
is in particular a cofibration and every symmetrizable acyclic cofibration
is in particular an acyclic cofibration.
We will now proceed to prove that in the category of orthogonal spaces,
all cofibrations and acyclic cofibrations in the positive global model structure
are symmetrizable with respect to the box product. 
The next proposition will be used to verify this 
for the generating acyclic cofibrations.
We recall from Construction <ref>
that given a morphism j:A→ B, the set (j) consists of all pushout product maps

    c(j) i_k  :   A× D^k ∪_A×∂ D^k Z(j)×∂ D^k →Z(j)× D^k

of the mapping cylinder inclusion c(j):A→ Z(j)
with the sphere inclusions for k≥ 0.



Let  be a symmetric monoidal topological model category. 
  
  
  * For every n≥ 1 the functor ^n preserves the homotopy
    relation on morphisms and it preserves homotopy equivalences.
  
  * Let j:A→ B be a symmetrizable acyclic cofibration between cofibrant objects.
    Then for every k≥ 0, the pushout product map
    
    j i_k  :  A× D^k∪_A×∂ D^k B×∂ D^k → 
        B× D^k

    is a symmetrizable acyclic cofibration.
  
  * Let j:A→ B be a morphism between cofibrant objects
    such that the morphism ^n(j):^n(A)→^n(B) is a weak equivalence
    for every n≥ 1. Then every morphism in the set (j)
    is a symmetrizable acyclic cofibration.
  


(i) This is the topological version of <cit.>.
For every object A of  and  every space K the morphism

    A^⊠ n× K  A^⊠ n×ΔA^⊠ n× K^n   (A× K)^⊠ n

is Σ_n-equivariant (with respect to the trivial Σ_n-action 
on K in the source) and factors over a natural morphism

    Δ̃ :  ^n(A)× K  =  ( A^⊠ n× K )/Σ_n  →(A× K)^⊠ n /Σ_n  =  ^n(A× K)  .

If H:A× [0,1]→ B is a homotopy from a morphism f=H(-,0)
to another morphism g=H(-,1), then the composite

    ^n(A)× [0,1]  Δ̃ ^n(A× [0,1])  ^n(H)^n(B)

is a homotopy from the morphism ^n(f) to ^n(g).
So ^n preserves the homotopy relation, and hence also homotopy equivalences.

(ii) We argue by induction on k. 
For k=0 the pushout product map j i_0
is isomorphic to j, hence a symmetrizable acyclic cofibration by hypothesis.
Now we assume the claim for some k, and deduce it for k+1.
Since j is a symmetrizable acyclic cofibration between cofibrant objects,
the morphism ^n(j) is a weak equivalence for every n≥ 1 
by <cit.>.  
Since the functors ^n preserve the homotopy relation and 
the projections A× D^k→ A and
B× D^k→ B are homotopy equivalences,
the morphism ^n(j× D^k) is a weak equivalence for every n≥ 1.
So j× D^k:A× D^k→ B× D^k is a symmetrizable acyclic cofibration,
again by <cit.>.  
We write ∂ D^k+1=D^k_+∪_∂ D^kD^k_- as the union of the upper and lower
hemisphere along the equator. The upper morphism in the pushout square

    D^k_+× A[r]^-D^k_+× j[d]     D^k_+× B [d]
    ∂ D^k+1× A [r]    ∂ D^k+1× A∪_D^k_+× A D^k_+× B

is a symmetrizable acyclic cofibration by the previous paragraph.
The class of symmetrizable acyclic cofibrations is closed under cobase
change by <cit.>;
the lower morphism is thus a symmetrizable acyclic cofibration.

The square 

    @C=20mm 
    A× D^k_- ∪_A×∂ D^k B×∂ D^k [d][r]^-j i_k    
    B× D^k_- [d] 
    
    A×∂ D^k+1∪_A× D^k_+ B× D^k_+ [r]_-jincl    
    B×∂ D^k+1

is a pushout. The upper morphism is a symmetrizable acyclic cofibration 
by the inductive hypothesis, 
hence so is the lower morphism,
again by stability under cobase change.
The morphism j×∂ D^k+1:A×∂ D^k+1→ B×∂ D^k+1 
is thus the composite of two symmetrizable acyclic cofibrations,
hence a symmetrizable acyclic cofibration itself, 
by <cit.>.
As a cobase change, the morphism

    A× D^k+1 → A× D^k+1∪_A×∂ D^k+1 B×∂ D^k+1

is then a symmetrizable acyclic cofibration
by <cit.>.
The induced morphism

    ^n(A×  D^k+1) → ^n( A× D^k+1∪_A×∂ D^k+1 B×∂ D^k+1)
 
is then a weak equivalence by <cit.>. 
Since ^n(j× D^k+1):^n( A× D^k+1)→^n(B× D^k+1) 
is a weak equivalence, so is the morphism

    ^n(j i_k+1) :  ^n( A× D^k+1∪_A×∂ D^k+1 B×∂ D^k+1)  →^n( B× D^k+1) .
 
One more time by <cit.>, this shows that j i_k+1 is
a symmetrizable acyclic cofibration. This completes the induction step.

(iii) Since A and B are cofibrant, the mapping cylinder inclusion  

    c(j)  :  A  → ( A× [0,1] )∪_j B = Z(j)

is a cofibration. Moreover, the projection Z(j)→ B is a homotopy
equivalence, hence so is ^n(Z(j))→^n(B) for every n≥ 1. 
Since ^n(j) is a weak equivalence by hypothesis, the morphism 
^n(c(j)):^n(A)→^n(Z(j)) is a weak equivalence for every n≥ 1. 
So c(j) is a symmetrizable acyclic cofibration by <cit.>. 
Applying (ii) to the morphism c(j) yields the claim.



Now we can verify the symmetrizability of cofibrations and acyclic cofibrations
for the positive global model structure of orthogonal spaces.
The cofibration part (i) is in fact slightly 
stronger in that it does not need any positivity hypothesis. 



  
  
  * Let i:A→ B be a flat cofibration  of orthogonal spaces.
    Then for every n≥ 1 the morphism 
    
    i^ n/Σ_n  :  Q^n(i)/Σ_n  → B^⊠ n/Σ_n

    is a flat cofibration. 
    In other words, all cofibrations in the global model structure
    of orthogonal spaces are symmetrizable.
  
  * Let i:A→ B be a positive flat cofibration of orthogonal spaces 
    that is also a global equivalence.
    Then for every n≥ 1 the morphism 
    
    i^ n/Σ_n  :  Q^n(i)/Σ_n  → B^⊠ n/Σ_n

    is a global equivalence.
    In other words, all acyclic cofibrations in the positive global model structure
    of orthogonal spaces are symmetrizable.
  


(i) We recall from the proof of Proposition <ref>
the set 

    I^ =  { G_m(  O(m)/H× i_k ) |  m,k ≥ 0, H≤ O(m)}

of generating flat cofibrations of orthogonal spaces, where i_k:∂ D^k→ D^k
is the inclusion.
The set I^ detects the acyclic fibrations 
in the strong level model structure of orthogonal spaces.
In particular, every flat cofibration is a retract of an I^-cell complex.
By <cit.> 
or <cit.>, 
it suffices to show that the generating flat cofibrations in I^ are symmetrizable.

The orthogonal space G_m( O(m)/H × K) is isomorphic to _H,^m× K,
so we show more generally that every morphism of the form

    _G,V× i_k   :  _G,V×∂ D^k  → _G,V× D^k

is a symmetrizable cofibration, 
where V is any representation of a compact Lie group G.
The symmetrized iterated pushout product

    (  _G,V× i_k )^ n / Σ_n  :  
    Q^n(_G,V× i_k ) /Σ_n →  (_G,V× i_k )^⊠ n/Σ_n

is isomorphic to 

    _Σ_n≀ G,V^n( i_k^ n)  :  _Σ_n≀ G,V^n ( Q^n ( i_k) )  → _Σ_n≀ G,V^n( (D^k)^n)  ,

where

    i_k^ n :  Q^n ( i_k )  → (D^k)^n

is the n-fold pushout product of the inclusion i_k:∂ D^k→ D^k, 
with respect to the cartesian product of spaces. 
Here the wreath product Σ_n≀ G acts on V^n by

    (σ;  g_1,…,g_n)· (v_1,…, v_n)  =  
    (g_σ^-1(1)v_σ^-1(1),…, g_σ^-1(n)v_σ^-1(n))  .

The map i_k^ n is Σ_n-equivariant, 
and we claim that i_k^ n is a cofibration of Σ_n-spaces.
One way to see this is to exploit that i_k is homeomorphic to
the geometric realization of the inclusion ι_k:∂Δ[k]→Δ[k]
of the boundary of the simplicial k-simplex.
So i_k^ n is Σ_n-homeomorphic to
the geometric realization of the 
inclusion ι_k^ n:Q^n(ι_k)→Δ[k]^n
of Σ_n-simplicial sets. The geometric realization of an equivariant
embedding of simplicial sets is always an equivariant cofibration of spaces,
so altogether this shows that i_k^ n is a cofibration of Σ_n-spaces.
Proposition <ref> (i)
then shows that i_k^ n is also a cofibration of (Σ_n≀ G)-spaces
via restriction along the projection Σ_n≀ G→Σ_n. 
So the morphism (<ref>) is a flat cofibration.

(ii) Theorem <ref> describes a set J^∪ K 
of generating acyclic cofibrations for the global model structure
on the category of orthogonal spaces.
From this we obtain a set J^+∪ K^+ 
of generating acyclic cofibration for the positive 
global model structure of Proposition <ref> 
by restricting to those morphisms in  J^∪ K 
that are positive cofibrations, i.e., homeomorphisms in level 0. 
So explicitly, we set

    J^+  =  { G_m(  O(m)/H × j_k) |  m≥ 1, k ≥ 0, H≤ O(m)} ,

where j_k:D^k×{0}→ D^k× [0,1] is the inclusion, and

    K^+  =  ⋃_G,V,W :   V 0(ρ_G,V,W)  ,

the set of all pushout products of sphere inclusions i_k
with the mapping cylinder inclusions of the global equivalences
ρ_G,V,W:_G,V⊕ W→_G,V.
Here (G,V,W) runs through a set of representatives
of the isomorphism classes of triples consisting of a compact Lie group G,
a non-zero faithful G-representation V and an arbitrary G-representation W.
By <cit.> or <cit.> 
it suffices to show that all morphisms in J^+∪ K^+
are symmetrizable acyclic cofibrations. 

We start with a morphism G_m( j_k× O(m)/H ) in J^+.
For every n≥ 1, the morphism

    (G_m(  O(m)/H× j_k ))^ n/Σ_n

is a flat cofibration by part (i), and a homeomorphism in level 0
because m≥ 1. Moreover, the map j_k is a homotopy
equivalence of spaces, so  G_m(  O(m)/H × j_k) is a homotopy equivalence
of orthogonal spaces; the morphism ^n(G_m(  O(m)/H× j_k )) 
is then again a homotopy equivalence for every n≥ 1,
by Proposition <ref> (i).
Then <cit.> shows that G_m( O(m)/H × j_k)
is a symmetrizable acyclic cofibration. 
This takes care of the set J^+.

Now we consider the morphisms in the set K^+.
Since G acts faithfully on the non-zero inner product space V, the action
of the wreath product Σ_n≀ G on V^n is again faithful.
So the morphism

    ρ_Σ_n≀ G,V^n, W^n :  _Σ_n≀ G, V^n⊕ W^n →_Σ_n≀ G, V^n

is a global equivalence by Proposition <ref> (ii).
By the isomorphism

    ^n(_G,V)  =  _G,V^⊠ n/Σ_n   _Σ_n≀ G,V^n ,

the morphism ρ_Σ_n≀ G,V^n, W^n is isomorphic to
 ^n(ρ_G,V,W) :^n(_G,V⊕ W) →^n(_G,V),
which is thus a global equivalence. 
Proposition <ref> (iii)
then shows that all morphisms in (ρ_G,V,W) 
are symmetrizable acyclic cofibrations. 


The hypothesis in Theorem <ref> (ii)
that i is a positive flat cofibration is really necessary. 
Indeed, the unique morphism ρ:_→∗ to the terminal orthogonal space
is a global equivalence, and source and target of ρ are flat, 
but only the source is positively flat.
Then the mapping cylinder inclusion c(ρ):_→ C(_)
is a global equivalence between flat orthogonal spaces, but it is not
a homeomorphism at 0. And indeed, for no n≥ 2 is the morphism
^n(_)→^n(C(_)) a global equivalence,
because the source is isomorphic to _Σ_n,^n=B_Σ_n,
whereas the target is homotopy equivalent to the terminal orthogonal space. 



Now we put all the pieces together and prove the 
global model structure for ultra-commutative monoids.
We call a morphism of ultra-commutative monoids a
global equivalence (respectively positive global fibration)
if the underlying morphism of orthogonal spaces is a
 global equivalence (respectively fibration in the positive global model structure).






  
  
  * The global equivalences and positive global fibrations are part of a 
   cofibrantly generated, proper, topological model structure
    on the category of ultra-commutative monoids,
    the global model structure.subjectglobal model structure!for ultra-commutative monoids
  
  * Let j:R→ S be a cofibration in the global model structure 
  of ultra-commutative monoids.
  
  
  * The morphism of R-modules underlying j is a cofibration
    in the global model structure of R-modules 
    of Corollary <ref> (i). 
  
  * The morphism of orthogonal spaces underlying j is an h-cofibration,
    and hence a closed embedding.
  
  * If the underlying orthogonal space of R is flat, then 
    j is a flat cofibration of orthogonal spaces.
 
  


(i) The positive global model structure of orthogonal spaces
established in Proposition <ref> is
 cofibrantly generated and monoidal 
(by Proposition <ref> (iv)).
The `unit axiom' also holds: we let f:I→∗ be any positive flat replacement
of the monoidal unit, the constant one-point orthogonal space.
Then for every orthogonal space Y the induced morphism
f⊠ X:I⊠ Y→∗⊠ Y is a global equivalence by
Theorem <ref> (ii).
The monoid axiom holds by Proposition <ref>.
Cofibrations and acyclic cofibrations are symmetrizable by
Theorem <ref>,
so the model structure satisfies the `commutative monoid axiom'
in the sense of <cit.>.
The symmetric algebra functor  commutes with filtered colimits
by Corollary <ref>.  
Theorem 3.2 of <cit.> thus shows that the positive global
model structure of orthogonal spaces lifts to the category of 
ultra-commutative monoids.

The global model structure is topological 
by Proposition <ref>,
where we take  as the set of free ultra-commutative monoids (L_H,^m)
for all m≥ 1 and all closed subgroups H of O(m),
and we take  as the set of acyclic cofibrations
(c(ρ_G.V.W)) for the mapping cone inclusions c(ρ_G,V,W) of
the global equivalences ρ_G,V,W:_G,V⊕ W→_G,V,
indexed by representatives as in the definition of the set K^+.
Since weak equivalences and fibrations of ultra-commutative monoids are
defined on underlying orthogonal spaces, and since pullbacks of
ultra-commutative monoids are created on underlying orthogonal spaces,
right properness is inherited from the positive global model structure
of orthogonal spaces (Proposition <ref>).
We defer the proof of left properness until after the proof of part (ii).

  (ii) For (a) we recall that the global model structure on the category of R-modules
  is lifted, via the free and forgetful adjoint functor pair,
  from the absolute global model structure of Theorem <ref>. 
  By Corollary <ref> (i) and (ii)
  this model structure of R-modules is a cofibrantly generated
  monoidal model category that satisfies the monoid axiom.
  Moreover, the unit object R is cofibrant; for this it is relevant that
  we have lifted the absolute model structure (as opposed to the
  positive model structure).
  We claim that all cofibrations in this model structure
  are symmetrizable with respect to the box product of R-modules.
  By <cit.>  or <cit.> 
  it suffices to show this for a set of generating cofibrations, which can be taken
  of the form  R⊠ i for i in a set of flat cofibrations
  of orthogonal spaces (for example the set I^ defined in
  the proof of the strong level model structure, 
  Proposition <ref>).
  A box product, over R, of free R-modules induced from orthogonal spaces
  is isomorphic to the free R-module generated by the box product of
  underlying orthogonal spaces:
  
    (R⊠ X)⊠_R (R⊠ Y)  R⊠ (X⊠ Y)

  Since R⊠- is a left adjoint, it commutes with
  pushouts and orbits by Σ_n-actions. Hence the analogous statement
  carries over to symmetrized iterated box products. In other words,
  for every morphism i:A→ B of orthogonal spaces there is a natural
  isomorphism in the arrow category of R-modules between
  
    (R⊠ i)^_R n/Σ_n  :  
      Q^n_R(R⊠ i)/Σ_n  → ^n_R(R⊠ B)

  and
  
    R⊠ (i^ n/Σ_n) :  R⊠ (Q^n(i)/Σ_n)  →  R⊠^n(B)  .

  If i is a flat cofibration of orthogonal spaces, then so is
  the morphism i^ n/Σ_n,
  by Theorem <ref> (i).
  So the morphism R⊠ (i^ n/Σ_n) is a cofibration of R-modules,
  hence so is the morphism (R⊠ i)^_R n/Σ_n.
  This completes the proof that all cofibrations in the global
  model structure for R-modules of Corollary <ref> (i)
  are symmetrizable with respect to ⊠_R. 

  Now we apply Corollary 3.6 of <cit.>;
  there is a slight caveat here, because the hypotheses
  ask for the validity of the `strong commutative monoid axiom'
  (Definition 3.4 of <cit.>), which requires
  the symmetrizability of both the cofibrations and the acyclic cofibrations.
  Since the model structure on R-modules was lifted from 
  an absolute model structure, it is not the case that all acyclic 
  cofibrations are symmetrizable. However, <cit.>
  and its proof are only about cofibrations, and don't involve the weak equivalences
  at all. So the proof of <cit.>
  only needs the symmetrizability of the cofibrations, which we just
  established for the global model category of R-modules of 
  Corollary <ref> (i).
  Since R is cofibrant as an R-module, 
  <cit.> shows that for every cofibrant
  commutative R-algebra S, the structure morphism i:R→ S
  is a cofibration of R-modules. Part (i) now follows because
  commutative R-algebras are morphisms of ultra-commutative monoids
  with source R. More precisely, the category of commutative R-algebras
  is isomorphic to the category of ultra-commutative monoids under R.
  Moreover, a commutative R-algebra S is cofibrant if and only if the
  structure morphism i:R→ S is a cofibration of ultra-commutative monoids.

  (b) This is a combination of part (a) and the fact, proved in 
  Corollary <ref> (i),
  that all cofibrations of R-modules are h-cofibrations of orthogonal spaces.
  
  (c) This is a combination of part (a) and the fact, also proved in 
  Corollary <ref> (i),
  that if R itself is flat, then all cofibrations of R-modules are 
  flat cofibrations of orthogonal spaces.

If remains to prove left properness of the model structure.
Pushouts in a category of commutative algebras are given
by the relative monoidal product. For ultra-commutative monoids this
means that a pushout square has the form

    @C=15mm
    R [d]_j [r]^-f_-≃    T[d]^j⊠_R T
     
    S[r]_-S⊠_R f    S⊠_R T

where S and T are
considered as R-modules by restriction along j respectively f.
For left properness we now suppose that j is a cofibration and f 
is a global equivalence.
By part (a) of (ii), the morphism j is then a cofibration of R-modules 
in the global model structure of Corollary <ref> (i). 
Since R is cofibrant in that model structure, also S is
cofibrant as an R-module.
Proposition <ref> then shows that the functor
S⊠_R - preserves global equivalences.
So the cobase change S⊠_R f of f is a global equivalence.
This shows that the global model structure of ultra-commutative monoids is left proper. 









§ GLOBAL POWER MONOIDS




In this section we investigate the algebraic structure that an
ultra-commutative multiplication produces on the -functor _0(R).
Besides an abelian monoid structure on π_0^G(R) for every
compact Lie group G, this structure includes power operations and
transfer maps. We formalize this algebraic structure under the name
`global power monoid', see Definition <ref>.
Theorem <ref> then says that global power monoids
are precisely the natural algebraic structure, i.e., they parametrize all
natural operations on _0(R) for ultra-commutative monoids.
In Construction <ref> we introduce the transfer maps,
which are an equivalent way of packaging the power operations in a global
power monoid; the main properties of the transfers are summarized in
Proposition <ref>. 



Given an orthogonal monoid space R (not necessarily commutative)
with multiplication morphism μ:R⊠ R→ R and a compact Lie group G,
we define a binary operation

    + : π_0^G(R) ×π_0^G(R) → π_0^G(R)

on the G-equivariant homotopy set of R as the composite

    π_0^G(R) ×π_0^G(R)   × π_0^G(R⊠ R)   μ_*  π_0^G(R) .

The pairing × was defined 
in Construction <ref>.
If we expand the definition, it boils down to the following explicit recipe:
if V and W are G-representations
and x∈ R(V)^G and y∈ R(W)^G are G-fixed points that
represent two classes in π^G_0(R), 
then [x]+[y] is represented by the G-fixed point

    μ_V,W(x,y) ∈ R(V⊕ W)  .


We write the pairing on the equivariant homotopy sets of R 
additively because we will mostly be interested in commutative
orthogonal monoid spaces.
Obviously, the additive notation is slightly dangerous for 
non-commu­tative orthogonal monoid spaces, because there the pairing
need not be commutative.



The following properties of the operation `+` are direct consequences 
of the corresponding properties of the pairings `×', 
compare Proposition <ref>;
a direct proof from the explicit definition of the operation `+' above
is also straightforward. 

 
  Let R be an orthogonal monoid space.
  
  
  * For every compact Lie group G 
    the binary operation + makes the set π_0^G(R) into a monoid.
  
  * If the multiplication of R is commutative, then so is the operation +.
  
  * The restriction map α^*:π_0^G(R)→π_0^K(R) associated to
    a continuous homomorphism α:K→ G between compact Lie groups
    is a monoid homomorphism.
  



Now we turn to special features that happen for ultra-commutative monoids.
If the multiplication on an orthogonal monoid space R is commutative,
then this does not only imply commutativity of the monoids π_0^G(R);
strict commutativity of the multiplication also gives rise to additional 
power operations that we discuss now.
An important special case will later be the 
multiplicative ultra-commutative monoid Ω^∙ R
arising from an ultra-commutative ring spectrum R.
In this situation the power operations satisfy further compatibility conditions
with respect to the addition and the transfer maps on _0(Ω^∙ R)=_0(R);
altogether this structure makes altogether makes the 0-th equivariant homotopy groups
of an ultra-commutative ring spectrum into a global power functor.



We let R be an ultra-commutative monoid, G a compact Lie group
and m≥ 1.subjectpower operation!in an ultra-commutative monoidsymbol[m] - m-th power operation in an ultra-commutative monoid 
We construct a natural power operation

    [m]  :  π_0^G ( R )  → π^Σ_m≀ G_0 ( R )

that is an equivariant refinement of the map x⟼ m· x.

We recall that the wreath product Σ_m≀ Gsubjectwreath productsymbolΣ_m≀ G - wreath product of a symmetric group Σ_m 
and a group G is the semidirect product

    Σ_m≀ G  =  Σ_m ⋉ G^m
 
formed with respect to the action of Σ_m by permuting the
factors of G^m. So the multiplication in Σ_m≀ G is given by

    (σ; g_1,…,g_m) · (τ; k_1,…, k_m)  =  
    (στ; g_τ(1)k_1,…,g_τ(m)k_m)  .

For every G-space E,
the wreath product Σ_m≀ G acts on the space E^m by

    (σ; g_1,…,g_m) · (e_1,…, e_m)  =  
    (g_σ^-1(1)e_σ^-1(1),…,g_σ^-1(m) e_σ^-1(m))  .

For every G-representation V, this action even makes V^m 
into a (Σ_m≀ G)-representation.
We let

    μ_V,…,V : R(V)×…× R(V)  → R(V⊕…⊕ V)

denote the (V,…,V)-component of the iterated multiplication map of R,
and we observe that this map is (Σ_m≀ G)-equivariant because the multiplication
of R is commutative. 
If x∈ R(V)^G is a G-fixed point representing a class in π^G_0 (R),
then (x,…,x) ∈ R(V)^m is a (Σ_m≀ G)-fixed point.
So its image under the map μ_V,…,V 
is a (Σ_m≀ G)-fixed point of R(V^m),
representing an element 

    [m]( [x] )  =  μ_V,…,V(x,…,x)  ∈ π_0^Σ_m≀ G ( R ) .

If we stabilize x along a G-equivariant
linear isometric embedding φ:V→ W
to R(φ)(x) ∈ R(W)^G, 
then μ_V,…,V(x,…,x) changes into 

    μ_W,…,W ( R(φ)(x),…, R(φ)(x))
     =  R(φ^m) (μ_V,…,V(x,…,x))  ∈  R(W^m)^Σ_m≀ G .

Since φ^m:V^m→ W^m is a (Σ_m≀ G)-equivariant
linear isometric embedding, 
this element represents the same class in π_0^Σ_m≀ G(R)
as μ_V,…,V(x,…,x), so the class 
[m]([x]) only depends on the class of x in π_0^G ( R ).
We have thus constructed a well-defined power operation (<ref>).

The power operations are clearly natural for homomorphisms φ:R→ S
of ultra-commutative monoids, i.e., for every compact Lie group G,
every m≥ 0 and all x∈π_0^G(R) the relation

    [m]( φ_* (x)) =  φ_* ( [m](x) )

holds in π_0^Σ_m≀ G(S).



The power operations [m] satisfy various properties reminiscent 
of the map x↦ m· x in an abelian monoid.
We formalize these properties into the concept of a global power monoid.
In the definition we need certain homomorphisms between different wreath products, 
so we fix notation for these now.
We use the plus symbol for the `concatenation' group monomorphism

    +   :  Σ_i×Σ_j  → Σ_i+j

defined by

    (σ+σ')(k)  =   σ(k)     for 1≤ k≤ i, and
    σ'(k-i)+i     for i+1≤ k≤ i+j.

This operation is strictly associative, so we will leave out parentheses.
The operation + is not commutative, but the permutations
σ+σ' and σ'+σ differ by conjugation with the (i,j)-shuffle.
An embedding of a product of wreath products is now defined by

    Φ_i,j :  (Σ_i≀ G) × (Σ_j≀ G)   →  Σ_i+j≀ G 
    
     ((σ;  g_1,…,g_i),  (σ';  g_i+1,…,g_i+j))     ⟼(σ+σ';  g_1,…,g_i+j) .

Another group monomorphism

    ♮ :  Σ_k≀Σ_m → Σ_k m

is defined bysymbol♮ - monomorphism Σ_k≀Σ_m→Σ_k m

    (σ♮(τ_1,…,τ_k))((i-1)m +j) =  (σ(i)-1)m + τ_i(j)  ,

for 1≤ i≤ k and 1≤ j≤ m.
This yields an  embedding of an iterated wreath product 

    Ψ_k,m :  Σ_k≀ (Σ_m≀ G)        →    Σ_k m≀ G 
    
     (σ;  (τ_1;  g^1),…,(τ_k;  g^k))     ⟼(σ♮(τ_1,…,τ_k);  g^1+…+g^k) .

Here each g^i=(g^i_1,…,g^i_m) is an m-tuple of elements of G, and 

    g^1+…+g^k =  
    (g^1_1,…,g^1_m,g^2_1,…,g^2_m,…,g^k_1,…,g^k_m)

denotes the concatenation of the tuples.


The formula for the homomorphism ♮:Σ_k≀Σ_m→Σ_k m
may seems slightly ad hoc, but it can be motivated 
in a more conceptual way as a composite 

    Σ_k≀Σ_m → Σ_{1,…,k}×{1,…,m}  Σ_k m .

The first monomorphism sends (σ; τ_1,…,τ_k) to the permutation
of the product set {1,…,k}×{1,…,m} defined by

    (i,j) ⟼ (σ(i),τ_i(j)) .

The second isomorphism is conjugation by the lexicographic ordering

    {1,…,k}×{1,…,m}  {1,…,k m} ,   
    (i,j) ⟼ (i-1)m +j .

The use of the lexicographic ordering (and hence the precise formula
for the homomorphism ♮) is not essential here:
if we use a different bijection between the sets
{1,…,k}×{1,…,m} and {1,…,k m}, then
the homomorphisms ♮ and Ψ_k,m change by inner automorphisms.
So the conjugacy classes of ♮ and Ψ_k,m
(but not the actual homomorphisms) are canonical.
Since we will always hit Ψ_k,m with functors that are invariant
under conjugation, this should motivate that the construction is reasonably natural.





 subjectglobal power monoid
A global power monoid is a functorsubjectabelian Rep-monoid

    M  :  ^ →  Mon

from the opposite of the category  of compact Lie groups 
and conjugacy classes of continuous homomorphisms to the category of abelian monoids,
equipped with monoid homomorphismssymbol  Mon - category of abelian monoids

    [m]  :  M(G)  → M(Σ_m≀ G)

for all compact Lie groups G and m≥ 1,
called power operations, that satisfy the following relations.


  * (Identity) The operation [1] is restriction along the 
preferred isomorphism Σ_1≀ G G, (1;g)↦ g.

  * (Naturality) 
For every continuous homomorphism α:K→ G between compact Lie groups
and every m≥ 1 the relation

    [m]∘α^*  =  (Σ_m≀α)^*∘ [m]

holds as homomorphisms M(G)→ M(Σ_m≀ K).

  * (Transitivity)
For all compact Lie groups G and all k,m≥ 1 the relation

    Ψ^*_k,m∘  [k m] =  [k]∘  [m]

holds as homomorphisms  M(G)→ M(Σ_k≀(Σ_m ≀ G)),
where Ψ_k,m is the monomorphism (<ref>). 

  * (Additivity)
For all compact Lie groups G, all m>i>0 and all x∈ M(G)
the relation

    Φ^*_i,m-i( [m](x) ) =  p_1^*([i](x))  +  p_2^*([m-i](x))

holds in M((Σ_i≀ G)×(Σ_m-i≀ G)),
where Φ_i,m-i is the monomorphism (<ref>) and

    p_1:(Σ_i≀ G)×(Σ_m-i≀ G)→Σ_i≀ G  
       and  
    p_2:(Σ_i≀ G)×(Σ_m-i≀ G)→Σ_m-i≀ G

are the two projections.  

A morphism of global power monoids is a natural transformation
of abelian monoid valued functors that also commutes with the power operations [m]
for all m≥ 1.subjectmorphism!of global power monoids





In any abelian Rep-monoid M we can define external pairings 

    ⊕ : M(G) × M(K) → M(G× K) 
       by  
     x⊕ y  =  p_G^*(x)+p_K^*(y)  ,

where p_G:G× K→ G and p_K:G× K→ K 
are the two projections.
In this notation, the additivity requirement in Definition <ref>
becomes the relation 

    Φ^*_i,m-i( [m](x) ) =  [i](x)  ⊕ [m-i](x) .

In a global power monoid, 
the power operations are also additive with respect to the external addition:
for all compact Lie groups G and K and all m≥ 1, and 
all classes x∈ M(G) and y∈ M(K) the relation

    [m](x⊕ y)  =  Δ^*( [m](x)⊕ [m](y))

holds in M(Σ_m≀(G× K)), where Δ is 
the `diagonal' monomorphism

    Δ : Σ_m≀(G× K)        →     (Σ_m≀ G)×(Σ_m≀ K)
    
     (σ;  (g_1,k_1),…,(g_m,k_m))     ⟼ 
    ((σ;  g_1,…,g_m),  (σ;  k_1,…,k_m))  .

Indeed, Δ factors as the composite

    Σ_m≀(G× K) Δ_Σ_m≀(G× K)       (Σ_m≀(G× K))× (Σ_m≀(G× K))
    (Σ_m≀ p_G)×(Σ_m≀ p_K)    (Σ_m≀ G)×(Σ_m≀ K) .

So

    [m](x⊕ y)    =    [m](p_G^*(x) + p_K^*(y)) =  
     [m](p_G^*(x)) + [m](p_K^*(y))
       =  (Σ_m≀ p_G)^*([m](x)) + (Σ_m≀ p_K)^*([m](y))  
       =  Δ_Σ_m≀(G× K)^*( (Σ_m≀ p_G)^*([m](x)) ⊕ (Σ_m≀ p_K)^*([m](y)))  
       =  Δ_Σ_m≀(G× K)^*( ((Σ_m≀ p_G)×(Σ_m≀ p_K))^*( [m](x) ⊕ [m](y))) 
       =  Δ^*( [m](x)⊕ [m](y)) .

The class [m](x) is an equivariant refinement of m· x=x+…+x
(m summands) in the following sense. 
Applying the relation (<ref>) repeatedly
shows that [m](x) restricts to the external m-fold sum

    x⊕…⊕ x  ∈ M(G^m)

on the normal subgroup G^m of Σ_m≀ G. Restricting further to the
diagonal takes the m-fold external sum to m· x in M(G).
  



We will soon discuss that the power operations of an ultra-commutative monoid
define a global power monoid. One aspect of this is the additivity 
of the power operations, which could be shown directly from the definition. 
However, we will use this opportunity to establish
a very general (and rather formal) additivity result 
that we will use several times in this book.
We let  be a category with a zero object and finite coproducts. 
We let X∨ Y be a coproduct of two objects X and Y
with universal morphisms i:X→ X∨ Y and j:Y→ X∨ Y.
If f:X→ A and g:Y→ A are two morphisms with common target,
we denote by f+g:X∨ Y→ A the unique morphism that satisfies
(f+g)i=f and (f+g)j=g.

We call a functor from  to the category of
abelian monoids reduced if it takes every zero object in 
to the trivial monoid. We call the functor F additive
if for every pair of objects X,Y of  the map

    ( F(_X+0), F(0+_Y))  :  F(X∨ Y) → F(X)× F(Y)

is bijective (and hence an isomorphism of monoids).


Let  be a category with a zero object and finite coproducts and 

    F,G :  →  Mon

two reduced functors to the category of
abelian monoids. Suppose that the functor G is additive.
Then every natural transformation of set valued functors from F to G
is automatically additive.


We let τ:F→ G be a natural transformation of set valued functors.
We consider two classes x, y∈ F(X).
We let i,j:X→ X∨ X be the two inclusions into the coproduct.
We claim that

    τ_X∨ X(F(i)(x)+ F(j)(y))  =   G(i)(τ_X(x)) + G(j)(τ_X(y))

in the abelian monoid G(X∨ X).
Indeed,

    G(_X+0) ( τ_X∨ X(F(i)(x)+ F(j)(y)))    =  τ_X( F(_X+0) (F(i)(x)+ F(j)(y)))
       =  τ_X( F(_X)(x)+ F(0)(y)) =  τ_X(x) 
       =  G(_X)(τ_X(x)) +  G(0)(τ_X(x))
       =   G(_X+0)( G(i)(τ_X(x)) + G(j)(τ_X(y)))

in G(X). Similarly, 

    G(0+_X) ( τ_X∨ X(F(i)(x)+ F(j)(y))) =  
    G(0+_X)( G(i)(τ_X(x)) + G(j)(τ_X(y))) .

Since G is additive, this shows the relation (<ref>).
We let ∇=(+):X∨ X→ X denote the fold morphism,
so that

    F(∇)(F(i)(x)+ F(j)(y))  =  F(∇ i)(x)+ F(∇ j)(y) =   x + y .

Then

    τ_X(x+y)     =  τ_X( F(∇)(F(i)(x)+ F(j)(y))) 
       =  G(∇)(τ_X∨ X(F(i)(x)+ F(j)(y))) 
    
    _(<ref>)    =  G(∇)( G(i)(τ_X(x)) + G(j)(τ_X(y)))
    
       =  G(∇ i)(τ_X(x)) + G(∇ j)(τ_X(y))  =  τ_X(x)  +  τ_X(y)  .





 
Let R be an ultra-commutative monoid.
Then the binary operations (<ref>)
and the power operations (<ref>) make the
Rep-functor _0(R) into a global power monoid.

 
Corollary <ref> 
shows that the binary operations (<ref>)
make the Rep-functor _0(R) into a functor to the category of
abelian monoids.
The coproduct of ultra-commutative monoids is given by the box product,
so the two reduced functors

    π_0^G ,  π_0^Σ_m≀ G :  umon  →  Mon

are additive by Corollary <ref>.
Since the power operation [m]:π_0^G(R)→π_0^Σ_m≀ G(R) 
is natural in R, 
Proposition <ref>, applied to the category of
ultra-commutative monoids, shows that [m] is additive.
The identity (i), naturality (ii), 
transitivity (iii) and  additivity property (iv)
in Definition <ref> of global power monoids 
are straightforward, and we omit the proofs.




 We let M be a commutative topological monoid. Then the
constant orthogonal space M is 
naturally an ultra-commutative monoid.  
Moreover, the equivariant homotopy functor _0(M)
is constant with value π_0(M), and monoid structure induced
from the multiplication of M. The power operation

    [m] : π_0(M) = π_0^G(M) →π_0^Σ_m≀ G(M) 
    = π_0(M)

then sends an element x to m· x.






[Naive units of an orthogonal monoid space]subjectunits!of an orthogonal monoid space!naive
Every orthogonal monoid space R contains an interesting  orthogonal monoid subspace
R^n ×, the naive units of R.symbolR^n× - naive units of the orthogonal monoid space R The value of R^n × 
at an inner product space V is the union of those path components of R(V) 
that are taken to invertible elements, with respect to the monoid structure 
on π_0(R), under the map

    R(V) → π_0(R(V))  → π^e_0(R)  .

In other words, a point x∈ R(V) belongs to R^n×(V) if and only
if there is an inner product space W and a point y∈ R(W) such that

    μ_V,W(x,y)  ∈ R(V⊕ W)
         and    μ_W,V(y,x)  ∈ R(W⊕ V)

are in the same path component as the respective unit elements.
We omit the verification that the subspaces R^n×(V) indeed form 
an orthogonal monoid subspace of R as V varies.
The induced map

    _0(R^n×) → _0(R)

is also an inclusion, and the value π_0^e(R^n×) at the trivial group is,
by construction, the set of invertible elements of π_0^e(R).
For a general compact Lie group G,

    π_0^G(R^n×)  =  {  x∈π_0^G(R)  | ^G_e(x) is invertible in π_0^e(R)}

is the submonoid of π_0^G(R) of elements that become invertible
when restricted to the trivial group. 
So contrary to what one could suspect at first sight, 
π_0^G(R^n×) may contain non-invertible elements
and the orthogonal monoid space R^n× is not necessarily group-like;
this is why we use the adjective `naive'.



[Units of a global power monoid]subjectunits!of a global power monoid
Every global power mo­noid M has a global power submonoid M^× of units.
The value M^×(G) at a compact Lie group G is the subgroup
of invertible elements of M(G). Since the restriction maps
and the power operations are homomorphisms, 
the sets M^×(G) are closed under restriction maps
and power operations. So for varying G, the subgroups M^×(G) 
indeed form a global power submonoid of M.

We call a global power monoid N group-likesubjectgroup-like!global power monoid
if the abelian monoid N(G) is a group for every compact Lie group G.
If f:N→ M is a homomorphism of global power monoids and N is group-like, 
then the image of f is contained in M^×. 
So the functor M↦ M^× is right adjoint
to the inclusion of the full subcategory of group-like global power monoids.

If R is an ultra-commutative monoid, then we introduce 
a global topological version R^× of the units 
in Construction <ref> below.subjectunits!of an ultra-commutative monoid
This construction comes with a homomorphism of ultra-commutative monoids
R^×→ R that realizes the inclusion of the units of _0(R),
compare Proposition <ref>.


[Group completion of a global power monoid]subjectgroup completion!of a global power monoid
A morphism j:M→ M^⋆ of global power monoids is a group completion
if for every group-like global power monoid N the map

    j^* :  (global power monoids)(M^⋆,N) → (global power monoids)(M,N)

is bijective. 
Since the pair (M^⋆,j) represents a functor,
it is unique up to preferred isomorphism under M.
Every global power monoid M has a group completion, 
which can be constructed `objectwise'.
We define a global power monoid M^⋆ at a compact Lie group G by
letting M^⋆(G) be a group completion (Grothendieck construction) 
of the abelian monoid M(G),
with j(G):M(G)→ M^⋆(G) the universal homomorphism.
Since the restriction maps α^*:M(G)→ M(K) and the power operations
[m]:M(G)→ M(Σ_m≀ G) are monoid homomorphisms, the universal property
provides unique homomorphisms α^*:M^⋆(G)→ M^⋆(K) and 
[m]:M^⋆(G)→ M^⋆(Σ_m≀ G) such that

    α^*∘ j(G) = j(K)∘α^*      and    
     [m]∘ j(G) = j(Σ_m≀ G)∘ [m] .

The functoriality of the restriction maps α^* and the additional 
relations required of a global power monoid are relations between
monoid homomorphism; so they are inherited by M^⋆ via the universal property
of group completion of abelian monoids.

If R is an ultra-commutative monoid, then 
in Construction <ref> below we introduce 
a global topological version R^⋆ of the group completion.subjectglobal group completion!of an ultra-commutative monoid
This construction comes with a homomorphism of ultra-commutative monoids
R→ R^⋆ that realizes the algebraic group completion,
compare Proposition <ref>.



[Free ultra-commutative monoid of a global classifying space]subjectfree ultra-commutative monoid!of a global classifying space
We look more closely at the free ultra-commutative monoid (B_G) 
generated by the global classifying space B_ G 
of a compact Lie group G.subjectglobal classifying space
For every G-representation V, Example <ref> 
provides an isomorphism of orthogonal spaces

    _G,V^⊠ m  _G^m,V^m .

At an inner product space W, 
the permutation  action of Σ_m on the left hand side becomes
the action on

    _G^m,V^m (W)  =  (V^m,W)/G^m

by permuting the summands in V^m. 
Passage to Σ_m-orbits gives an isomorphism
  
    ^m(_G,V)  =  _G,V^⊠ m/Σ_m   _G^m,V^m /Σ_m   _Σ_m≀ G,V^m .

Thus

    (_G,V)  =  ∐_m≥ 0 _Σ_m≀ G,V^m .

If G acts faithfully on V and V 0, then the action of
Σ_m≀ G on V^m is again faithful. So in terms of global classifying spaces
the free ultra-commutative monoid generated by B_G is given by

    ( B_ G)  =  ∐_m≥ 0   B_(Σ_m≀ G)  .

 The tautological class u_G∈π_0^G(B_ G) is represented by the orbit of the
identity of V in

    (_G,V(V))^G  =  ((V,V)/G)^G ,

compare (<ref>).
So the class [m](u_G)∈π_0^Σ_m≀ G( ( B_ G))
is represented by the orbit of the identity of V^m in

    (_Σ_m≀ G,V^m(V^m))^G  =  ((V^m,V^m)/Σ_m≀ G)^Σ_m≀ G ;

so with respect to the identification (<ref>)
we have

    [m](u_G) =  u_Σ_m≀ G .




[Coproducts of ultra-commutative monoids]
The category of ultra-commutative monoids is cocomplete;
in particular, every family {R_i}_i∈ I
of ultra-commutative monoids has a coproduct that we denote
⊠_i∈ I'  R_i.
We claim that the functor

    _0 :   umon → (global power monoids)

preserves coproducts.
Indeed, if the indexing set I is finite, then the underlying orthogonal space
of the coproduct ⊠_i∈ I'  R_i is simply the iterated
box product of the underlying orthogonal spaces.
The functor _0 takes box products 
of orthogonal spaces to the objectwise product of Rep-functors,
by Corollary <ref>.
In the category of abelian monoids, finite products are also finite coproducts.
Coproducts of global power monoids are formed objectwise,
so a finite product of global power monoids is also a coproduct.
This proves the claim whenever the indexing set I is finite.

In any category, an infinite coproduct is the filtered colimit of the finite coproducts.
Moreover, filtered colimits of ultra-commutative monoids are
formed on underlying orthogonal spaces,
compare Corollary <ref>.
So if the set I is infinite, then the underlying orthogonal space of
the coproduct is the `infinite box product'
in the sense of Construction <ref>,subjectbox product!of orthogonal spaces!infinite i.e., the filtered colimit, 
formed over the poset of finite subsets of I, of the finite coproducts,

    ⊠_i∈ I'  R_i  _J⊂ I, J finite ⊠_j∈ J R_j .

Proposition <ref>
thus provides a bijection

    ∏'_i∈ I π_0^G(R_i) → π_0^G(⊠'_i∈ I R_i)

from the weak product of the abelian monoids π_0^G(R_i)
to the abelian monoid π_0^G(⊠'_i∈ I R_i).
For abelian monoids, the weak product is also the direct sum, i.e.,
the categorical coproduct.
Since colimits of global power monoids are calculated objectwise,
this proves the claim in general.




Here is another family of global power monoids, with underlying Rep-func­tor
represented by an abelian compact Lie group A.
In fact, the next proposition shows that (-,A) is
a free global power monoid subject to a specific set of explicit `power relations'.
In Construction <ref> below we exhibit a 
multiplicative model for the global classifying space of A, 
i.e., an ultra-commutative monoid that realizes 
the global power monoid (-,A) on _0.



subjectcompact Lie group!abelian
For every abelian compact Lie group A, the Rep-functor (-,A)
has a unique structure of global power monoid. 
The monoid structure of (G,A) is given by
pointwise multiplication of homomorphisms. The power operation

    [m] :  (G,A) → (Σ_m≀ G,A)

is given by

    [m](α) =  p_m∘ (Σ_m≀α) ,

where p_m:Σ_m≀ A→ A is the homomorphism defined by
p_m(σ; a_1,…,a_m) = a_1·…· a_m.
Moreover, for every global power monoid M the map  

    (global power monoids)((-,A),M)  →  M(A) ,
       f ⟼ f(A)(_A)

is injective with image those x∈ M(A) that satisfy [m](x)=p_m^*(x) 
for all m≥ 1.


Since A is abelian, conjugate homomorphisms into A are already equal,
i.e., we can ignore the difference between homomorphisms and conjugacy classes.
We establish the monoid structure and power opera­tions first, 
which also shows the uniqueness.
Since (e,A) has only one element, it is the additive unit.  
Since restriction maps are monoid homomorphisms,
the trivial homomorphism is the neutral element of (G,A).
The sum

    q_1 + q_2  ∈ (A× A,A)

of the two projections is a homomorphism from A× A to A
whose restriction along the two maps (-,1),(1,-):A→ A× A
is the identity. The only such homomorphism is the multiplication
μ:A× A→ A, so we conclude that

    q_1 + q_2  =  μ .

Naturality now gives

    α+β =   (α,β)^*(q_1) +(α,β)^*(q_2)  =  
    (α,β)^*(q_1+q_2)  =  (α,β)^*(μ)  =  μ∘(α,β)   .

Since power operations refine power maps, the element

    [m](_A) ∈ (Σ_m≀ A,A)

restricts to the sum of the m projections on A^m≤Σ_m≀ A. 
We let 1:e→ A be the unique homomorphism and claim that the composite

    Σ_m≀ e  Σ_m≀ 1 Σ_m≀ A [m](_A) A

is trivial. Indeed,

    [m](_A) ∘ (Σ_m≀ 1) =  
     (Σ_m≀ 1)^*([m](_A)) =   [m](1^*(_A)) =  [m](1) =  1  ,

using that the operation [m] is a monoid homomorphism. Thus 

    [m](_A)(σ; a_1,…,a_m)     =  
    [m](_A)(σ; 1,…,1) · [m](_A)(1; a_1,…,a_m) 
       =  a_1·…· a_m .

In other words, [m](_A)=p_m. Naturality now gives

    [m](α)  =   [m](α^*(_A))     =  (Σ_m≀α)^*([m](_A))
       =  (Σ_m≀α)^*(p_m) = p_m∘ (Σ_m≀α)  .


It remains to show the existence of the global power monoid structure.
Clearly, pointwise multiplication of homomorphisms makes (G,A) into
an abelian monoid (even an abelian group), and the monoid structure is
contravariantly functorial in G.
When we define [m](α) by the formula of the proposition,
then the remaining axioms of a global power monoid 
(compare Definition <ref>) are similarly straightforward.
The identity property (i) is clear, and naturality (ii) follows from the relation

    [m](α) =  (Σ_m≀α)^*([m](_A))  .

The transitivity relation (iii) holds by inspection:

    Ψ_k,m^*([k m](α))(σ;(τ_1;h^1),   …,(τ_k,h^k)) 
       =   
    ([k m](α))(Ψ_k,m(σ;(τ_1;h^1),…,(τ_k,h^k))) 
       =  ∏_i=1^k (∏_j=1^m h_j^i) =  ∏_i=1^k ([m](α))(τ_1;h^i) )
       =  ([k]( [m](α)))(σ;(τ_1;h^1),…,(τ_k,h^k))  ,

and so does the additivity relation (iv):

    Φ^*_i,m-i(  [m](α) )    ((σ; g_1,…,g_i),(σ'; g_i+1,…,g_m)) 
       =  ([m](α)) (σ+σ'; g_1,…,g_m) 
       =  α(g_1)·…·α(g_m) 
       =  ([i](α)) (σ; g_1,…,g_i) ·
    ([m-i](α)) (σ'; g_i,…,g_m) 
       = ([i](α)⊕ [m-i](α) )((σ; g_1,…,g_i),(σ'; g_i+1,…,g_m)) 
     .

It remains to identify the global power morphisms out of (-,A).
Since the class _A generates (-,A) as a -functor, 
the evaluation map in injective.
For surjectivity we consider a class x∈ M(A) such that [m](x)=p_m^*(x)
for all m≥ 1. The Yoneda lemma then provides a unique morphism of -functors

    f  :  (-,A) → M

such that f(A)(_A)=x,
and this morphism is given by f(G)(α)=α^*(x), for α:G→ A.
We need to show that f is a morphism of global power monoids, 
i.e., additive and compatible with power operations. 
For additivity we recall that

    α+β =  μ∘ (α,β)  =  
    p_2∘incl_A^2^Σ_2≀ A∘ (α,β)  .

Hence

    f(G)(α+β)    =  
    (p_2∘incl_A^2^Σ_2≀ A∘ (α,β))^*(x) 
       =  (α,β)^*(_A^2^Σ_2≀ A(p_2^*(x)))  
    =  (α,β)^*(_A^2^Σ_2≀ A([2](x))) 
       =  (α,β)^*(x⊕ x) =  α^*(x)+ β^*(x)  =  f(G)(α) + f(G)(β)  .

This shows that f is a morphism of abelian Rep-monoids.
Finally, given a continuous homomorphism α:G→ A, we have

    [m](f(G)(α))    =  [m](α^*(x)) =  (Σ_m≀α)^*([m](x))  =  (Σ_m≀α)^*(p_m^*(x))
       = (p_m∘(Σ_m≀α))^*(x) 
    = ( [m](α) )^*(x) =  f(Σ_m≀ G)([m](α)) .

So the morphism f is also compatible with power operations.



Now we are going to show that the restriction maps along continuous group
homomorphisms and the power operations give all natural operations between
equivariant homotopy sets of ultra-commutative monoids. 
The strategy is the same as in the analogous situation for orthogonal spaces, 
see Corollary <ref>:
natural operations for ultra-commutative monoids from the functor π_0^G 
to the functor π_0^K biject with the K-equivariant homotopy set 
of (B_G), the free ultra-commutative monoids
generated by a global classifying space of G. 
So ultimately we need to calculate π_0^K((B_G)).

The tautological class u_G in π_0^G(B_G)
was defined in (<ref>).
We set

    u_G^umon =  η_*(u_G ) ∈ π_0^G((B_ G)) ,

where η:B_ G→(B_ G) is the adjunction unit, i.e., the
inclusion of the homogeneous summand for m=1. 
The next theorem says in particular that 
the global power monoid _0( (B_ G)) is freely generated 
by the element u_G^umon.



  Let G and K be compact Lie groups.subjectglobal classifying space
  
  
  * Every class in π_0^K((B_G)) is of the form α^*([m](u_G^umon))
    for a unique m≥ 0 and a unique conjugacy class of continuous homomorphisms
    α:K→Σ_m≀ G.
  
  * For every global power monoid M and every x∈ M(G)
    there is a unique morphism of global power monoids 
    f:_0((B_ G))→ M such that 
    
    f(G)(u_G^umon) =  x  .

  
  * Every natural transformation π_0^G →π_0^K
  of set valued functors on the category of ultra-commutative monoids 
  is of the form α^*∘ [m] for a unique m≥ 0 and
  a unique conjugacy class of continuous group homomorphisms α:K→Σ_m≀ G.



  For the course of the proof we abbreviate u=u_G^umon. 

  (i) By (<ref>) the underlying orthogonal space of ( B_ G)
  is the disjoint union of global classifying spaces for the wreath product
  groups Σ_m≀ G; moreover, the class [m](u)
  lies in the m-th summand of ( B_ G)
  and is a universal element for B_(Σ_m≀ G), by (<ref>). 
  So part (i) follows from Proposition <ref> (ii)
  and the fact that π_0^K commutes with disjoint unions. 
  
  
  (ii) By (i) every element of π_0^K((B_ G)) is of the form α^*([m](u));
  every morphism of global power monoids
  f:_0((B_ G))→ M satisfies
  
    f(K)(α^*([m](u)))  =  α^*([m](f(G)(u)))  ,

  so f is determined by its value on the class u.
  This shows uniqueness.

  Conversely, if x∈ M(G) is given we define f(K):π_0^K( (B_G))→ M(K)
  by the formula
  
    f(K)(α^*([m](u)))  =  α^*([m](x))  .

  Then f(G)(u)=x.
  It remains to show that f is indeed a morphism of global power monoids. 
  This is a routine  – but somewhat lengthy – calculation as follows.
  Given α:K→Σ_m≀ G and  α̅:K→Σ_n≀ G,
  we have
  
    α^*([m](u)) +  α̅^*([n](u))     =  (α,α̅)^*(p_1^*([m](u)) + p_2^*([n](u))) 
       =  (α,α̅)^*(Φ_m,n^*([m+n](u))) 
       =  (Φ_m,n∘(α,α̅))^*([m+n](u))  .

  So f(K) is additive because
  
    f(K)(α^*([m](u)) +  α̅^*([n](u)))     =  
        f(K)((Φ_m,n∘(α,α̅))^*([m+n](u)) )
       =  (Φ_m,n∘(α,α̅))^*([m+n](x))
       =  (α,α̅)^*(Φ_m,n^*([m+n](x)))
       =  (α,α̅)^*( p_1^*([m](x)) + p_2^*([n](x)))
       =  α^*([m](x)) +  α̅^*([n](x))
       =  f(K)(α^*([m](u))) +  f(K)(α̅^*([n](u))) .

  For a continuous homomorphism β:L→ K we have
  
    (β^*∘  f(K))(α^*([m](u)))     =  β^*(α^*([m](x)))  
        =  (α∘β)^*([m](x)) 
       =  f(L)( (α∘β)^*([m](u)))  
        =  (f(L)∘β^*)( α^*([m](u))) ;

  so the homomorphisms f(K) form a natural transformation of -functors.
  For k≥ 1 we have
  
    [k](α^*([m](u)))    =  
        (Σ_k≀α^*)([k]([m](u)))
       =   (Σ_k≀α^*)(Ψ_k, m^*([k m](u))) 
        =   (Ψ_k,m∘(Σ_k≀α))^*([k m](u)) ;

  hence
  
    f(Σ_k≀ K)([k](α^*   ([m](u)))) 
        =  (Ψ_k,m∘ (Σ_k≀α))^*([k m](x)) 
       =  (Σ_k≀α)^*(Ψ_k,m^*( [k m](x)))  
        =  (Σ_k≀α)^*([k]([m](x))) 
       =  [k](α^*([m](x)))  
        = [k]( f(K)(α^*([m](u))))  .

  So the homomorphisms f(K) are compatible with power operations.

  (iii) 
  We apply the representability result 
  of Proposition <ref>
  to the category of ultra-commutative monoids and
  the free and forgetful adjoint functor pair: 
  
    @C=10mm :   @<.4ex>[r]      umon :  U
        @<.4ex>[l]

  If G is a compact Lie group, V a G-representation and W
  a non-zero faithful G-representation, then the restriction morphism
  ρ_G,V,W:_G,V⊕ W→_G,W is a global equivalence
  between positive flat orthogonal spaces. We showed in the proof of
  Theorem <ref> (ii)
  that the induced morphism of free ultra-commutative monoids
  (ρ_G,V,W):(_G,V⊕ W)→(_G,W) is a global equivalence;
  in particular, the morphism of -functors
  _0((ρ_G,V,W)) is an isomorphism. So 
  Proposition <ref> applies and shows that
  evaluation at the tautological class is a bijection
  
    ^umon(π_0^G,π_0^K)   → π_0^K((B_ G))  ,   τ ⟼ τ(u) .

  Part (i) then completes the argument.



[Natural n-ary operations]
By similar arguments as in the previous theorem
we can also identify the natural n-ary operations on equivariant homotopy sets
of ultra-commutative monoids. For every n-tuple G_1,…, G_n 
of compact Lie groups the functor

    (umon) → (sets) ,  
    X  ⟼ π_0^G_1(X)×…×π_0^G_n(X)

is represented by the free ultra-commutative monoid
(B_ G_1 ⨿…⨿  B_ G_n).
So the set of natural transformations from the functor
π_0^G_1×…×π_0^G_n to the functor π_0^K,
for another compact Lie group K, bijects with the K-equivariant homotopy set
of this representing object.
Because

    (B_ G_1 ⨿…⨿  B_ G_n)    (B_ G_1)⊠…⊠( B_ G_n)
        ∐_j_1,…,j_n≥ 0 
    B_ (Σ_j_1≀ G_1)⊠…⊠ B_ (Σ_j_n≀ G_n)
        ∐_j_1,…,j_n≥ 0 
    B_ ( (Σ_j_1≀ G_1)×…×  (Σ_j_n≀ G_n)) ,

the group
π_0^K( (B_ G_1 ⨿…⨿  B_ G_n))
bijects with the disjoint union of the sets

    π_0^K(B_ ( (Σ_j_1≀ G_1)×…×  (Σ_j_n≀ G_n)))  (K, (Σ_j_1≀ G_1)×…×  (Σ_j_n≀ G_n)) .

So every natural operation from π_0^G_1×…×π_0^G_n to π_0^K
is of the form

    (x_1,…,x_n)  ⟼ α^*( [j_1](x_1) ⊕…⊕ [j_n](x_n))

for a unique tuple (j_1,…, j_n) of non-negative integers
and a unique conjugacy class of continuous homomorphisms
α:K→  (Σ_j_1≀ G_1)×…×  (Σ_j_n≀ G_n).
In particular, the n-ary operations are generated by unary operations
and external sum.





We describe an alternative (but isomorphic) way 
to organize the book keeping of the natural operations between the
0-th equivariant homotopy sets of ultra-commutative monoids.
We denote by ^umon the category whose objects are all compact Lie groups
and where the morphism set ^umon(G,K) is the set of all natural
transformations, of functors from the ultra-commutative monoids to sets, 
from π_0^G to π_0^K.
We define an isomorphic algebraic category 𝔸^+, 
the effective Burnside category.subjecteffective Burnside category
Both ^umonand 𝔸^+ are
`pre-preadditive' in the sense that all morphism sets are abelian monoids 
and composition is biadditive. In ^umon, the monoid structure is
objectwise addition of natural transformations.

The category 𝔸^+ has the same objects as ^umon, 
namely all compact Lie groups. 
In the effective Burnside category, the morphism set
^+(G,K)symbol^+(G,K) - monoid of isomorphism classes of G-free K-G-sets 
is the set of isomorphism classes of those K-G-spaces
that are disjoint unions of finitely many free right G-orbits.
This set is an abelian monoid via disjoint union of K-G-spaces.
Composition
is induced by the balanced product over K:

    ∘ :  ^+(K,L) ×^+(G,K)  → ^+(G,L) ,   
     [T]∘ [S]  =   [ T×_K S]  .
 
Here T has a left L-action and a commuting free right K-action,
whereas S has a left K-action and a commuting free right G-action.
The balanced product T×_K S then inherits a left L-action from T
and a free right G-action from S.
We define a functor

    B  :  ^umon → 𝔸^+

as the identity on objects; 
on morphisms, the functor is given by

    B  :  ^umon(G,K)  → 𝔸^+(G,K) ,    
    B(α^*∘[m]) =  [ α^* ({1,…,m}× G)_G]  .

In the definition we use the characterization of the natural operations
given by Theorem <ref> (iii).
Also, we consider 
{1,…,m}× G as a right G-space by translation;
the wreath product Σ_m≀ G acts from the left on 
{1,…,m}× G by 

    (σ; g_1,…,g_m)· (i,γ) =  (σ(i),  g_i·γ)  .

Then we let K act by restriction of the (Σ_m≀ G)-action along α.




 The functor B: ^umon→𝔸^+ is additive and an isomorphisms of categories.


We start by showing that the map B:^umon(G,K)→^+(G,K) is additive.
The map

    ({1,…,k}× G)  ⨿ ({1,…,m}× G)    → Φ_k,m^*({1,…,k+m}× G)

that is the inclusion on the first summand and given by
(j,g)↦(k+j,g) on the second summand
is an isomorphism of ((Σ_k≀ G)× (Σ_m≀ G))-G-bispaces.
Restriction along the homomorphism 
(β,α):K→ (Σ_k≀ G)× (Σ_m≀ G)
provides an isomorphism of K-G-spaces between

    β^*({1,…,k}× G)_G  ⨿ α^*({1,…,m}× G)_G  and 
    (Φ_k,m∘ (β,α))^*({1,…,k+m}× G)_G   .

This shows that

    B((β^*∘[k])+(α^*∘[m])) =  
     B((Φ_k,m∘(β,α))^*∘[k+m] ) =  
     B(β^*∘[k]) + B(α^*∘[m]) .

The identity operation in ^umon(G,G) can be written as _G^*∘[1];
on the other hand, the G-bispace _G^*({1}× G)_G is isomorphic to G
under left and right translation. Since the isomorphism class of _G G_G is the
identity of G in 𝔸^+, the construction B preserves identities.

For the compatibility of B with composition we consider another
operation β^*∘[k]∈^umon(K,M) and observe that 

    (β^*∘[k])∘(α^*∘[m]) =  
    (Ψ_k,m∘(Σ_k≀α)∘β)^*∘[ k m]  .

Moreover, an isomorphism of M-G-spaces

    β^* ({1,…,k}× K) ×_K 
    α^* ({1,…,m}× G)   
    (Ψ_k,m∘ (Σ_k≀α)∘β)^*({1,…,k m}× G)

is given by

    [ ( i, κ), (j, γ) ]  ⟼ 
    ( (i-1)m + σ(j)-1, g_j·γ)  ,

where

    α(κ) =  (σ; g_1,…,g_m) ∈ Σ_m≀ G .

So B is a functor, which is bijective on objects by definition.

To see that B is full we let S be any K-G-space that is 
a disjoint union of m free right G-orbits.
We choose a G-equivariant homeomorphism

    ψ :  S  → {1,…,m}× G  .

We transport the left K-action from S to 
{1,…,m}× G along ψ, so that ψ becomes an
isomorphism of K-G-spaces. The (Σ_m≀ G)-action 
on {1,…,m}× G defined in (<ref>)
identifies the wreath product with the group of 
G-equivariant automorphisms of {1,…,m}× G;
so the K-action on {1,…,m}× G corresponds to a continuous homomorphism
α:K→Σ_m≀ G. Altogether, S is isomorphic 
to α^*({1,…,m}× G)_G.

If the K-G-spaces constructed from α^*∘[m] and β^*∘[n]
are isomorphic, then we must have m=n. Moreover, a K-G-isomorphism

    α^*({1,…,m}× G)_G  β^*({1,…,m}× G)_G

is given by the action of a unique element ω∈Σ_m≀ G,
and then the homomorphisms α,β:K→Σ_m≀ G
are conjugate by ω. So the functor B is faithful.



Now we define transfer maps
_H^G:M(H)→ M(G) in global power monoids,
for every subgroup H of finite index in a compact Lie group G.
As we will see in Proposition <ref> 
below, the set of operations from π_0^G to π_0^K 
is a free abelian monoid with an explicit basis involving transfers.


[Transfer maps]
In the following we let M be a global power monoid,
G a compact Lie group and H a closed subgroup of G
of finite index m.
We choose an `H-basis' of G, i.e., an ordered m-tuple 
g̅=(g_1,…,g_m) of elements in disjoint H-orbits such that

    G  =  ⋃_i=1^m  g_i H  .

The wreath product Σ_m≀ H acts freely and transitively 
from the right on the set of all such H-bases of G, by the formula

    (g_1,…,g_m)· (σ; h_1,…,h_m) =  
    (g_σ(1) h_1,…,g_σ(m)h_m) .

We obtain a continuous homomorphism Ψ_g̅:G→Σ_m≀ H by requiring that

    γ·g̅ =  g̅·Ψ_g̅(γ)   .

We define the transfer _H^G:M(H)→ M(G)subjecttransfer!in global power monoidssymbol_H^G - transfer map as the composite

    M(H)   [m]  M(Σ_m≀ H)   Ψ_g̅^*  M(G)  .

Any other H-basis is of the form
g̅ω for a unique ω∈Σ_m≀ H. 
We have Ψ_g̅ω=c_ω∘Ψ_g̅ as maps G→Σ_m≀ H,
where c_ω(γ)=ω^-1γω. Since inner automorphisms
induce the identity in any Rep-functor, we conclude that

    Ψ_g̅^* = Ψ_g̅ω^*  :  M(Σ_m≀ H)  → M(G) .

So the transfer _H^G does not depend on the choice of basis g̅. 






The various properties of the power operations
imply certain properties of the transfer maps.
Moreover, the last item of the following proposition shows that
power operations in a global power monoid are determined by the transfer
and restriction maps.
 
 
The transfer homomorphisms of a global power monoid M satisfy the following relations,
where H is any subgroup of finite index in a compact Lie group G.


  * (Transitivity) 
We have _G^G=_M(G) and for nested subgroups
H⊆ G⊆ F of finite index the relation

    _G^F ∘_H^G  =  _H^F

holds as maps  M(H)→ M(F).

  * (Double coset formula)
For every subgroup K of G (not necessarily of finite index)
the relationsubjectdouble coset formula!for transfer map

    ^G_K∘_H^G   =  ∑_[g]∈ K\ G/H^K_K∩^g H∘ (c_g)^* ∘^H_K^g∩ H

holds as homomorphisms M(H)→ M(K).
Here [g] runs over a set of representatives of the finite
set of K-H-double cosets.

  * (Inflation)
For every continuous epimorphism  α:K→ G of compact Lie groups
the relation

    α^*∘_H^G  = _L^K∘ (α|_L)^*

holds as maps from M(H) to M(K), where L=α^-1(H).

  * For every m≥ 1 the m-th power operation can be recovered as

    [m]  = _K^Σ_m≀ G∘ q^*  ,

where K is the subgroup of Σ_m≀ G
consisting of all (σ; g_1,…,g_m) such that σ(m)=m
and q:K→ G is defined by q(σ; g_1,…,g_m)=g_m.



(i)
For G=H we can choose the unit 1 as the G-basis,
and with this choice Ψ_1:G→Σ_1≀ G 
is the preferred isomorphism that sends g to (1;g). 
The restriction of [1](x) along this isomorphism is x,
so we get _G^G(x)=x.

For the second claim we choose a G-basis f̅=(f_1,…,f_k) of F
and an H-basis g̅=(g_1,…,g_m) of G.
Then 

    f̅g̅ =  (f_1 g_1,…,f_1 g_m, f_2 g_1,…, f_2 g_m,
     …, f_k g_1,…,f_k g_m)

is an H-basis of F. With respect to this basis, the homomorphism 
Ψ_f̅g̅:F→Σ_k m≀ H equals the composite

    F  Ψ_f̅ Σ_k≀ G
     Σ_k≀Ψ_g̅Σ_k≀(Σ_m≀ H)    Ψ_k,m Σ_k m≀ H

where the monomorphism Ψ_k,m was defined in (<ref>). 
So 

    _H^F     =  Ψ_f̅g̅^*∘  [k m]  =  Ψ_f̅^*∘ (Σ_k≀Ψ_g̅)^*∘Ψ_k,m^*∘[k m] 
       =  Ψ_f̅^*∘ (Σ_k≀Ψ_g̅)^*∘ [k] ∘[m] =  Ψ_f̅^*∘ [k]∘Ψ_g̅^* ∘[m] =  _G^F∘_H^G  .
 


(ii) We choose representatives g_1,…,g_r for the K-H-double cosets
in G. Then we choose, for each 1≤ i≤ r, a (K∩^g_iH)-basis

    k̅^i  =  (k^i_1,…,k^i_s_i)

of K, where s_i=[K:K∩^g_iH]. 
Then s_1+…+s_r=m=[G:H] is the index of H in G, 
and this data provides an H-basis of G, namely

    g̅ =  (k^1_1 g_1, …,k^1_s_ig_1, 
    k^2_1 g_2, …,k^2_s_2g_2, …,
    k^r_1 g_r, …,k^r_s_rg_r) .

The following diagram of group homomorphisms then commutes:

    @C=10mm 
    K [r]^-incl[d]_(Ψ_k̅^1,…,Ψ_k̅^r)   
    G [r]^-Ψ_g̅   Σ_m≀ H 
    ∏_i=1^r  Σ_s_i≀(K∩^g_iH)
    [rr]_-∏Σ_s_i≀ c_g_i      ∏_i=1^r Σ_s_1≀ H[u]_Φ_s_1,…,s_r

The right vertical morphism is the generalization of
the embedding (<ref>) to multiple factors.
From here the double coset formula is straightforward:

    ^G_K∘_H^G      =  ^G_K∘Ψ_g̅^*∘[m] 
       =  ((Σ_s_1≀ c_g_1)∘Ψ_k̅^1,…,(Σ_s_r≀ c_g_r)∘Ψ_k̅^r)^*∘Φ_s_1,…,s_r^*∘[m] 
     
    _(<ref>)   =   ((Σ_s_1≀ c_g_1)∘Ψ_k̅^1,…,(Σ_s_r≀ c_g_r)∘Ψ_k̅^r)^*∘( [s_1]⊕…⊕[s_r] )  
       =  ∑_i=1^r Ψ_k̅^i^*∘ (Σ_s_i≀ c_g_i)^*∘ [s_i]  
       =  ∑_i=1^r Ψ_k̅^i^*∘ [s_i]∘ (c_g_i)^*  
    =  ∑_i=1^r ^K_K∩^g_i H∘ (c_g_i)^* ∘^H_K^g_i∩ H .


(iii) If k̅=(k_1,…,k_m) is an L-basis of K,
then α(k̅)= (α(k_1),…,α(k_m)) is an H-basis of G.
With respect to these bases we have

    Ψ_α(k̅)∘α =  (Σ_m≀ (α|_L)) ∘Ψ_k̅ :  K  → Σ_m≀ H  .

So 

    α^*∘_H^G  =  α^*∘Ψ^*_α(k̅)∘ [m]
        =  Ψ_k̅^*∘(Σ_m≀ (α|_L))^* ∘  [m] 
       =  Ψ_k̅^*∘  [m]∘ (α|_L)^*  =  _L^K ∘ (α|_L)^*   .


(iv) The subgroup K has index m in Σ_m≀ G and a K-basis
of Σ_m≀ G is given by the elements τ_j=((j, m); 1,…,1)
for j=1,…,m. 
In order to determine the monomorphism Ψ_τ̅:Σ_m≀ G→Σ_m≀ K
associated to this K-basis,
we consider any element (σ; g_1,…,g_m) of Σ_m≀ G.
The permutation (σ(j), m)·σ· (j, m)∈Σ_m 
fixes m, so the element

    l_j  =  τ_σ(j)· (σ; g_1,…,g_m)·τ_j 
     ∈ Σ_m≀ G

in fact belongs to the subgroup K.
Then 

    (σ; g_1,…,g_m)·τ_j  =  τ_σ(j)· l_j

in the group Σ_m≀ G, by definition. This means that

    (σ; g_1,…,g_m)· (τ_1,…,τ_m)  =  
    (τ_1,…,τ_m) · (σ; l_1,…,l_m)  ,

and hence

    Ψ_τ̅(σ; g_1,…,g_m)  =  (σ; l_1,…,l_m)  .

Because

    (Σ_m≀ q)(Ψ_τ̅(σ; g_1,…,g_m)) =  
    (Σ_m≀ q)(σ; l_1,…,l_m)  =(σ; g_1,…,g_m)  ,

we conclude that the composite (Σ_m≀ q)∘Ψ_τ̅
is the identity of Σ_m≀ G. So

    _K^Σ_m≀ G∘ q^*  =  Ψ_τ̅^*∘ [m]∘ q^*     =  Ψ_τ̅^*∘ (Σ_m≀ q)^*∘ [m] 
       =  ((Σ_m≀ q)∘Ψ_τ̅)^*∘ [m]  =  [m] .




Theorem <ref> (iii) gives a description of the
set of natural operations, on the category of ultra-commutative monoids, 
from π_0^G to π_0^K. The next proposition gives an alternative
description that also captures the monoid structure given by pointwise
addition of operations.


  Let G and K be compact Lie groups.
  The monoid ^umon(π_0^G,π_0^K) is a free abelian monoid
  generated by the operations _L^K∘α^*
  where (L,α) runs over all (K× G)-conjugacy classes of pairs 
  consisting of
  
  
  * a subgroup L≤ K of finite index, and
  
  * a continuous group homomorphism α:L→ G.
      


This a straightforward algebraic consequence of the calculation
of the category ^umon given in Proposition <ref>.
Every K-G-space with finitely many free G-orbits is the disjoint union
of transitive K-G-spaces with the same property. So 𝔸^+(G,K)
is a free abelian monoid with basis the isomorphism classes of the
transitive K-G-spaces. A transitive K-G-space 
with finitely many free G-orbits is isomorphic to one of the form

    K×_α G  =  (K× G)/ (k l,g)∼ (k,α(l) g)

for a pair (L,α:L→ G), with L of finite index in K.
Moreover, K×_α G is isomorphic to
K×_α' G if and only if (L,α) is conjugate to
(L',α') by an element of K× G.
So 𝔸^+(G,K) is freely generated by the classes of the
K-G-spaces K×_α G, where (L,α) 
runs through the (K× G)-conjugacy classes of the relevant pairs.

The claim then follows from the verification that the isomorphism
of categories B:Nat^umon→^+ 
established in Proposition <ref>
takes the operation _L^K∘α^* to the class of K×_α G. 
Indeed, if k̅=(k_1,…,k_m) is an L-basis of K and
Ψ_k̅:K→Σ_m≀ L the classifying homomorphism, then 

    B(_L^K∘α^*)  =  B(Ψ^*_k̅∘[m]∘α^*)
     =   B(Ψ^*_k̅∘(Σ_m≀α)^*∘[m]) .

On the other hand, the map

    ((Σ_m≀α)∘Ψ_k̅)^*({1,…,m}× G)_G  → 
    K×_α G  ,    (i,γ) ⟼ [k_i, γ]

is an isomorphism of K-G-bispaces, 
so B(_K^K∘α^*)=[K×_α G] in 𝔸^+(G,K).  






§ EXAMPLES


In this section we discuss various examples of ultra-commutative monoids,
mostly of a geometric nature, and several geometrically defined morphisms between them.
We start with the ultra-commutative 
monoids  and  (Example <ref>), 
 and  (Example <ref>), 
 (Example <ref>) 
and  and ^c (Example <ref>),
all made from the corresponding families of classical Lie groups.
All of these are examples of 
`symmetric monoid valued orthogonal spaces' 
in the sense of Definition <ref>,
a more general source of examples of ultra-commutative monoids.
The additive Grassmannian  (Example <ref>),
the oriented variant ^ (Example <ref>)
and the complex and quaternionic analogs ^ and ^ 
(Example <ref>)
consist – as the names suggest – of Grassmannians with monoid
structure arising from direct sum of subspaces.
The multiplicative Grassmannian _ (Example <ref>)
is globally equivalent as an orthogonal space to , but the 
monoid structure arises from tensor product of subspaces;
the global projective space  is the ultra-commutative submonoid
of _ consisting of lines (1-dimensional subspaces).
The global projective space  is a multiplicative model of
a global classifying space for the cyclic group of order 2, and
Example <ref> describes multiplicative models
of global classifying spaces for all abelian compact Lie groups.
Example <ref> introduces
the ultra-commutative monoid  of unordered frames, with monoid structure
arising from disjoint union.
The ultra-commutative `multiplicative monoid of the sphere spectrum'
Ω^∙ is introduced in Example <ref>;
this a special case of the multiplicative monoid of an ultra-commutative
ring spectrum, and we return to the more general construction 
in Example <ref>. 


[Orthogonal monoid spaces from monoid valued
orthogonal spaces]
Our first series of examples involves orthogonal spaces made from
the infinite families of classical Lie groups, 
namely the orthogonal, unitary and symplectic groups,
the special orthogonal and unitary groups, and the pin, pin^c, spin and spin^c groups. 
These orthogonal spaces have the special feature that they are group valued;
we will now explain that a group valued (or even just a monoid valued)
orthogonal space automatically leads
to an orthogonal monoid space. In the cases of the orthogonal, special orthogonal,
unitary, special unitary, spin, spin^c and symplectic groups, 
these multiplications are symmetric, so those examples yield ultra-commutative monoids.

 
A monoid valued orthogonal spacesubjectorthogonal space!monoid valued 
is a monoid object in the category of orthogonal spaces.


Since orthogonal spaces are an enriched functor category, monoid valued orthogonal 
spaces are the same thing as continuous functors from the category 
to the category of topological monoids 
and continuous monoid homomorphisms
(i.e., monoid objects in the category  of compactly generated spaces). 

Now we let M be a monoid valued orthogonal space. 
In (<ref>)
we introduced a lax symmetric monoidal natural transformation

    ρ_X,Y :  X⊠ Y → X× Y

from the box product to the cartesian product of orthogonal spaces.
For X=Y=M we can form the composite 

    M⊠ M  ρ_M,M  M× M     multiplication M

with the objectwise multiplication of M.
Since ρ_X,Y is lax monoidal, this composite makes M into
an orthogonal monoid space with unit the multiplicative unit 1∈ M(0).
The morphism (<ref>) corresponds, 
via the universal property of ⊠, to the bimorphism
whose (V,W)-component is the composite

    M(V)× M(W) M(i_V)× M(i_W) M(V⊕ W)× M(V⊕ W) multiply M(V⊕ W) ,

where i_V:V→ V⊕ W and i_W:W→ V⊕ W are the direct summand embed­dings. 
If f:M→ M' is a morphism of monoid valued orthogonal spaces
(i.e., a morphism of orthogonal spaces that is objectwise a monoid homomorphism),
then f is also a homomorphism of orthogonal monoid spaces
with respect to the multiplications (<ref>).



If M is a commutative monoid valued orthogonal space, then the associated
⊠-multiplication is also commutative, simply because the transformation
ρ_X,Y is symmetric monoidal. However, there is a more general
condition that provides M with the structure of an ultra-commutative monoid.

 
A monoid valued orthogonal space M 
is symmetricsubjectorthogonal space!monoid valued!symmetric 
if for all inner product spaces V and W the images of the two homomorphisms

    M(i_V) :  M(V) →  M(V⊕ W)   and  
    M(i_W) :  M(W) →  M(V⊕ W)

commute. 


We emphasize that the objectwise multiplications in a symmetric 
monoid valued orthogonal space need not be commutative – 
we'll discuss many interesting examples of this kind below.
The proof of the following proposition is straightforward from the definitions,
and we omit it.



Let M be a symmetric monoid valued orthogonal space.
Then the multiplication (<ref>)
makes M into an ultra-commutative monoid.



[Orthogonal group ultra-commutative monoid]
subjectorthogonal group ultra-commutative monoid
We denote by symbol - ultra-commutative monoid of orthogonal groups 
the orthogonal space that sends an inner product space V
to its orthogonal group O(V).subjectorthogonal group
A linear isometric embedding φ:V→ W induces a continuous group
homomorphism (φ):(V)→(W) by conjugation
and the identity on the orthogonal complement of the image of φ.
Construction <ref> then gives  the
structure of an orthogonal monoid space.
The (V,W)-component of the bimorphism 

    μ_V,W :  (V)×(W) → (V⊕ W)

is direct sum of orthogonal transformations.
The unit element is the identity of the trivial vector space, 
the only element of (0).
So  is a symmetric group valued orthogonal space,
and hence it becomes an ultra-commutative monoid.

If G is a compact Lie group and V a G-representation, 
then the G-action on the group (V) is by conjugation,
so the fixed points (V)^G are the group of G-equivariant orthogonal
automorphisms of V.
Moreover, (_G) is the orthogonal group of _G, 
i.e., -linear isometries of _G
(not necessarily G-equivariant) that are the identity on the orthogonal 
complement of some finite-dimensional subspace;
the G-action is again by conjugation.
Any G-equivariant isometry preserves the decomposition of _G
into isotypical summands, and the restriction to
almost all of these isotypical summands must be the identity.
The G-fixed subgroup is thus given by

    (_G)^G =  O^G(_G)  =  ∏'_[λ]  O^G(_λ) ,

where the weak product is indexed by the isomorphism classes of irreducible
orthogonal G-representations λ, and _λ 
is the λ-isotypical summand. 
If the compact Lie group G is finite, 
then there are only finitely many isomorphism classes 
of irreducible G-representations,
so in that case the weak product coincides with the product.

Irreducible orthogonal representations come in three different
flavors, and the group  O^G(_λ) has one of three different forms.
If λ is an irreducible orthogonal G-representation, then
the endomorphism ring ^G_(λ,λ) is a finite-dimensional
skew field extension of , so it is isomorphic to either ,  
or ; the representation λ is accordingly called `real', `complex'
respectively `quaternionic'. We have

    O^G(_λ)   O^G(λ^∞)   O     if λ is real,
     U     if λ is complex, and
    
      Sp     if λ is quaternionic.

So we conclude that the G-fixed point space (_G)^G
is a weak product of copies of the infinite orthogonal, unitary
and symplectic groups, indexed by the different types 
of irreducible orthogonal representations of G.
Since the infinite unitary and symplectic groups are connected, 
only the `real' factors contribute to π_0((_G)^G)= π_0^G(),
which is a weak product of copies of π_0(O)=/2 indexed
by the irreducible G-representations of real type.

There is a straightforward `special orthogonal' analog:
the property of having determinant 1 is preserved under conjugation by
linear isometric embeddings and under direct sum of linear isometries,
so the spaces SO(V)subjectspecial orthogonal groupsubjectorthogonal group!special|seespecial orthogonal group
form an ultra-commutative 
submonoid symbol - ultra-commutative monoid of special orthogonal groups of .
Here, (_G) is the group of -linear isometries of _G
(not necessarily G-equivariant) that have determinant 1 on some finite
dimensional subspace V of _G and are the identity on the orthogonal 
complement of V.


[Unitary group ultra-commutative monoid]
subjectunitary group ultra-commutative monoid
There is a straightforward unitary analog symbol - ultra-commutative monoid of unitary groups 
of , defined as follows. 
We recall that

    V_ =  _ V

denotes the complexification of an inner product space V.
The euclidean inner product -,- on V induces a hermitian
inner product (-,-) on V_, defined
as the unique sesquilinear form that satisfies

    (1 v, 1 w)  =  v,w

for all v, w∈ V.
We now define an orthogonal space  by

    (V) =  U(V_)  ,

the unitary group of the complexification of V.subjectunitary group
The complexification of every -linear isometric embedding φ:V→ W 
preserves the hermitian inner products, 
so we can define a continuous group homomorphism

    (φ) : (V) → (W)

by conjugation with φ_:V_→ W_
and the identity on the orthogonal complement of the image of φ_.
The ⊠-multiplication on 
produced by Construction <ref>
is by direct sum of unitary transformations;
this multiplication is symmetric, and hence ultra-commutative.
There is a straightforward `special unitary' ultra-commutative
submonoid  of ; the value (V) is the group of
unitary automorphisms of V_ of determinant 1.symbol - ultra-commutative monoid of special unitary groupssubjectspecial unitary groupsubjectunitary group!special|seespecial unitary group

If G is a compact Lie group, then the identification of
the G-fixed points of  also works much like the orthogonal analog.
The outcome is an isomorphism between (_G)^G and U^G(_G^).
Here _G^=__G is the complexified complete universe
for G, which happens to be a `complete complex G-universe' in
the sense that every finite-dimensional complex G-representation
embeds into it.
The complete complex G-universe breaks up into (unitary)
isotypical summands _λ^,
indexed by the isomorphism classes of irreducible
unitary G-representations λ, and the group U^G(_G^)
breaks up accordingly as a weak product.
In contrast to the orthogonal situation above, there is only one `type'
of irreducible unitary representation, and the group U^G(^_λ) 
is always isomorphic to the infinite unitary group U, independent of λ.
So in the unitary context, we get a decomposition

    (_G)^G =  U^G(_G^)  =  ∏'_[λ] U^G(_λ^)  ∏'_[λ]  U   .

This weak product is indexed by the isomorphism classes of irreducible
unitary G-representations. Since the unitary group U is connected,
the set π_0^G()=π_0((_G)^G) has only one element,
and so  is globally connected.

The orthogonal monoid space  comes with an involution

    ψ :   →

by complex conjugationsubjectcomplex conjugation!on 
that is an automorphism of ultra-commutative monoids.
The value of ψ at V is the map

    ψ(V)  :  U(V_)  → U(V_)  ,    
    A  ⟼ ψ_V∘ A∘ψ_V  ,

where 

    ψ_V :  V_ → V_ ,   λ v ⟼ λ̅ v

is the canonical -semilinear conjugation map on V_.

The complexification morphismsubjectcomplexification morphism!from  to 
is the homomorphism of ultra-commutative monoids given by

    c  :   →  ,   
     c(V) :  O(V) → U(V_)  ,   φ ⟼ φ_ .

Complexification is an isomorphism onto the ψ-invariant ultra-commutative submonoid 
of , and it takes  to .

Every hermitian inner product space W has an underlying 
-vector space equipped with the euclidean inner product
defined by

    v, w =  Re ( v,w )  ,

the real part of the given hermitian inner product. 
Every -linear isometric embedding is in particular
an -linear isometric embedding of underlying euclidean vector spaces.
So the unitary group U(W) is a subgroup of the special orthogonal group
of the underlying euclidean vector space of W. We can thus define 
the realification morphismsubjectrealification morphism!from  to 

    r  :   → ^_ (  )

at V as the inclusion
r(V): U(V_)→ O( V).
Here ^_ denotes the multiplicative shift by 
as defined in Example <ref>.



[Symplectic group ultra-commutative monoid]
subjectsymplectic group ultra-commutative monoidsubjectsymplectic group
There is also a quaternionic analog of  and ,
the ultra-commutative monoid  made from symplectic groups.symbol - ultra-commutative monoid of symplectic groups 
Given an -inner product space V, we denote by

    V_ =  _ V

the extension of scalars to the skew field  of quaternions.
The extension comes with a -sesquilinear form 

    [-,-] :  V_× V_ →    characterized by  
     [1 v, 1 w] =  v,w

for all v,w∈ V. The symplectic group

    (V) =  Sp(V_)

is the compact Lie group of -linear automorphisms A:V_→ V_
that leave the form invariant, i.e., such that

    [A x, A y]  =  [x,y]

for all x,y∈ V_.
The -linear extension φ_=_φ:V_→ W_ 
of an -linear isometric embedding φ:V→ W 
preserves the new inner products, so we can define a continuous group homomorphism

    (φ) : (V) → (W)

by conjugation with φ_
and the identity on the orthogonal complement of the image of φ_.
As for  and , the ⊠-multiplication on 
produced by Construction <ref>
is by direct sum of symplectic automorphisms;
so  is symmetric, hence ultra-commutative.

If G is a compact Lie group, then the identification of
the G-fixed points of  also works much like the orthogonal case.
Quaternionic representations decompose canonically into isotypical
summands, and this results in a product decomposition for the 
G-fixed subgroup

    (_G)^G =  ( Sp(_G^))^G  =  ∏'_[λ]  (Sp(_λ^))^G ,

where the weak product is indexed by the isomorphism classes of irreducible
quaternionic G-representations λ, and _λ^ 
is the λ-isotypical summand in _G^=__G. 
As in the real case, irreducible quaternionic representations λ 
come in three different flavors, depending on whether 
^G_(λ,λ) – again a finite-dimensional
skew field extension of  – is isomorphic to ,  or .
As for , the G-fixed point space (_G)^G
is a weak product of copies of the infinite orthogonal, unitary
and symplectic groups, indexed by the different types 
of irreducible quaternionic representations of G.



[Pin and Spin group orthogonal spaces]
Given a real inner product space V we denote by (V)subjectClifford algebra|(symbol(V) - Clifford algebra
the Clifford algebra of the negative definite quadratic form on V, 
i.e., the quotient of
the -tensor algebra of V by the ideal generated by
v v + |v|^2· 1 for all v∈ V.
The Clifford algebra is /2-graded
with the even (respectively odd) part generated by an even
(respectively odd) number of vectors from V.   
The composite

    V  linear summand T V  proj (V)

is -linear and injective, and we denote it by v↦[v].

We recall that orthogonal vectors of V anti-commute in the Clifford algebra:
given v, v̅∈ V with v,v̅=0, then

    [v][v̅] + [v̅][ v]  =  
    (|v|^2· 1 + [v]^2)  +[v] [v̅]    + [v̅] [v]  + ( |v̅^2|· 1 + [v̅]^2)
       =  |v+v̅|^2· 1  + [v+v̅]^2   
     = 0  .

In (V) every unit vector v∈ S(V) satisfies [v]^2=-1,
so all unit vectors of V become units in (V).
The pin groupsubjectpin group of V is the subgroup

    (V)  ⊂ (V)^×

generated inside the multiplicative group of (V)
by -1 and all unit vectors of V.
A linear isometric embedding φ:V→ W induces a morphism
of /2-graded -algebras (φ):(V)→(W)
that restricts to φ on V.
So (φ) restricts to a continuous homomorphism

    (φ) :  (φ)|_(V) :  (V) → (W)

between the pin groups. The map (φ) depends continuously
on φ and satisfies (ψ)∘(φ)=(ψ∘φ),
so we have defined a group-valued orthogonal space . 
Construction <ref> then gives  the
structure of an orthogonal monoid space.
subjectpin group orthogonal monoid spacesymbol - orthogonal monoid space of pin groups 

Since the group (V) is generated by homogeneous elements of
the Clifford algebra, all of its elements are homogeneous.
The /2-grading of (V) provides a continuous homomorphism
(V) →/2 whose kernel 

    (V) =  (V)_∩(V)

is the spin groupsubjectspin group of V.
The map (φ) induced by a linear isometric
embedding φ:V→ W is homogeneous, so it restricts to a homomorphism

    (φ) =  (φ)|_(V) :  (V) → (W)

between the spin groups. 
The spin groups thus form an orthogonal monoid subspace 
of .subjectspin group ultra-commutative monoidsymbol - ultra-commutative monoid of spin groups   
We claim that  is symmetric, i.e.,
for all inner product spaces V and W the images of
the two group homomorphisms

    (V) (i_V) (V⊕ W)  (i_W) (W)

commute.
Indeed, (V) is generated by -1 and the elements [v][v']
for v,v'∈ S(V), and similarly for W. 
The elements -1 map to the central element -1 in (V⊕ W),
and

    [v,0]·[v',0]·[0,w]· [0,w'] =  
    [0,w]· [0,w']· [v,0]·[v',0]

because [v,0] and [v',0] anti-commute with [0,w] and [0,w'].
So  is an ultra-commutative monoid with respect to
the ⊠-multiplication of Construction <ref>.



In contrast to , the group valued orthogonal space  is not
symmetric; equivalently, the continuous map

    μ_V,W :  (V)× P(W) → (V⊕ W)

is not a group homomorphism.
The issue is that for v∈ S(V) and w∈ S(W)
the elements [v,0] and [0,w] anti-commute in (V⊕ W).




Now we turn to the groups pin^c and spin^c,
the complex variations on the pin and spin groups.
These arise from the complexified  Clifford algebra
_(V), where V is again a euclidean inner product space.
The complexified Clifford algebra is again /2-graded, and functorial 
for -linear isometric embeddings.
The pin^c groupsubjectpin^c group of V is the subgroup

    ^c(V)  ⊂ (_(V))^×

generated inside the multiplicative group
by the unit scalars λ 1 for all λ∈ U(1) 
and the elements 1 [v] for all unit vectors v∈ S(V).
The pin^c-groups form a group valued orthogonal space ^c,
and hence an orthogonal monoid space, in much the same way
as for the pin groups above. As for , the monoid valued orthogonal space ^c
is not symmetric, 
so the associated ⊠-multiplication of ^c is not commutative.
subjectpin^c group orthogonal monoid spacesymbol^c - orthogonal monoid space of pin^c groups 

Since the group ^c(V) is generated by homogeneous elements of
the complexified Clifford algebra, all of its elements are homogeneous.
So the /2-grading of _(V) provides a continuous homomorphism
^c(V)→/2 whose kernel 

    ^c(V) =   (_(V))_∩^c(V)

is the spin^c groupsubjectspin^c group of V.
As V varies, the spin^c groups from a group valued orthogonal subspace ^c
of ^c.subjectspin^c group ultra-commutative monoidsymbol^c - ultra-commutative monoid of spin^c groups   
As for , the images of the homomorphisms
^c(i_V) and ^c(i_W) commute, so
^c is even an ultra-commutative monoid. 

We embed the real Clifford algebra (V)
as an -subalgebra of its complexification by

    ι(V) = 1 -  :  (V) → _(V)  .

This homomorphism restricts to embeddings

    ι(V) :  (V) → ^c(V)   and  ι(V) :  (V) → ^c(V) ;

as V varies, these maps
form morphisms of group valued orthogonal spaces ι:→^c
and ι:→^c, and hence morphisms of
orthogonal monoid spaces.
These morphisms extend to isomorphisms 

    ×_{± 1} U(1)  ^c      and    ×_{± 1} U(1)  ^c

of orthogonal monoid spaces.

We let α:(V)→(V) denote the unique -algebra automorphism
of the Clifford algebra such that α[v]=-[v] for all v∈ V.
The map α is the grading involution, i.e., it is the
identity on the even part and -1 on the odd part of the Clifford algebra.
We also denote by α the automorphism  
of the complexification _(V) obtained by complexifying the real version.
For every element x∈^c(V) the twisted conjugation map

    c_x  :  _(V) → _(V)  ,    
    c_x(y) =  α(x)y x^-1

is an automorphism of /2-graded -algebras.
We let v∈ S(V) be a unit vector. Then twisted conjugation by [v] 
takes [v] to -[v], and it fixes the elements [w] for all
w∈ V that are orthogonal to v.
So twisted conjugation by [v] `is' reflection
in the hyperplane orthogonal to v, hence a linear isometry of V
of determinant -1.
The complex scalars are central in _(V), 
so conjugation by elements λ 1 with λ∈ U(1) is the identity.
So for all elements x∈^c(V), the conjugation map c_x 
restricts to a linear isometry on V, in the sense that 
there is a unique (x)∈ O(V) satisfying

    α(x)[v] x^-1 = [(x)(v)]

for all v∈ V. We thus obtain a continuous group homomorphism

    (V)  :  ^c(V) → O(V) ,    x  ⟼ (x)  ,

the twisted adjoint representation.subjecttwisted adjoint representation!of the pin group
The kernel of the twisted adjoint representation
is the subgroup U(1)· 1, compare <cit.>.
This homomorphism takes the spin^c group to the special orthogonal group,
and restricts to the adjoint representationsubjectadjoint representation!of the spin^c group

    (V)  :  ^c(V) → SO(V) .

These homomorphisms form morphisms of group valued orthogonal spaces

    :  ^c →      and     :  ^c →  .

Via Construction <ref> we can interpret these as
morphisms of orthogonal monoid spaces.
subjectClifford algebra|) 

There is yet another interesting morphism of group valued orthogonal spaces 

    l  :   → ^_(^c)

that lifts the forgetful realification morphism (<ref>)
through

    ^_() : ^(^c) → ^_() .

The definition of l(V_) is a coordinate free description
of the homomorphism U(n)→^c(2n) that is defined 
for example in <cit.>.
Since we don't need this, we won't go into any details.
Since both  and ^c are symmetric group valued
orthogonal spaces, l:→^_(^c)
is also a homomorphism of ultra-commutative monoids.
The morphism l takes the special unitary group S U(V_) 
to the group ( V), so it restricts to a morphism of ultra-commutative 
monoids l:→^_.

Most of the examples discussed so far 
can be summarized in the commutative diagram of orthogonal monoid spaces:

    [r]^-incl@–>[d]_l    @–>[d]_l @–>[dr]^-r   
    [d]_incl[r]^-ι   ^c [d]^incl[r]_-   [d]^incl
    [r]_-ι   ^c [r]_-

The two dotted arrows mean that the actual morphism goes to a multiplicative
shift of the target.
With the exception of  and ^c, all the orthogonal monoid spaces
are ultra-commutative.





[Additive Grassmannian]
We define an ultra-commutative monoid ,
the additive Grassmannian.subjectadditive GrassmanniansubjectGrassmannian!additive|seeadditive Grassmannian
The value of symbol - additive Grassmannian
at an inner product space V is

    (V) =  ∐_m≥ 0  Gr_m(V)  ,

the disjoint union of all Grassmannians in V.
The structure map induced by a linear isometric embedding φ:V→ W
is given by  (φ)(L) = φ(L).
A commutative multiplication on  is given by direct sum:

    μ_V,W :  ( V ) × ( W )  → ( V⊕ W)  ,  
    (L,L') ⟼ L⊕ L'

The unit is the only point {0} in (0).
The orthogonal space  is naturally -graded, with degree m
part given by ^[m](V)=Gr_m(V). The multiplication is graded
in that it sends ^[m]( V ) ×^[n] ( W ) to ^[m+n]( V⊕ W). 

As an orthogonal space,  is the disjoint union of global
classifying spaces of the orthogonal groups.subjectglobal classifying space!of O(n) 
Indeed, the homeomorphisms

    (^m,V)/O(m)   ^[m](V)  ,   φ· O(m) ⟼ φ(^m)

show that the summand ^[m] is isomorphic to the 
semifree orthogonal space _O(m),^m.
Since the tautological action of O(m) on ^m is faithful, 
this is a global classifying space for the orthogonal group.
So as orthogonal spaces,

    =  ∐_m≥ 0  B_ O(m) .

Proposition <ref> (ii) identifies the
equivariant homotopy set π_0^G ( B_ O(m))
with the set of conjugacy classes of continuous homomorphisms from G to O(m);
by restricting the tautological O(m)-representation on ^m, this
set bijects with the set of isomorphism classes of 
m-dimensional G-representations.
In the union over all m≥ 0, this becomes a bijection between
π_0^G() and ^+(G), the set of isomorphism classes of 
orthogonal G-representations.

We make the bijection more explicit, showing at the same time that it is an 
isomorphism of monoids.
We let V be a finite-dimensional orthogonal G-representation. 
The G-fixed points of (V) are the G-invariant subspaces of V, 
i.e., the G-subrepresentations.
We define a map

    (V)^G  =  ∐_m≥ 0 ( Gr_m(V) )^G 
     → ^+(G)

from this fixed point space to the monoid of isomorphism classes
of G-re­presentations
by sending L ∈(V)^G to its isomorphism class.
The isomorphism class of L only depends on the path component of L 
in (V)^G, and the resulting maps 
π_0 ( (V)^G) →^+(G)
are compatible as V runs through the finite-dimensional G-subrepresentations
of _G. So they assemble into a map

    π_0^G (  ) =  _V∈ s(_G) π_0( (V)^G )  → ^+(G) ,

and this map is an isomorphism of monoids with respect to the direct sum
of representations on the target.
Moreover, the isomorphism is compatible with restriction maps,
and it takes the transfer maps induced by the commutative
multiplication of  to induction of representations on the right hand side;
so as G varies, the maps (<ref>) form an isomorphism
of global power monoids.

An interesting morphism of ultra-commutative monoids

    τ :   →
subjectorthogonal group ultra-commutative monoid
is defined at an inner product space V as the map  

    τ(V)  :  (V)  → (V)  ,    L  ⟼ 
    p_L^⊥-p_L  ,

sending a subspace L⊂ V 
to the difference of the orthogonal projection onto L^⊥=V-L
and the orthogonal projection onto L.
Put differently, τ(V)(L) is the linear isometry that
is multiplication by -1 on L and the identity on the orthogonal complement L^⊥.
We omit the straightforward verification that these maps
do define a morphism of ultra-commutative monoids.
The induced monoid homomorphism

    π_0^G(τ) :  π_0^G() → π_0^G()

is easily calculated. The isomorphism (<ref>)
identifies the source with the monoid ^+(G) of isomorphism classes
of orthogonal G-representations, under direct sum.
By Example <ref>
the group π_0^G() is a direct sum of copies of /2,
indexed by the isomorphism classes of irreducible
orthogonal G-rep­resentations of real type. 
If λ is any irreducible orthogonal G-representation,
then π_0^G(τ) sends its class to the automorphism -_λ. 
The group (λ)^G is isomorphic to O(1), U(1)
or Sp(1) depending on whether λ is of real, complex or
quaternionic type. In the real case, the map -_λ lies in the
non-identity path component; in the complex and quaternionic cases,
the group (λ)^G is path connected.
So under the previous isomorphisms, π_0^G(τ) becomes the homomorphism

    ^+(G)  → ⊕_[λ] real/2

that sends the class of λ to the generator of the λ-summand
if λ is of real type, and to the trivial element 
if λ is of complex or quaternionic type.
Since the classes of irreducible representations freely generate
^+(G) as an abelian monoid, this determines the morphism π_0^G(τ).
Moreover, this also shows that π_0^G()→π_0^G() is surjective.




[Oriented Grassmannian]
A variation of the previous example is 
the orthogonal monoid space ^ of oriented Grassmannians.
subjectoriented GrassmanniansubjectGrassmannian!oriented|seeoriented Grassmanniansymbol^ - oriented Grassmannian
The value of ^ at an inner product space V is

    ^(V) =  ∐_m≥ 0  Gr_m^(V)  ,

the disjoint union of all oriented Grassmannians in V.
Here a point in Gr_m^(V) is a pair (L,[b_1,…,b_m]) consisting
of an L∈ Gr_m(V) and an orientation, 
i.e., a G L^+(L)-equivalence class  [b_1,…,b_m] of bases of L.
The structure map induced by φ:V→ W sends
(L,[b_1,…,b_m]) to (φ(L),[φ(b_1),…φ(b_m)]).
A multiplication on ^ is given by direct sum:

    μ_V,W :  ^( V ) ×^ ( W )     → ^( V⊕ W) 
     
    ((L,[b_1,…,b_m]),  (L',[b_1',…,b_n']) )    ⟼
     
    (L⊕ L',    [(b_1,0),…,(b_m,0),(0,b_1'),…,(0,b_n')])  .

The unit is the only point ({0},∅) in ^(0).

As an orthogonal space, ^ is the disjoint union of global
classifying spaces of the special orthogonal groups, via the homeomorphisms

    (B_ S O(m))(V) =  (^m,V)/S O(m)         Gr_m^(V) 
    φ· S O(m)     ⟼ 
    (φ(^m),[φ(e_1),…φ(e_m)] ) .

The multiplication of ^ is not commutative. The issue is
that when pushing a pair around the two ways of the square

    @C=12mm 
    Gr_m^(V)× Gr^_n(W) [r]^-μ_V,W[d]_twist    
    Gr_m+n^(V⊕ W)[d]^Gr_m+n^(τ_V,W)
    
    Gr_n^(W)× Gr_m^(V) [r]_-μ_W,V    
    Gr_n+m^(W⊕ V)

then we end up with the same subspaces of W⊕ V, but they
come with different orientations if m and n are both odd.

We can arrange commutativity of the multiplication by passing
to the orthogonal submonoid ^, of even-dimensional
oriented Grassmannians, defined as

    ^,(V) =  ∐_n≥ 0  Gr_2 n^(V)  ;

the multiplication of ^, is then commutative.
Moreover, the forgetful map ^,→ 
to the additive Grassmannian is a homomorphism
of ultra-commutative monoids.



[Complex and quaternionic Grassmannians]
The complex additive Grassmannian ^subjectadditive Grassmannian!complexsubjectGrassmannian!complex additive|seeadditive Grassmannian
symbol^ - complex additive Grassmannian
and the quaternionic additive Grassmannian ^subjectadditive Grassmannian!quaternionicsubjectGrassmannian!quaternionic additive|seeadditive Grassmannian
symbol^ - quaternionic additive Grassmannian
are two more ultra-commutative monoids, the complex respectively quaternionic analogs 
of Example <ref>.
The underlying orthogonal spaces send an inner product space V to

    ^(V) =  ∐_m≥ 0  Gr_m^ (V_) 
       respectively  ^(V) =  ∐_m≥ 0  Gr_m^ (V_)  ,

the disjoint union of all complex (respectively quaternionic) Grassmannians 
in the complexification V_=_V 
(respectively in V_=_V).
As in the real analog, the structure maps are given by taking images 
under (complexified respectively quaternified) linear isometric embeddings;
direct sum of subspaces, plus identification 
along the isomorphism V_⊕ W_(V⊕ W)_
(respectively V_⊕ W_(V⊕ W)_),
provides an ultra-commutative multiplication on ^ and on ^.

The homogeneous summand ^,[m] is a global classifying space
for the unitary group U(m).   
Indeed, in Construction <ref> we introduced the 
`complex semifree' orthogonal space ^_U(m),^m 
associated to the tautological U(m)-representation on ^m.
This orthogonal space  is isomorphic to ^,[m] via

    ^(^m, V_) / U(m)   ^,[m](V) ,  φ· U(m) ⟼ φ(^m) .

Proposition <ref> (i) then exhibits a global equivalence

    B_ U(m) =  _U(m),u(^m)  ∼  ^_U(m),^m ^,[m] .

Although Proposition <ref> (i) 
does not literally apply to ^,[m], it has an -linear
analog for symplectic representations, 
showing that the homogeneous summand ^,[m] is a global classifying space
for the symplectic group S p(m).subjectglobal classifying space!of U(n)    
subjectglobal classifying space!of Sp(n) 

The ultra-commutative monoid ^ comes with an involutive automorphism

    ψ :  ^ → ^

given by complex conjugation.subjectcomplex conjugation!on ^
Here we exploit that the complexification of an -vector space V
comes with a preferred -semilinear morphism

    ψ_V  :  V_ → V_ ,   λ v  ⟼λ̅ v  .

The value of ψ at V takes a -subspace L⊂ V_
to the conjugate subspace L̅=ψ_V(L).
Complexification of subspaces defines a morphism of ultra-commutative monoidssubjectcomplexification morphism!from  to ^

    c  :   → ^ ,   (V) → ^(V) ,    L  ⟼ L_

from the real to the complex additive Grassmannian.
A complex subspace of V_ is invariant under ψ_V if and
only if it is the complexification of an -subspace of V
(namely the ψ_V-fixed subspace of V).
So the morphism c is an isomorphism of  onto
the ψ-invariant ultra-commutative submonoid (^)^ψ. 

Realification defines a morphism of ultra-commutative monoids
subjectrealification morphism!from ^ to ^,

    r  :  ^ → _^ (^, )

to the multiplicative shift (see Example <ref>)
of the even part of the oriented Grassmannian of Example <ref>.
The value r(V):^(V)→^,(V_) 
takes a complex subspace of V_ to the underlying real vector space,
endowed with the preferred orientation [x_1, i x_1,…, x_n,i x _n],
where (x_1,…,x_n) is any complex basis.

The isomorphism (<ref>)
between π_0^G ( ) and ^+(G) has an obvious complex analog.
For every compact Lie group G an isomorphism of commutative monoids

    π_0^G ( ^ ) =  _V∈ s(_G) π_0( ^(V)^G ) 
      ^+(G)

is given by sending the class of a G-fixed point in ^(V)^G, 
i.e., a complex G-subrepresentation of V_, to its isomorphism class.
The isomorphism is compatible with restriction maps,
takes the involution π_0^G(ψ) of π_0^G ( ^ )
to the complex conjugation involution of ^+(G),
and it takes the transfer maps induced by the commutative
multiplication of ^ to induction of representations.
So as G varies, the maps form an isomorphism of global power monoids
_0( ^ )^+. The isomorphisms are also compatible
with complexification and realification, in the sense of the commutative diagram:

    @C6mm_0() [r]^-_0(c)[d]_(<ref>)^   _0(^) [r]^-_0(r)[d]^(<ref>)_   _0(_^( ^, )) [r]^-   _0( ^,) [r]^-forget   _0(  ) [d]^(<ref>)_
    ^+[r]_-c    ^+ [rrr]_-r          ^+

The lower horizontal maps are complexification 
and realification of representations.
The isomorphism in the upper row is inverse to the one induced by the homomorphism
^,∘ i:^,→_^( ^, ), i.e.,
precomposition with the natural linear isometric embedding

    i_V :  V → V_ ,    v  ⟼ 1 v  ;

the morphism ^,∘ i is a global equivalence 
by Theorem <ref>.




[Multiplicative Grassmannians]
We define an ultra-commuta­tive monoid _,
the multiplicative Grassmannian.subjectmultiplicative GrassmanniansubjectGrassmannian!multiplicative|seemultiplicative Grassmannian 
We let 

    (V)  =  ⊕_i≥ 0 ^i(V) =  ⊕_i≥ 0  V^ i/Σ_i

denote the symmetric algebra of an inner product space V.
If W is another inner product space, then the two direct summand inclusions
induce algebra homomorphisms

    (V)    (V⊕ W)     (W) .

We use the commutative multiplication on (V⊕ W)
to combine these into an -algebra isomorphism

    (V)(W)  (V⊕ W)  .

These isomorphisms are natural for linear isometric embeddings in V and W.
The value of _symbol_ - multiplicative Grassmannian 
at an inner product space V is then

    _(V) =  ∐_n≥ 0  Gr_n((V))  ,

the disjoint union of all Grassmannians 
in the symmetric algebra of V.
The structure map _(φ):_(V)→_(W)
induced by a linear isometric embedding φ:V→ W
is given by  

    _(φ)(L)  =  (φ)(L) ,

where (φ):(V)→(W) is the induced map of symmetric algebras.
A commutative multiplication on _ is given by tensor product, i.e.,

    μ_V,W :  _( V ) ×_ ( W )  → _( V⊕ W)

sends (L,L')∈_( V ) ×_ ( W ) to the image of
L⊗ L' under the isomorphism (<ref>). 
The multiplicative unit is the point  in _(0)=.
As the additive Grassmannian , the multiplicative Grassmannian _ 
is -graded, with degree n
part given by ^[n]_(V)=Gr_n((V)). 
The multiplication sends _^[m]( V ) ×_^[n] ( W ) to 
_^[m· n](V⊕ W). 




As orthogonal spaces, the additive and multiplicative Grassmannians are
globally equivalent. 
For an inner product space V we let
i:V→(V) be the embedding as the linear summand of the
symmetric algebra. Then as V varies, the maps

    (V) =  ∐_n≥ 0 Gr_n(V)  → ∐_n≥ 0 Gr_n((V))  =  _(V)

sending L to i(L) form a global equivalence →_.
Indeed, for each n≥ 0, _^[n](V) is a sequential colimit, 
along closed embeddings, of a sequence of orthogonal spaces 

    ^[n] → ^[n]_≤ 1 → ^[n]_≤ 2 → … → ^[n]_≤ k → … ,

where _≤ k^[n](V)=G r_n(⊕_i=0^k ^i(V)).
Each of the morphisms ^[n]→_≤ k^[n] is a global equivalence 
by Theorem <ref>; so all morphisms in the
sequence are global equivalences, hence so is the map from ^[n]
to the colimit _^[n], by
Proposition <ref> (ix).
This global equivalence induces a bijection

    π_0^G()   π_0^G(_)

for every compact Lie group G, hence both are isomorphic
to the set ^+(G) of isomorphism classes of orthogonal G-representations. 
The commutative mo­noid structures and transfer maps
induced by the products of  respectively _ are
quite different though: the monoid structure of π_0^G() 
corresponds to direct sum of representations, and the transfer maps
are additive transfers;
the monoid structure of π_0^G(_) 
corresponds to tensor product of representations, and the transfer maps
are multiplicative transfers, also called norm maps.

The orthogonal subspace =_^[1] 
of the multiplicative Grassmannian _ 
is closed under the product and contains the multiplicative unit, 
hence  is an ultra-commutative monoid 
in its own right. Because

    (V) =  _^[1](V)  =  P((V))

is the projective space of the symmetric algebra of V,
we use the symbol  and refer to it as the 
global projective space.subjectglobal projective spacesymbol - global projective space
The multiplication is given by tensor product of lines,
and application of the isomorphism (<ref>).
Since =_^[1] is globally equivalent
to the additive variant ^[1], it is a global classifying space
for the group O(1), a cyclic group of order 2,

    ≃ ^[1] ≃ B_ O(1)  =  B_ C_2  .

In other words,  is an ultra-commutative  
multiplicative model for B_ C_2.subjectglobal classifying space!of C_2 

There is a straightforward complex analog of the multiplicative Grassmannian,
the ultra-commutative monoid _^ with value at V given by

    ^_(V) =  ∐_n≥ 0  Gr_n^((V)_)  ,

the disjoint union of all Grassmannians 
in the complexified symmetric algebra of V.
The structure maps and multiplication (by tensor product) are as in the real case.
The orthogonal subspace ^=_^,[1] 
consisting of 1-dimensional subspaces
is closed under the product and contains the multiplicative unit; 
hence ^ is an ultra-commutative monoid 
in its own right, the complex global projective space.
As an orthogonal space, ^ is globally equivalent
to the additive variant ^,[1], and hence a global classifying space
for the group U(1),subjectglobal classifying space!of U(1)  

    ^ ≃ ^,[1] ≃ B_ U(1)  .



Since the multiplication in the skew field of quaternions  
is not commutative, there is no
tensor product of -vector spaces; so there is no multiplicative
version of the quaternionic Grassmannian ^.


[Bar construction]
For the next class of examples we quickly recall the bar construction
of a topological monoid M.subjectbar construction!of a topological monoid
This is the geometric realization of the simplicial space M^∙ whose space 
of n-simplices is M^n, the n-fold cartesian power of M.
For n≥ 1 and 0≤ i ≤ n, the face map d_i:M^n→ M^n-1
is given by

    d_i(x_1,… x_n)  = {[                          (x_2,…,x_n)                                     ; (x_1,…,x_i-1,x_i· x_i+1,x_i+2,…,x_n)                                     ;                        (x_1,…,x_n-1)                                      ].

For n≥ 1 and 0≤ i ≤ n-1 the degeneracy map  s_i:M^n-1→ M^n
is given by

    s_i(x_1,…, x_n-1)  =  (x_1,…,x_i,1,x_i+1,…,x_n-1)  .

The bar construction is the geometric realization 
(see Construction <ref>)

    B M  =  | M^∙|  =  |[n]↦ M^n|

of this simplicial space; the construction M↦ B M 
is functorial in continuous monoid homomorphisms. 
The bar construction commutes with products in the sense that
for a pair of topological monoids M and N, the canonical map

    (B p_M, B p_N)  :  B(M× N) → B M× B N

is a homeomorphism, where p_M:M× N→ M and p_N:M× N→ N
are the projections. Indeed, this map factors as the composite of two maps

    | (M× N)^∙ |  |(p_M^∙,p_N^∙)| 
    |  M^∙× N^∙ |  (|p_M^∙|,|p_N^∙|) 
    |  M^∙|× | N^∙ |  .

The first map is the realization of an isomorphism of simplicial spaces,
given levelwise by shuffling the factors.
The second map is a homeomorphism because realization commutes with products,
see Proposition <ref> (ii).




[Multiplicative global classifying spaces]
As we discuss now, all abelian compact Lie groups admit
multiplicative models of their global classifying spaces.
We use the bar construction, giving a non-equivariant classifying space, followed by the
cofree functor R (see Construction <ref>).
The cofree functor takes a space A to the orthogonal space R A with
values

    (R A)(V) =  ((V,^∞),A)  .

We endow the cofree functor with a lax symmetric monoidal transformation

    μ_A,B :  R A ⊠ R B  → R(A× B) .

To construct μ_A,B we start from the continuous maps

    ((V,^∞), A) ×((W,^∞), B) 
      ×    ((V,^∞)×(W,^∞), A× B) 
     (_V,W)^*    ((V⊕ W,^∞), A× B )

that constitute a bimorphism from (R A,R B) to R(A× B).
Here 

    _V,W : (V⊕ W,^∞) → (V,^∞)×(W,^∞)

restricts an embedding of V⊕ W to the summands V and W.
The morphism μ_A,B is associated to this bimorphism
via the universal property of the box product.

The bar construction preserves products in the sense that for every
pair of compact Lie groups G and K the canonical map

    (B p_G, B p_K) :   B(G× K) → B G× B K

is a homeomorphism, compare (<ref>).
So the composite G↦ R(B G) is a lax symmetric
monoidal functor via the morphism of orthogonal spaces

    R(B G) ⊠ R(B K)   μ_B G,B K  
    R(B G× B K)   R(B(G× K)) .

The bar construction is functorial in continuous group homomorphisms, so for
an abelian compact Lie group A the composite

    R(B A) ⊠ R(B A)  →  R(B(A× A)) R(Bμ_A) R(B A)

is an ultra-commutative and associative multiplication on the orthogonal space
R(B A), where μ_A:A× A→ A is the multiplication of A.
Theorem <ref> shows that for abelian A
the cofree orthogonal space R(B A) is a global classifying space for A.
subjectcofree orthogonal space
In particular, the Rep-functor _0(R(B A)) is representable by A.
We saw in Proposition <ref>
that there is then a unique structure of global power monoid on
_0(R(B A)), and the power operations are characterized by the relation

    [m](u_A) =  p_m^*(u_A)

where u_A∈π_0^A(R(B A)) is a tautological class 
and p_m:Σ_m≀ A→ A is the homomorphism defined by

    p_m(σ; a_1,…,a_m)  =  a_1·…· a_m  .





[Unordered frames]
The ultra-commutative monoid  of unordered frames
subjectunordered framessymbol - ultra-commutative monoid of unordered frames  
sends an inner product space V to

    (V) =  { A ⊂ V | A is orthonormal} ,

the space of all unordered frames in V, i.e., subsets of V that consist of
pairwise orthogonal unit vectors. Since V is finite-dimensional, such a subset
is necessarily finite. The topology on (V) is as the disjoint union,
over the cardinality of the sets, of quotient spaces of Stiefel manifolds.
The structure map induced by a linear isometric embedding φ:V→ W
is given by  (φ)(A) = φ(A).
A commutative multiplication on  is given, essentially, by disjoint union:

    μ_V,W :  ( V ) × ( W )  → ( V⊕ W)  ,  
    (A,A') ⟼ i_V(A) ∪ i_W(A') ;

here i_V:V→ V⊕ W and i_W:W→ V⊕ W are the direct summand embed­dings.
The unit is the empty set, the only point in (0).
The orthogonal space  is naturally -graded, with degree m
part ^[m] given by the unordered frames of cardinality m; 
the multiplication sends ^[m]( V ) ×^[n] ( W ) 
to ^[m+n]( V⊕ W). 

As an orthogonal space,  is the disjoint union of global
classifying spaces of the symmetric groups. 
We let Σ_m act on ^m by permuting the coordinates,
which is also the permutation representation of the tautological
Σ_m-action on {1,…,m}.
This Σ_m-action is faithful, so the semifree orthogonal space _Σ_m,^m
is a global classifying space for the symmetric group.
The homeomorphisms

    (^m,V)/Σ_m   ^[m](V)  ,   φ·Σ_m ⟼ {φ(e_1),…,φ(e_m)}

show that the homogeneous summand ^[m] is isomorphic to
_Σ_m^m=B_Σ_m; 
here e_1,…,e_m is the canonical basis of ^m.
So as orthogonal spaces,subjectglobal classifying space!of Σ_n 

    =  ∐_m≥ 0  B_Σ_m .

Proposition <ref> (ii) identifies the
equivariant homotopy set π_0^G ( B_Σ_m)
with the set of conjugacy classes of continuous homomorphisms from G to Σ_m;
by restricting the tautological Σ_m-representation on {1,…,m}, 
this set bijects with the set of isomorphism classes of 
finite G-sets of cardinality m.

As m varies, this gives an isomorphism of monoids from π_0^G() 
to the set ^+(G)symbol^+(G) - monoid of isomorphism classes of finite G-sets of isomorphism classes of finite G-sets 
that we make explicit now.
We let V be a G-representation. 
An unordered frame A∈(V) is a G-fixed point if and only if it is G-invariant.
So for such frames, the G-action restricts to an action on A,
making it a finite G-set.
We define a map

    (V)^G  → ^+(G)  ,    A  ⟼ [A]

from this fixed point space to the monoid of isomorphism classes of finite G-sets.
The isomorphism class of A as a G-set 
only depends on the path component of A in (V)^G, and the resulting maps 
π_0 ( (V)^G) →^+(G)
are compatible as V runs through the finite-dimensional G-subrepresentations
of _G. So they assemble into a map

    π_0^G (  ) =  _V∈ s(_G) π_0( (V)^G )  → ^+(G) ,

and this map is a monoid isomorphism with respect to the disjoint union
of G-sets on the target.
Moreover, the isomorphisms are compatible with restriction maps,
and they take the transfer maps induced by the commutative
multiplication of  to induction of equivariant sets on the right hand side.



It goes without saying that actions of compact Lie groups
are required to be continuous and that the use of the term `set'
(as opposed to `space') implies the discrete topology on the set; 
so the identity path component G^∘ acts trivially
on every G-set.
Hence the monoids π_0^G (  ) and ^+(G)
only see the finite group π_0(G)=G/G^∘=G̅ of path components, i.e.,
for every compact Lie group G, the inflation maps

    p^*  :  π_0^G̅ (  )  → π_0^G (  )
         and    
    p^*  :  ^+(G̅)  → ^+(G)

along the projection p:G→G̅ are isomorphisms. 
So if G has positive dimension, then 
the group completion of the monoid ^+(G) need not be isomorphic 
to what is sometimes called the Burnside ring of G 
(the 0-th G-equivariant stable stem).



A morphism of -graded ultra-commutative monoids

    :   →      is defined by    (V)(A) =  (A) ,

i.e., a frame is sent to its linear span.
The induced morphism of global power monoids
is linearization: the square of monoid homomorphisms

    @C=18mmπ_0^G()[d]_(<ref>)^[r]^-π_0^G()   π_0^G()[d]^(<ref>)_
    ^+(G)[r]_-[S] ⟼  [ S]   ^+(G)

commutes, where the lower map sends the class of a G-set to the class of its
permutation representation.



[Multiplicative monoid of the sphere spectrum]
We define an ultra-commutative monoid Ω^∙,
the `multiplicative monoid of the sphere spectrum'.subjectsphere spectrum 
The notation and terminology indicate that this is a special case
of a more general construction that associates to an ultra-commutative
ring spectrum R its multiplicative monoid Ω^∙ R,
see Example <ref> below.

The values of the orthogonal space Ω^∙ are given by

    ( Ω^∙)(V)  =  _*(S^V,S^V) ,

the space of continuous based self-maps of the sphere S^V.
A linear isometric embedding φ:V→ W acts by conjugation
and extension by the identity, i.e., the map 

    (Ω^∙)(φ) :  _*(S^V,S^V) → _*(S^W,S^W)

sends a continuous based map f:S^V→ S^V to the composite

    S^W  S^V S^W-φ(V) f S^W-φ(V) S^V S^W-φ(V)   S^W .

The two unnamed homeomorphisms between S^V S^W-φ(V) and S^W
use the map φ on the factor S^V.
In particular, the orthogonal group O(V) acts on _*(S^V,S^V)
by conjugation.

The multiplication of Ω^∙ is by smash product, i.e., the map

    μ_V,W :   ( Ω^∙)(V) × ( Ω^∙)(W)  → 
    ( Ω^∙)(V⊕ W)

smashes a self-map of S^V with a self-map of S^W and
conjugates with the canonical homeomorphism between S^V S^W and S^V⊕ W.
The unit is the identity of S^V.

The equivariant homotopy set
π_0^G(Ω^∙) is equal 
to the stable G-equivariant 0-stem π_0^G(),
compare Construction <ref> below.
The monoid structure on π_0^G(Ω^∙) 
arising from the multiplication on Ω^∙
is the multiplicative (rather than the additive) 
monoid structure of π_0^G().
The set π_0^G(Ω^∙)
thus bijects with the underlying set of the Burnside ring (G) of the group G
(compare Example <ref>), which is
additively a free abelian group with basis 
the conjugacy classes of closed subgroups of G with finite Weyl group.
The multiplication on π_0^G(Ω^∙) corresponds to the multiplication 
(not the addition !) in the Burnside ring (G).
When G is finite, π_0^G(Ω^∙) bijects with the
underlying set of the Grothendieck group of finite G-sets,
and the multiplication corresponds to the product of G-sets.
The power operations in  _0(Ω^∙) 
are thus represented by `raising a G-set to the cartesian power',
and the transfer maps are known as `norm maps' or `multiplicative induction'.



[Exponential homomorphisms]
The classical J-homomor­phismsubjectJ-homomorphism@J-homomorphism
fits in nicely here, in the form of a global refinement 
to a morphism of ultra-commutative monoidssubjectorthogonal group ultra-commutative monoid

    J  :   → Ω^∙

defined at an inner product space V as the map

    J(V)  :  (V)  → _*(S^V,S^V)

sending a linear isometry φ:V→ V
to its one-point compactification S^φ:S^V→ S^V.
The fact that these maps are multiplicative and 
compatible with the structure maps is straightforward.
The induced map

    π_0^G(J) :  π_0^G() → π_0^G(Ω^∙)  = π_0^G() ,

for G a compact Lie group, can be described as follows.
By Example <ref>, 
the group π_0^G() is a direct sum of copies of /2,
indexed by the isomorphism classes of irreducible
orthogonal G-representations of real type. 
If λ is such an irreducible G-representation,
then the image of the λ-indexed copy of /2
is represented by the antipodal map of S^λ.

In (<ref>) we defined 
a morphism of ultra-commutative monoids τ:→. 
The composite morphism

    τ     J  Ω^∙

realizes an `exponential' homomorphismsubjectexponential homomorphism!from (G) to (G)^×  
from the real representation ring to the  multiplicative group
of the Burnside ring of a compact Lie group G.
The exponential homomorphism was studied by tom Dieck 
and is sometimes called the `tom Dieck exponential map'.
Tom Dieck's definition of the exponential homomorphism 
in <cit.>
is completely algebraic: we start from the homomorphism

    s : ^+(G) → Cl(G,)^× ,     s[V](H) =  (-1)^ (V^H)

that sends an orthogonal representation V to the class function s(V) 
that records the parities of the dimensions of the fixed point spaces.
The Burnside ring embeds into the ring of class functions by `fixed point counting'

    Φ :  (G) → (G,) ,  Φ[S](H) =  |S^H| ,

i.e., a (virtual) G-set S is sent to the class function that counts the number
of fixed points. This map is injective, and for finite groups the image
can be characterized by an explicit system of congruences <cit.>.
The image of s satisfies the congruences, so there is
a unique map exp:^+(G)→(G)^×
such that Φ∘exp=s.
The map s sends the direct sum of representations to the product
of the parity functions, so exp is a homomorphism of abelian monoids,
and thus extends to a homomorphism
from the orthogonal representation ring (G).
The morphism of ultra-commutative monoids (<ref>) 
realizes the exponential morphism in the sense that the following diagram 
of monoid homomorphisms commutes:

    @C=6mmπ_0^G()[d]_(<ref>)^[rr]^-π_0^G(J∘τ)      π_0^G(Ω^∙) @=[r]     (π_0^G())^×
    ^+(G)[rrr]_exp         (G)^×[u]_^α

To show the commutativity of this diagram we may compose with
the degree monomorphism 

    : (π_0^G())^×→Cl(G,) ,   [f](H)  =  (f^H:S^V^H→ S^V^H )

that takes the class of an equivariant self-map f:S^V→ S^V
of a representation sphere to the class function that records the
fixed point dimensions. The composite ∘α:(G)→Cl(G,)
coincides with the fixed point counting map Φ, so 

    ∘α∘exp =  Φ∘exp =  s  : 
    ^+(G) → Cl(G,) .

On the other hand, the map π_0^G(J∘τ) 
sends the class of a G-representation V to
the involution S^-_V:S^V→ S^V. The diagram thus commutes because

    ( (S^-_V)^H ) =  (-1)^(V^H) =  s[V](H) .








§ GLOBAL FORMS OF BO



In this section we discuss different orthogonal monoid spaces whose
underlying non-equivariant homotopy type is B O, 
a classifying space for the infinite orthogonal group.
Each example is interesting in its own right, 
and as a whole, the global forms of B O are a great illustration of how
non-equivariant homotopy types `fold up' into many different global
homotopy types.
The different forms of B O have associated orthogonal Thom spectra
with underlying non-equivariant stable homotopy type M O;
we will return to these Thom spectra in Section <ref>.
The examples we discuss here all come with multiplications, some ultra-commutative, 
but some only E_∞-commutative; 
so our case study also illustrates the different degrees of commutativity 
that arise `in nature'.

We can name five different global homotopy types that all have the same underlying
non-equivariant homotopy type, namely that of a classifying space of 
the infinite orthogonal group:


  * the constant orthogonal space B O with value a classifying space
of the infinite orthogonal group;

  * the `full Grassmannian model' ,
the degree 0 part of the periodic global 
Grassmannian  (Example <ref>);

  * the bar construction model 𝐁^∘ (Construction <ref>);

  * the `restricted Grassmannian model'  that is also
a sequential homotopy colimit of the global classifying spaces B_ O(n)
(Example <ref>);

  * the cofree orthogonal space R(BO) associated to a classifying space
of the infinite orthogonal group (Construction <ref>).

These global homotopy types are related by weak morphisms of orthogonal spaces:

    B O[r][d]    [d] 
    𝐁^∘[r]    [r]     R(BO)

The orthogonal spaces 𝐁^∘ and  come with
ultra-commutative multiplications.
The global homotopy type of R(BO) also admits an ultra-commutative multiplication;
we will not elaborate this point, but one way to see it is to
extend the cofree functor R to a lax symmetric monoidal functor
on the category of orthogonal spaces, so that R()
is an ultra-commutative monoid within this global homotopy type.
The orthogonal spaces B O and  admit E_∞-multiplications; 
for B O this is a consequence of the non-equivariant E_∞-structure of BO.
All weak morphisms above can be arranged to preserve the E_∞-multiplications,
so they induce additive maps of abelian monoids on π_0^G
for every compact Lie group G. 

As we explain in Example <ref>,
the bar construction model makes sense more generally for
monoid valued orthogonal spaces; in particular, applying the bar construction
to the ultra-commutative monoids made from the families of classical Lie groups
discussed in the previous section provides ultra-commutative monoids 
𝐁^∘, 𝐁^∘, 𝐁^∘
𝐁^∘, 𝐁^∘ and 𝐁^∘^c.
Example <ref> introduces the complex and quaternionic
analogs of  and , i.e., the ultra-commutative monoids  and 
and the E_∞-orthogonal monoid spaces  and .

[Periodic Grassmannian] 
We define an ultra-commutative monoid symbol - periodic global BOsubjectperiodic global BO that is a global refinement
of the non-equivariant homotopy type × BO,
and at the same time a global group completionsubjectgroup completion
of the additive Grassmannian  introduced in Example <ref>. 
The orthogonal space  comes with tautological vector bundles whose Thom spaces
form the periodic Thom spectrum , 
discussed in Example <ref> below.

For an inner product space V we set

    (V)  =  ∐_m≥ 0  Gr_m(V^2) ,

the disjoint union of the Grassmannians 
of m-dimensional subspaces in V^2=V⊕ V.
The structure map associated to a linear isometric embedding φ:V→ W
is given by 

    (φ)(L)  =  φ^2(L)  + ((W-φ(V))⊕ 0) ,

the internal orthogonal sum of the image of L under φ^2:V^2→ W^2
and the orthogonal complement of the image of φ:V→ W, viewed as sitting
in the first summand of W^2=W⊕ W.
In particular, the orthogonal group O(V) acts on (V)
through its diagonal action on V^2.

We make the orthogonal space  into an ultra-commutative monoid 
by endowing it with multiplication maps

    μ_V,W :  (V) ×(W)  → (V⊕ W)  ,  
    (L,L')  ⟼ κ_V,W(L⊕ L') ,

where 

    κ_V,W :  V^2⊕ W^2  (V⊕ W)^2    is defined by  κ_V,W(v,v',w,w')=(v,w,v',w') .

The unit is the unique element {0} of (0).

The orthogonal space  is naturally -graded: for m∈ we let

    ^[m](V) ⊂ (V)

be the path component consisting of all subspaces L⊂ V^2
such that (L)=(V) + m. For fixed m the spaces ^[m](V)
form a subfunctor of , i.e., ^[m] is an orthogonal subspace
of . The multiplication is graded in the sense that μ_V,W
takes  ^[m](V) ×^[n](W) to ^[m+n](V⊕ W).
We write =^[0]symbol - global BO 
for the homogeneous summand of  of degree 0,
which is thus an ultra-commutative monoid in its own right.
The underlying non-equivariant homotopy type of  is that of
a classifying space of the infinite orthogonal group.



While  and the additive Grassmannian 
are both made from Grassmannians, one should beware of the different nature 
of their structure maps.
There is a variation ' of the additive Grassmannian with values
'(V)=∐_n≥ 0 Gr_n(V^2) and structure maps
'(φ)(L)=φ^2(L). This orthogonal space
is a `multiplicative shift' of  
in the sense of Example <ref>,
it admits a commutative multiplication in much the same way as , and the maps

    (V) → '(V)  ,    L  ⟼ L⊕ 0

form a global equivalence of ultra-commutative monoids
by Theorem <ref>.
A source of possible confusion is the fact that
'(V) and (V) are equal as spaces, but they come with
very different structure maps making them into two different global homotopy types.


[ versus ]subjectadditive Grassmannian
In Example <ref> we explained that 
for every compact Lie group G, the monoid π_0^G() 
is isomorphic to the monoid ^+(G), under direct sum, 
of isomorphism classes of orthogonal G-represen­tations. 
In Theorem <ref> we will identify
the monoid π_0^G() with the orthogonal representation ring (G).
The latter is the algebraic group completion of the former, and this group
completion is realized by a morphism of ultra-commutative monoids 

    i  :   →  .

The morphism i is given at V by the map

    (V) =  ∐_m≥ 0   Gr_m( V)     → ∐_n≥ 0  Gr_n (V^2) = (V)    ,   L  ⟼ V⊕ L .

The morphism is homogeneous in that it takes ^[m] to ^[m].



As we will now show, the morphism i:→
induces an algebraic group completion of abelian monoids
upon taking equivariant homotopy sets from any equivariant space.
This fact is the algebraic shadow of a more refined relationship: 
as we will show in Theorem <ref> below,
the morphism i:→ is a group completion
in the world of ultra-commutative monoids, i.e.,
`homotopy initial', in the category of ultra-commutative monoids,
among morphisms from  to group-like ultra-commutative monoids.



We recall from Definition <ref>
the equivariant homotopy set
  
    [A,R]^G  =  _V∈ s(_G)  [A, R(V)]^G  ,

where R is  an orthogonal space, 
G is a compact Lie group and A a G-space.
If R is an ultra-commutative monoid, then this set inherits an
abelian monoid structure defined as follows. We let
α:A→ R(V) and β:A→ R(W) be two G-maps
that represent classes in [A,R]^G. Then their sum is defined as

    [α] +[β]  =  [μ_V,W(α,β)]  ,

where μ_V,W:R(V)× R(W)→ R(V⊕ W) is the
(V,W)-component of the  multiplication of R.
The monoid structure is contravariantly functorial for G-maps in A,
and covariantly functorial for morphisms of ultra-commutative monoids in R.
When A=∗ is a one-point G-space, then [A,R]^G becomes π_0^G(R), 
and the addition (<ref>) reduces to the
addition as previously defined in (<ref>). 


For every compact Lie group G and every G-space A, the homomorphism

    [A,i]^G  :  [A,]^G  → [A,]^G

is a group completion of abelian monoids.


We start by showing that the abelian monoid [A,]^G is a group.
We consider a G-representation V.
For a linear subspace L⊆ V^2 
we consider the 1-parameter family of linear isometric embeddings

    H_L  :  [0,1]× L^⊥ → V^2⊕ V^2  ,    (t,x) ⟼( t· x,  √(1-t^2)· x )  .

For every t∈[0,1], the image of H_L(t,-) is isomorphic to L^⊥
and orthogonal to the space L⊕ 0⊕ 0.
We can thus define a G-equivariant homotopy

    K  :  [0,1]× G r(V^2)  → G r(V^2⊕ V^2)    by   
    K(t,L)  =  (L⊕ 0⊕ 0) +  H_L(t,L^⊥) .

Then

    K(0,L) = (L⊕ 0⊕ 0) + H_L(0,L^⊥)  =  
    (L⊕ 0) +(0⊕ L^⊥)  =  L⊕ L^⊥

and

    K(1,L)    = (L⊕ 0⊕ 0) + H_L(1,L^⊥)
       = (L⊕ 0⊕ 0) + (L^⊥⊕ 0⊕ 0)  =  V⊕ V⊕ 0⊕ 0  .

We recall that the multiplication of  is given by

    μ_V,V^ :  (V)×(V) →(V⊕ V)  ,   μ_V,V^(L,L') =  κ_V,V(L⊕ L') ,

where κ_V,V(v,v',w,w')=(v,w,v',w').
So the equivariant homotopy G r(κ_V,V)∘ K interpolates between the composite

    (V) (,(-)^⊥) (V)×(V) μ_V,V^(V⊕ V)

and the constant map with value

    κ_V,V(V⊕ V⊕ 0⊕ 0) =  V⊕ 0⊕ V⊕ 0  .

The subspace V⊕ 0⊕ V⊕ 0 lies in the same path component
of (V^2)^G= (G r(V^2⊕ V^2))^G as the 
subspace V⊕ V⊕ 0⊕ 0.
So altogether this shows that the composite μ_V,V^∘ (,(-)^⊥)
is G-equivariantly homotopic to the constant map with value
V⊕ V⊕ 0⊕ 0.

Now we let α:A→(V) be a G-map, representing a class in [A,]^G.
The composite of α and the orthogonal complement map
(-)^⊥:(V)→(V) represents another class in [A,]^G,
and

    [α]+ [(-)^⊥∘α]    =  
    [μ^_V,V∘(, (-)^⊥)∘α] =  
    [c_V⊕ V⊕ 0⊕ 0∘α] =  0

in the monoid structure of [A,]^G,
because the subspace V⊕ V⊕ 0⊕ 0 
is the neutral element in (V⊕ V).
So the class [α] has an additive inverse, 
and this concludes the proof that the abelian monoid [A,]^G is a group.


To show that the homomorphism [A,i]^G is a group completion we 
show two separate statements that amount to the surjectivity, respectively injectivity,
of the extension of [A,i]^G to a homomorphism on the Grothendieck group
of the monoid [A,]^G.

(a) We show that every class in [A,]^G is the difference of two
classes in the image of i_*=[A,i]^G:[A,]^G→[A,]^G. 
To see this, we represent a given class x∈[A,]^G
by a G-map α:A→(V), for some G-representation V.
Because (V)=(V⊕ V), the same map α also represents 
a class in [A,]^G; to emphasize the different role, we write this map as
α^♯:A→(V⊕ V). 
We let c_V:A→(V) denote the constant map with value V
and χ:V^4→ V^4 the linear isometry defined by

    χ(v_1,v_2,v_3,v_4) =  (v_2,v_3,v_1,v_4) .

We observe that the following diagram commutes:

    @C=10mm(V)@=[d]  [rr]^-(i(V)∘ c_V,)      (V)×(V)[d]^μ_V,V^
    (V⊕ V)[r]_-i(V⊕ V)   (V⊕ V)[r]_-G r(χ)   (V⊕ V)

Since χ is equivariantly homotopic, through linear isometries,
to the identity, this shows that the composite
μ_V,V^∘ (i(V)∘ c_V,) is G-equivariantly homotopic
to i(V⊕ V). Thus

    i_*[c_V] + x     =  [μ_V,V^∘(i(V)∘ c_V,α)]
       =  
    [μ_V,V^∘(i(V)∘ c_V,)∘α]=  [i(V⊕ V)∘α^♯]   =  i_*[α^♯]  .

Thus x = i_*[α^♯] - i_*[c_V], which shows the claim.

(b) Now we consider two classes a,b ∈ [A,]^G such that
i_*(a)=i_*(b) in [A,]^G. We show that there exist another
class c∈ [A,]^G such that c+a=c+b.
We can represent a and b by two G-maps
α:A→(V) and β:A→(V) such that the two composites

    i(V)∘α ,    i(V)∘β :  A → (V)

are equivariantly homotopic.
As before we let c_V:A→(V) be the constant map with value V.
The map i(V):(V)→(V)=(V⊕ V)
factors as the composite

    (V) (c_V,) (V)×(V) μ_V,V^ (V⊕ V)  ,

so

    [c_V] + a     =  [c_V]+[α] =  [μ_V,V^∘ (c_V,α)] 
       =   [μ_V,V∘ (c_V,)∘α]  =  [i(V)∘α]  .

Similarly, [c_V]+ b = [i(V)∘β]. So [c_V] + a =[c_V] + b
in [A,]^G, as claimed.


Our next aim is to show that the ultra-commutative monoid 
represents equivariant vector bundles, and  
represents equivariant K-theory, at least for compact G-spaces.



We let G be a compact Lie group and A a G-space.
We recall that a G-vector bundle over Asubjectequivariant vector bundle
consists of a vector bundle ξ:E→ A 
and a continuous G-action on the total space E such that


  * the bundle projection ξ:E→ A is a G-map,

  * for every g∈ G and a∈ A the map 
g·-:E_a→ E_g a is -linear.

We let _G(A)symbol_G(A) - monoid of isomorphism classes
of G-vector bundles over A 
be the commutative monoid, under Whitney sum, 
of isomorphism classes of G-vector bundles over A.
We define a homomorphism of monoids

    - :  [A,]^G  =  _V∈ s(_G)  [A, (V) ]^G   → _G(A)

that will turn out to be an isomorphism 
for compact A and that specializes 
to the isomorphism (<ref>) from π_0^G() to ^+(G)
when A is a one-point G-space.
We let f:A→(V) be a continuous G-map, for some G-representation V.
We pull back the tautological G-vector bundle γ_V over (V) and obtain
a G-vector bundle f^⋆(γ_V):E→ A over A with total space

    E =  { (v,a)∈ V× A |  v∈ f(a)} .

The G-action and bundle structure are as 
a G-subbundle of the trivial bundle V× A.
Since the base (V) of the tautological bundle is a disjoint union of
compact spaces, the isomorphism class of the bundle f^⋆(γ_V) 
depends only on the G-homotopy class of f, 
see for example <cit.>. 
So the construction yields a well-defined map

    [A, (V) ]^G  → _G(A)  ,   
    [f] ⟼ [f^⋆(γ_V)] .

If φ:V→ W is a linear isometric embedding of G-representations, 
then the restriction along (φ):(V)→(W)
of the tautological G-vector bundle over (W)
is isomorphic to the tautological G-vector bundle over (V).
So the two G-vector bundles f^⋆ and ((φ)∘ f)^⋆ 
over A are isomorphic. We can thus pass to the colimit over the poset s(_G)
and get a well-defined map (<ref>).
The map (<ref>) is a monoid homomorphism
because all additions in sight arise from direct sum of inner product spaces.

Now we `group complete' the picture.
We denote by _G(A)symbol_G(A) - equivariant K-group of Asubjectequivariant K-theorysubjectK-theory@K-theory!equivariant|seeequivariant K-theory
the G-equi­variant K-group of A, i.e.,
the group completion (Grothendieck group) of the abelian monoid _G(A).
The composite

    [A,]^G   -  _G(A) →  _G(A)

of (<ref>) and the group completion map
is a monoid homomorphism into an abelian group. 
The morphism [A,i]^G: [A,]^G → [A,]^G
is a group completion of abelian monoids by Proposition <ref>, 
where i: → was defined in (<ref>).
So there is a unique homomorphism of abelian groups 

    - :  [A,]^G   =  _V∈ s(_G)  [A, (V) ]^G  → _G(A)

such that the following square commutes:

    @C=20mm 
    [A,]^G [d]_[A,i]^G[r]^--   _G(A)[d]
    
    [A,]^G [r]_--   _G(A)

We can make the homomorphism (<ref>) more explicit as follows.
We let f:A→(V) be a G-map for some G-representation V.
We pull back the tautological G-vector bundle over (V)=Gr(V^2) and obtain
a G-vector bundle f^⋆(γ_V^2):E→ A over A with total space

    E =  { (v,a)∈ V^2 × A |  v∈ f(a)} .

Again the G-action and bundle structure are as 
a G-subbundle of the trivial bundle V^2× A.
The homomorphism (<ref>) then sends the G-homotopy class of f
to the class of the virtual G-vector bundle

    [f^⋆(γ_V^2)] - [ V× A] ∈ _G(A) .

In contrast to the definition 
of the earlier map (<ref>), we now subtract the class
of the trivial G-vector bundle V× A over A.
The class in _G(A) only depends on the class of f in [A,]^G,
and this recipe defines the map (<ref>).




For every compact Lie group G and every compact G-space A 
the homomorphisms

    - :  [A,]^G   → _G(A)      and  - :  [A,]^G   → _G(A)

defined in (<ref>) respectively (<ref>) are isomorphisms.
As G varies the isomorphisms are compatible with
restriction along continuous homomorphism. 


The Grassmannian  is the disjoint union of the homogeneous pieces ^[n],
and the latter is isomorphic to the semifree orthogonal space _O(n),^n, via

    (^n,V)/ O(n)  → ^[n](V) ,   φ· O(n) ⟼φ(^n) .

Since the tautological action of O(n) on ^n is faithful,
_O(n),^n is a global classifying space for O(n);
Example <ref> thus provides a bijection

    [A, ^[n] ]^G  → _(G,O(n))(A)

to the set of isomorphism classes of G-equivariant principal O(n)-bundles over A, 
by pulling back the (G,O(n))-principal bundle (^n,V)→^[n](V),
the frame bundle of the tautological vector bundle over ^[n](V).
On the other hand, we can consider the map

    _(G,O(n))(A)  → ^[n]_G(A)

to the set of isomorphism classes of G-vector bundles of rank n over A,
sending a (G,O(n))-bundle γ:E→ A to the associated 
G-vector bundle with total space E×_O(n)^n.
Since A is compact, every G-vector bundle admits a G-invariant
euclidean inner product, so it arises from a 
(G,O(n))-bundle; hence the latter map is bijective as well.
Altogether this shows that map

    [A, ^[n]]^G  → ^[n]_G(A)

given by pulling back the tautological vector bundles is bijective.

A general G-vector bundle need not have constant rank, 
so it remains to assemble the results for varying n.
We let ξ be any G-vector bundle over A, not necessarily
of constant rank.
Then the subset

    A_(n) =  {a∈ A |  (ξ_a)= n}

of points over which ξ is n-dimensional is open
by local triviality of vector bundles.
So A is the disjoint union of the sets A_(n) for n≥ 0,
and each subset A_(n) is also closed and hence compact
in the subspace topology.
Moreover, A_(n) is G-invariant,
so the restriction ξ_(n) of the bundle to A_(n) is classified
by a G-map f_(n):A_(n)→^[n](V_n) for some
finite-dimensional G-representation V_n.
Since A is compact, almost all A_(n) are empty, so by increasing
the representations, if necessary, we can assume that the
classifying maps have target in ^[n](V) for a fixed finite-dimensional
G-representation V, independent of n. Then

    ⨿_n≥ 0 f_(n) :  ⨿_n≥ 0 A_(n) =   A  → ⨿_n≥ 0 ^[n](V)  =  (V)

is a classifying G-map for the original bundle ξ.
This shows that the map (<ref>) is surjective. 

The argument for injectivity is similar.
Any pair of classes in [A, ]^G 
can be represented by G-maps f,f̅:A→(V)
for some finite-dimensional G-representation V.
Since (V) is the disjoint union of the subspaces ^[n](V)
for n≥ 0, their inverse images under f and f̅
provide disjoint union decompositions of A by fiber dimension.
If the bundles f^⋆(γ_V) and f̅^⋆(γ_V)
are isomorphic, the decompositions of A induced by f and f̅
must be the same.
The rank n summands f_(n),f̅_(n):A_(n)→^[n](V)
become equivariantly homotopic after increasing the representation V,
because the map (<ref>) is injective.
Moreover, almost all summands are empty, one more time by compactness.  
So there is a single finite-dimensional representation W
and a G-equivariant linear isometric embedding φ:V→ W
such that f,f̅:A→(V) 
become equivariantly homotopic after composition with
(φ):(V)→(W). Hence f and f̅
represent the same class in [A,]^G, so the map (<ref>) is injective.
This completes the proof that the map is an isomorphism for compact A.

The left vertical map in the commutative square (<ref>)
is a group completion by Proposition <ref>, 
and the right vertical map is a group completion by definition.
So the lower horizontal map (<ref>) is also an isomorphism.


We take the time to specialize Theorem <ref>
to the one-point G-space.
This special case identifies the global power monoid
_0() with the global power 
monoid symbol - orthogonal representation ring global functorsubjectrepresentation ring!orthogonal
of orthogonal representation rings.
For every compact Lie group G the abelian monoid (G) 
is the Grothendieck group, under direct sum,
of finite-dimensional G-representations.
The restriction maps are induced by restriction of representations,
and the power operation [m]:(G)→(Σ_m≀ G)
takes the class of a G-representation V 
to the class of the (Σ_m≀ G)-representation V^m.
The resulting transfer _H^G:(H)→(G)
of Construction <ref>, for H of finite index in G,
is then the transfer (or induction), sending
the class of an H-representation V 
to the class of the induced G-representation ^H(G,V).
A G-vector bundle over a one-point space `is' a G-representation and the map

    (G) → _G(∗)

that considers a (virtual) representation as a (virtual) vector bundle is an 
isomorphism of groups and compatible with restriction along continuous homomorphisms
of compact Lie groups.

For easier reference we spell out the isomorphism (<ref>)
in the special case A=∗ more explicitly.
We let V be a finite-dimensional orthogonal G-representation. 
The G-fixed points of (V) are the G-invariant subspaces 
of V^2, i.e., the G-subrepresentations W of V^2.
Representations of compact Lie groups are discrete
(compare the example after <cit.>), 
so two fixed points in the same path component of (V)^G
are isomorphic as G-representations. Hence we obtain a well-defined map

    π_0( (V)^G)  → (G)

by sending W ∈(V)^G to [W]-[V], 
the formal difference in (G) of the classes of W and V. 
These maps are compatible as V runs through 
the finite-dimensional G-subrepresentations
of _G, so they assemble into a map

    π_0^G (  ) =  _V∈ s(_G) π_0( (V)^G )  → (G) .



 
For every compact Lie group G
the map (<ref>) is an isomorphism of groups. 
As G varies, these isomorphisms form an isomorphism of global power monoids

    _0()    .



The special case A=∗ of Theorem <ref>
shows that the map (<ref>) is an isomorphism
and compatible with restriction along continuous homomorphisms.
We have to argue that in addition, the maps (<ref>) 
are also compatible with transfers (or equivalently, with power operations).
The compatibility with transfers can either be deduced directly from
the definitions; equivalently it can be formally deduced from the 
compatibility of the isomorphisms _0() ^+
with transfers by the universal property of a group completion.




The bijection (<ref>) sends elements of 
π_0^G( ^[k]) to virtual representations of dimension k,
so we can also identify the global power monoid of the homogeneous degree 0
part =^[0]. Indeed, the map (<ref>) restricts
to an isomorphism of abelian groups

    π_0^G()   (G)

to the augmentation ideal (G)⊂(G) of the orthogonal representation ring,
compatible with restriction maps, power operations
and transfer maps.subjectaugmentation ideal!of the orthogonal representation ring


[Bar construction model 𝐁^∘]
Using the functorial bar construction we define another global refinement
𝐁^∘ of the classifying space of the infinite orthogonal group.
This ultra-commutative monoid is globally connected, and it
comes with a weak homomorphism to  that `picks out' the path components
of the neutral element in the G-fixed point spaces (_G)^G. 

We define 𝐁^∘ by applying the bar construction 
(see Construction <ref>)subjectbar construction!of a topological monoid 
objectwise to the monoid valued orthogonal 
space  of Example <ref>.subjectorthogonal group ultra-commutative monoid
So the value at an inner product space V is

    (𝐁^∘)(V)  =   B ( O(V) ) ,

the bar construction of the orthogonal group of V.
The structure map of a linear isometric embedding φ:V→ W is 
obtained by applying the bar construction to the continuous homomorphism
(φ):(V)→(W).
We make 𝐁^∘ into an ultra-commutative monoid by endowing 
it with multiplication maps

    μ_V,W :  (𝐁^∘)(V) × (𝐁^∘)(W) 
     → (𝐁^∘)(V⊕ W)

defined as the composite

    B ( O(V) ) × B ( O(W) )     
      B(O(V) × O(W) )   B⊕  B ( O( V⊕ W ) )  ,

where the first map is inverse to the homeomorphism (<ref>).

Now we let G be a compact Lie group and V an orthogonal G-representation.
Taking fixed points commutes with geometric realization
(see Proposition <ref> (iv)) 
and with products, so

    ( (𝐁^∘)(V) )^G  =  |O(V)^∙|^G   
    | (O(V)^G)^∙|  =   B( O^G(V) )  .

Taking colimit over the poset s(_G) gives

    ((𝐁^∘)(_G))^G   _V∈ s(_G)   B( O^G(V) )    B( O^G(_G) )   ∏_[λ]'  B(O^G(_λ))  .

Here the last weak product is indexed by isomorphism classes of irreducible 
G-representations, and each of the groups O^G(_λ) 
is either an infinite orthogonal, unitary or symplectic group, depending
on the type of the irreducible representation,
compare Example <ref>.
In particular, the space ((𝐁^∘)(_G))^G is connected,
so the equivariant homotopy set π_0^G (𝐁^∘) has one
element for every compact Lie group G;
the global power monoid structure is then necessarily trivial.
In particular, 𝐁^∘ is not globally equivalent to .

However, the difference seen by _0 is the only difference between
𝐁^∘ and , as shall now explain. We construct a weak morphism of
ultra-commu­tative monoids
that exhibits 𝐁^∘ as the `globally connected component' of .
We define an ultra-commutative monoid 𝐁' by combining the
constructions of 𝐁^∘ (bar construction) and  (Grassmannians)
into one definition. 
The value of 𝐁' at an inner product space V is

    (𝐁')(V) =  |B_∙((V,V^2),(V),∗)|  ,

the two-sided bar construction (homotopy orbit construction) of the right O(V)-action
on the space (V,V^2) by precomposition.
Here B_∙((V,V^2),(V),∗) is the simplicial space
whose space of n-simplices is (V,V^2)×(V)^n.
For n≥ 1 and 0≤ i ≤ n, the face map d_i
is given by

    d_i(φ,A_1,… A_n)  = {[                     (φ∘ A_1, A_2,…,A_n)                                        ; (φ, A_1,…,A_i-1,A_i∘ A_i+1,A_i+2,…,A_n)                                        ;                        (φ, A_1,…,A_n-1)                                         ].

For n≥ 1 and 0≤ i ≤ n-1 the degeneracy map  s_i
is given by

    s_i(φ,A_1,…, A_n-1)  =  (φ,A_1,…,A_i,,A_i+1,…,A_n-1)  .

Then (𝐁')(V) is the realization of the
simplicial space B_∙((V,V^2),(V),∗).

To define the structure map 
associated to a linear isometric embedding φ:V→ W we recall that
the structure map (φ):(V)→(W)  of the orthogonal space 
is given by conjugation by φ and direct sum with the identity on W-φ(V).
We define a continuous map

    φ_♯ :  (V,V^2) → (W,W^2)

by

    (φ_♯ψ)(φ(v)+ w) =  φ^2( ψ(v) + (w,0) ) ;

here v∈ V and w∈ W-φ(V) is orthogonal to φ(V).
The map φ_♯ is compatible with the actions of the orthogonal groups,
i.e., the following square commutes:

    @C=18mm(V,V^2)×(V) [r]^-φ_♯×(φ)[d]_∘   (W,W^2)×(W) [d]^∘
    (V,V^2)[r]_-φ_♯   (W,W^2)

This equivariance property of φ_♯ ensures that it passes to
the two-sided bar construction, i.e., we can define the structure map 
(𝐁')(φ): (𝐁')(V) → (𝐁')(W)
as the geometric realization of the morphism of simplicial spaces

    B_∙(φ_♯,(φ),∗)  :  
    B_∙((V,V^2),(V),∗)  → B_∙((W,W^2),(W),∗)  .

A commutative multiplication

    μ_V,W :  (𝐁')(V)× (𝐁')(W) → 
    (𝐁')(V⊕ W)

is obtained by combining the multiplications of 𝐁^∘ and .
The construction comes with two collections of continuous maps:

    (𝐁^∘)(V) α(V)  B((V,V^2),(V),∗)    = (𝐁')(V)
       β(V) (V,V^2)/(V) = (V)

The left map α(V) is defined by applying the bar construction
to the unique map from (V,V^2) to the one-point space.
The right map β(V) is the canonical map from homotopy orbits to
strict orbits. As V varies, the α and β maps
form morphisms of ultra-commutative monoids

    𝐁^∘  𝐁'  β   ,

essentially by construction.
As we shall now see, the morphism α is a global equivalence;
so we can view the chain as a weak morphism of ultra-commuta­tive
monoids from 𝐁^∘ to . 
The ultra-commutative monoid 𝐁^∘ 
is globally connected, whereas  is not, so the morphism β cannot
be a global equivalence. However, the second part of the next proposition shows that
it is as close to a global equivalence as it can be.



  
  
  * The morphism α:𝐁'→𝐁^∘ is a global equivalence 
    of ultra-commutative monoids.
  
  * For every compact Lie group G, the morphism β:𝐁'→ 
    induces a weak equivalence 
    from the G-fixed point space (𝐁'(_G))^G
    to the path component of the unit element in 
    the G-fixed point space ((_G))^G.
  


(i) We let V be a representation of a compact Lie group G and 
compare the two-sided bar constructions for the O(V)-equivariant map
from (V,V^2) to the one-point space

    α̃(V) :  |B_∙ ((V,V^2),O(V), O(V))|  → | B_∙(∗,O(V), O(V))| = E O(V)  .

The group G acts by conjugation on (V,V^2) and on O(V). 
The group O(V) acts freely from the right on the
last factor in the bar construction,
and this right O(V)-action then commutes with
the G-action. The map α̃(V) is (G× O(V))-equivariant.
The map α(V):𝐁'(V)→(V)
is obtained from α̃(V) by passage to O(V)-orbits.
So Proposition <ref>
allows us to analyze and compare the G-fixed points of α(V).
Indeed, the proposition shows that the G-fixed points of 
𝐁'(V)=|B_∙((V,V^2),O(V), O(V))|/O(V)
are a disjoint union, indexed by conjugacy classes of continuous homomorphisms
γ:G→ O(V) of the spaces

    |B_∙((V,V^2),O(V), O(V))| ^Γ(γ) / C(γ)  ,

where Γ(γ) is the graph of γ.
Since fixed points commute with geometric realization
(see Proposition <ref> (iv))
and with products, we have

    | B((V,V^2),O(V), O(V)) |^Γ(γ) =  
     | B_∙(^G(V,V^2),O^G(V), ^G(γ^*(V),V))|  .

If γ^*(V) and V are not isomorphic as G-representations, then
^G(γ^*(V),V) and hence also the bar construction is empty. 
So there is in fact only one summand in the disjoint union
decomposition of ( 𝐁'(V))^G, namely the one indexed
by the representation homomorphism G→ O(V)
that specifies the given G-action on V. We conclude that the 
inclusions of G-fixed points

    ^G(V,V^2) → (V,V^2)     and     O^G(V) →  O(V)

induce a homeomorphism

    |B_∙(^G(V,V^2),O^G(V),∗)|    
    |B_∙((V,V^2),O(V),∗)|^G =  ( 𝐁'(V))^G  .

The same argument identifies the G-fixed points of
the bar construction B( O(V))= 𝐁^∘(V), and we arrive at a commutative square

    |B_∙(^G(V,V^2),O^G(V),∗)|[d]_-[r]   
     B(O^G(V))[d]^
    
    ( 𝐁'(V))^G  [r]_-α(V)^G    (𝐁^∘(V))^G

in which both vertical maps are homeomorphisms.
The space ^G(V,V^2) becomes arbitrarily highly connected as V exhausts 
the complete G-universe _G.
So the upper horizontal quotient map also becomes 
arbitrarily highly connected as V grows.
Hence the map α(V)^G becomes an equivalence

    _i α(V_i)^G :  _i  (𝐁'(V_i))^G  → _i  (𝐁^∘(V_i))^G

on the mapping telescopes over an exhaustive sequence {V_i}_i≥ 1
of G-represen­tations.
Fixed points commute with mapping telescopes, so we conclude that the map 

    _i α(V_i) :  _i 𝐁'(V_i)  → _i 𝐁^∘(V_i)

is a G-weak equivalence.
The mapping telescope criterion of Proposition <ref> 
thus shows that
the morphism α:𝐁'→𝐁^∘ is a global equivalence. 

(ii) We let V be a G-representation, specified by a continuous
homomorphism ρ:G→ O(V).
We show that the map

    β(V)^G  :   (𝐁'(V))^G =   
    | B( (V,V^2), O(V),∗) |^G  → (V)^G

is a weak equivalence of the source onto the path component of (V)^G
that contains the neutral element of the addition.
The claim then follows by passing to colimits over V in s(_G).

We showed in part (i) that the 
inclusions of G-fixed points ^G(V,V^2)→(V,V^2)
and O^G(V)→ O(V) induce a homeomorphism

    | B_∙(^G(V,V^2),O^G(V),∗)|    
    |B_∙((V,V^2),O(V),∗)|^G =  ( 𝐁'(V))^G  .

Since the precomposition action of O^G(V) on ^G(V,V^2) is free, 
and ^G(V,V^2) is cofibrant as an O^G(V)-space,
the homotopy orbits map by a weak equivalence to the strict orbits, 

    (𝐁'(V))^G  =  
     | B( (V,V^2), O(V),∗) |^G   ≃  ^G(V,V^2) /  O^G(V)  .

The map β(V)^G factors as the composite

    (𝐁'(V))^G   ≃  ^G(V,V^2) /  O^G(V)  → ( (V,V^2) / O(V) )^G  =  (V)^G   ,

where the second map is induced by the inclusion ^G(V,V^2) →(V,V^2).
The space (V)^G is the Grassmannian of G-invariant subspaces of V^2
of the same dimension as V, 
and the space ^G(V,V^2) /  O^G(V) consists of 
those subspaces that are G-isomorphic to V.
This is precisely the path component of (V)^G containing
the neutral element.




We have now identified the G-equivariant path components of the three ultra-commutative
monoids 𝐁^∘,  and ,
and they are isomorphic to the trivial group, the augmentation ideal (G)
respectively the real representation ring (G).
Now we determine the entire homotopy types of the
G-fixed point spaces of the three ultra-commutative
monoids 𝐁^∘,  and .

 
Let G be a compact Lie group.
  
  
  * The G-fixed point space of 𝐁^∘ is a
  classifying space of the group O^G(_G) 
  of G-equivariant orthogonal isometries of the complete G-universe:
  
    (𝐁^∘(_G))^G  ≃  B( O^G(_G) )

  
  * The G-fixed point space of  is a disjoint union,
    indexed by the augmentation ideal (G), subjectaugmentation ideal!of the orthogonal representation ring 
    of classifying spaces of the group O^G(_G): 
        
    ((_G))^G  ≃ (G)× B( O^G(_G) )

  
  * The G-fixed point space of  is a disjoint union,
    indexed by (G), of classifying spaces of the group O^G(_G): 
    
    ((_G))^G  ≃ (G)× B( O^G(_G) )

  


(i) This is almost a tautology.
Since G-fixed points commute with geometric realization
(see Proposition <ref> (iv))
and with products, they commute with the bar construction.
So the space (𝐁^∘(V))^G is homeomorphic to B (O^G(V)) 
for every finite-dimensional G-representation V.
Since G-fixed points also commute with the filtered colimit at hand
(see Proposition <ref> (ii)), we have

    (𝐁^∘(_G))^G     =  ( _V∈ s(_G)𝐁^∘(V) )^G
      
        _V∈ s(_G)( 𝐁^∘(V) )^G
        _V∈ s(_G)  B ( O^G(V)) 
         B (_V∈ s(_G)  O^G(V))  
      = B  (  O^G(_G) )   .


(ii) 
As we explained in Remark <ref>,
the commutative multiplication of  makes the G-space (_G) 
into an E_∞-G-space; so the fixed points (_G)^G
come with the structure of a non-equivariant E_∞-space.
The abelian monoid of path components π_0((_G)^G)
is isomorphic to π_0^G()(G), hence an abelian group.
So all path components of the space (_G)^G
are homotopy equivalent. 
Proposition <ref> identifies the zero path component of (_G)^G
with 𝐁^∘(_G)^G, so part (i) finishes the proof.

The proof of part (iii) is the same as for part (ii), the only difference being
that _0((_G)^G) is isomorphic to the abelian group (G).


The fixed point spaces described in Corollary <ref> 
can be decomposed even further.
As we explained in Example <ref>,
the group O^G(_G) is a weak product of infinite 
orthogonal, unitary and symplectic groups, indexed by the
isomorphism classes of irreducible G-representations λ.
The classifying space construction commutes with weak products, 
which gives a weak equivalence

    B (O^G(_G))  ≃ ∏' B( O^G(_λ))   .

Moreover, the group O^G(_λ) is isomorphic to an infinite orthogonal, unitary
or symplectic group, depending on the type of the irreducible representation λ.


[More bar construction models]
Since the bar construction is functorial and continuous for 
continuous homomorphisms between topological monoids, we can apply it 
objectwise to every monoid valued orthogonal space M 
in the sense of Definition <ref>;
the result is an orthogonal space 𝐁^∘ M.subjectbar construction!of a monoid valued orthogonal space
The bar construction is symmetric monoidal, so if M is symmetric
(and hence an ultra-commutative monoid),
then 𝐁^∘ M inherits an ultra-commutative multiplication.
By the same argument as for 𝐁^∘, the orthogonal space 𝐁^∘ M 
is globally connected.

So the ultra-commutative monoid 𝐁^∘
has variations with  replaced by , , , ,
, , ^c and ^c.subjectunitary group ultra-commutative monoidsubjectsymplectic group ultra-commutative monoidsubjectspin group ultra-commutative monoidsubjectspin^c group ultra-commutative monoidsubjectpin group orthogonal monoid spacesubjectpin^c group orthogonal monoid space
We can also apply the bar construction to morphisms of
monoid valued orthogonal spaces, i.e., morphism of orthogonal 
spaces that are objectwise monoid homomorphisms. 
So hitting all the previous examples with the bar construction
yields a commutative diagram of globally connected orthogonal spaces:

    @C=15mm𝐁^∘[r]^-𝐁^∘incl@–>[d]_𝐁^∘ l   𝐁^∘@–>[d]_𝐁^∘ l@–>[dr]^-𝐁^∘ r   
    𝐁^∘[d]_-𝐁^∘incl[r]^-𝐁^∘ι   𝐁^∘^c [d]_𝐁^∘incl[r]_-𝐁^∘   𝐁^∘[d]^𝐁^∘incl
    𝐁^∘[r]_-𝐁^∘ι   𝐁^∘^c [r]_-𝐁^∘   𝐁^∘

As before, the two dotted arrows mean that the actual morphism goes 
to a multiplicative shift of the target.
With the exception of 𝐁^∘ and 𝐁^∘^c, 
all these orthogonal spaces inherit ultra-commutative multiplications.






We define an orthogonal space symbol - global BO;
its values come with tautological vector bundles whose Thom spaces
form the global Thom spectrum , 
compare Example <ref> below.
For finite and abelian compact Lie groups G, the equivariant
homotopy groups of  are isomorphic to the bordism groups
of smooth closed G-manifolds, compare Theorem <ref>;
so  and  are also geometrically relevant.
In Remark <ref> we define an E_∞-multiplication
on  and show, using power operations, that this E_∞-multiplication
cannot be refined to an ultra-commutative multiplication.

For an inner product space V of dimension n we set

    (V)  =   Gr_n(V⊕^∞) ,

the Grassmannian of n-dimensional subspaces in V⊕^∞.
The structure map (φ):(V)→(W) is given by

    (φ)(L)  =  (φ⊕^∞)(L) + ( (W-φ(V))⊕ 0) ,

the internal orthogonal sum of the image of L under
φ⊕^∞:V⊕^∞→ W⊕^∞
and the orthogonal complement of the image of φ:V→ W, viewed as sitting
in the first summand of W⊕^∞.




We want to describe the equivariant homotopy sets π_0^G()
and the homotopy types of the fixed point spaces (_G)^G,
for every compact Lie group G.
We denote by ^♯(G) the abelian submonoid
of ^+(G) consisting of the isomorphism classes of G-representations
with trivial G-fixed points.
We let V be a G-representation. 
The G-fixed points of (V) are the G-subrepresentations L
of V⊕^∞ of the same dimension as V.
Since G acts trivially on ^∞, the `non-trivial summand' 
L^⊥=L-L^G is contained in V^⊥=V-V^G. 
So V^⊥-L^⊥ is a G-representation with trivial fixed points.
We can thus define a map

    ((V))^G  =  ( Gr_|V|(V⊕^∞) )^G  → ^♯(G)

from this fixed point space by sending L ∈(V)^G to [V^⊥-L^⊥].
As before, the isomorphism type of L only depends on the path component of L 
in (V)^G. Moreover, for every linear isometric embedding φ:V→ W
the relation

    ((φ)(L))^⊥    =  ( (φ⊕^∞)(L) +(W-φ(V))⊕ 0 )^⊥
       =  (φ(L^⊥) + (W^⊥-φ(V^⊥)))⊕ 0  =  ( W^⊥- φ(V^⊥-L^⊥) )⊕ 0

shows that

    [W^⊥  - ((φ)(L))^⊥]  =   [φ(V^⊥-L^⊥)]
     =   [V^⊥-L^⊥]  .

So the class in ^♯(G) depends only on the class of L
in π_0^G(), and the assignments assemble into a well-defined map

    π_0^G (  ) =  _V∈ s(_G) π_0( (V)^G ) 
     → ^♯(G) .

For the description of the G-fixed points of  we introduce the abbreviation

    G r_j^G,⊥ =  ( G r_j(_G^⊥) )^G

for the space of j-dimensional G-invariant subspaces of _G^⊥=_G-(_G)^G.
The space G r_j^G,⊥ can be decomposed further:
before taking G-fixed points, G r_j(_G^⊥) 
is G-equivariantly homeomorphic to (^j,_G^⊥)/O(j).
So Proposition <ref> provides a decomposition of
G r_j^G,⊥ as the disjoint union, indexed over conjugacy classes of
continuous homomorphisms α:G→ O(j), of the spaces

    ^G(α^*(^j),_G^⊥)/ C(α)  ,

where C(α) is the centralizer, in O(j), of the image of α.
Conjugacy classes of homomorphism from G to O(j) biject – by restriction
of the tautological O(j)-representation –
with isomorphism classes of j-dimensional G-representations. 
If V=α^*(^j) is such a G-representation, then the space
^G(V,_G^⊥) is empty if V has non-trivial G-fixed points,
and contractible otherwise. Moreover, the centralizer C(α)
is precisely the group of G-equivariant linear self-isometries of V,
which acts freely on ^G(V,_G^⊥).
So if V^G=0, then the orbit 
space ^G(V,_G^⊥)/C(α) is 
a classifying space for the group ^G(V,V)=O^G(V).
So altogether,

    G r_j^G,⊥ ≃ ∐_[V]∈^♯(G), |V|=j B( O^G(V)) .

Every G-representation V is the direct sum of its isotypical components V_λ, 
indexed by the isomorphism classes of irreducible orthogonal G-representations.
If V^G=0, then only the non-trivial irreducibles occur,
and the group O^G(V) decomposes accordingly as a product

    O^G(V)  ∏_[λ]  O^G(V_λ)  ,

indexed by non-trivial irreducible G-representations. 
The irreducibles come in three flavors (real, complex or quaternionic),
and so the group O^G(V_λ) is isomorphic to
one of the groups O(m), U(m), and S p(m), where m 
is the multiplicity of λ in V.
So altogether, G r_j^G,⊥ is a disjoint union
of products of classifying spaces of orthogonal, unitary and symplectic groups.



Let G be a compact Lie group.


  * The G-fixed point space (_G)^G is weakly equivalent to the space

    ∐_j≥ 0   G r_j^G,⊥×  B O  .


  * The map (<ref>) is a bijection from
π_0^G (  ) to ^♯(G).

  * If U is a G-representation with trivial fixed points,
then the path component of (_G)^G indexed by U 
is a classifying space for the group O^G(U)× O.



(i)
We let V be a G-representation with V^G=0 and W a trivial G-representation.
Then every G-invariant subspace L of V⊕ W⊕^∞ is the 
internal direct sum of the fixed part L^G 
(which is contained in W ⊕^∞)
and its orthogonal complement L^⊥=L-L^G (which is contained in the summand V).
This canonical decomposition provides a homeomorphism

    (V⊕ W)^G  =  (G r_|V|+|W|(V⊕ W⊕^∞))^G  ∐_j=0,…,|V|( G r_j(V) )^G ×  G r_j+|W|(W⊕^∞)

sending L to the pair (V- L^⊥, L^G).
Every G-invariant subspace of _G is the direct sum of its fixed
points and their orthogonal complement, so the poset s(_G)
is the product of the two posets s(_G^⊥) and s((_G)^G).
We can thus calculate the colimit over s(_G) in two steps.
For fixed W, passing to the colimit over s(_G^⊥) gives a homeomorphism

    _V∈ s(_G^⊥)(V⊕ W)^G   ∐_j≥ 0  G r_j^G,⊥×  G r_j+|W|(W⊕^∞)  .

The factor G r_j+|W|(W⊕^∞) is a classifying space
for the group O(j+|W|). Passing to the colimit over s((_G)^G)
then provides a weak equivalence

    (_G)^G     =  _W∈ s( (_G)^G)_V∈ s(_G^⊥)(V⊕ W)^G 
       ≃ _W∈ s( (_G)^G)( ∐_j≥ 0  G r_j^G,⊥×  B O(j+|W|) ) 
       ≃ ∐_j≥ 0  G r_j^G,⊥×  B O  .


(ii) Since the orthogonal space  is closed, we can calculate π_0^G() 
as the set of path components of the space (_G)^G, by
Corollary <ref> (i). 
Since the space B O is path connected, part (i) 
allows us to identify π_0((_G)^G) with
the disjoint union, over j≥ 0, of the path components of G r_j^G,⊥.
As we explained in the weak equivalence (<ref>),
the set π_0(G r_j^G,⊥ ) bijects with the set of isomorphism classes of
j-dimensional G-representations with trivial fixed points.
Altogether this identifies π_0((_G)^G) with ^♯(G),
and unraveling all definitions shows 
that the combined bijection between π_0^G() and ^♯(G)
is the map (<ref>).

(iii) This is a direct consequence of part (i) and the
description (<ref>) of the path components of the space
G r_j^G,⊥.




If H is a closed subgroup of a compact Lie group G 
and V a G-represen­tation with V^G=0,
then V may have non-zero H-fixed points.
So the restriction homomorphism ^G_H:^+(G)→^+(H)
does not in general take ^♯(G) to ^♯(H).
So the monoids ^♯(G) do not form a sub Rep-functor
of , and Proposition <ref> does not describe
_0() as a Rep functor.
We will give a description of _0() as sub-Rep monoid
of the augmentation ideal global power monoid  in 
Proposition <ref> below.

As we shall now explain, the global homotopy type of the orthogonal space 
is that of a sequential homotopy colimit, in the category of orthogonal spaces, 
of global classifying spaces of the orthogonal groups O(m):

    ≃ _m≥ 1  B_ O(m)

The homotopy colimit is taken over morphisms 
B_ O(m)→ B_ O(m+1) that classify the homomorphisms
O(m)→ O(m+1) given by A↦ A⊕.
To make this relation rigorous, we define a filtration 

    ∗  _(0) ⊂ _(1) ⊂ … ⊂ _(m) ⊂ …

of  by orthogonal subspaces. At an inner product space V we define

    _(m)(V)  =   G r_|V|(V⊕^m) ;

here we consider ^m as the subspace 
of vectors of the form (x_1,…,x_m,0,0,…)
in ^∞.
The inclusion of _(m) into _(m+1) is a closed embedding,
so the global invariance property of Proposition <ref> (viii)
entitles us to view the union  as a global homotopy colimit of
the filtration.

The tautological action of O(m) on ^m is faithful, 
so the semifree orthogonal space B_ O(m) = _O(m),^m
is a global classifying space for O(m). We define a morphism 
γ_m:B_ O(m)→_(m) by

    γ_m(V) :  (^m,V)/ O(m)     →   G r_|V|(V⊕^m) =  _(m)(V)
     , 
    φ· O(m)     ⟼ (V-φ(^m))⊕^m  .

We omit the straightforward verification that these maps indeed form
a morphism of orthogonal spaces.
The semifree orthogonal space B_O(m) comes with a
tautological class u_O(m),^m, defined in (<ref>),
which freely generates the Rep-functor _0(B_ O(m)).
We denote by

    u_m  =  (γ_m)_*( u_O(m),^m ) ∈ π_0^O(m)( _(m))

the image  in _(m) of the tautological class.
The following proposition justifies the claim 
that  is a homotopy colimit of the orthogonal spaces B_ O(m).


For every m≥ 0 the morphismsubjectglobal classifying space!of O(n) 

    γ_m   :  B_ O(m) → _(m)

is a global equivalence of orthogonal spaces.
The inclusion _(m)→_(m+1)
takes the class u_m to the class

    ^O(m+1)_O(m)(u_m+1) ∈ π_0^O(m)(_(m+1))  .



  The morphism γ_m factors as the composite of two morphisms of
  orthogonal spaces
  
    _O(m),^m _O(m),^m∘ i _⊕^^m ( _O(m),^m)   [] (-)^⊥  _(m) .

  For the first morphism we let i:V→ V⊕^m denote the embedding 
  of the first summand, and _⊕^^m is the additive shift by ^m 
  as defined in Example <ref>;
  the first morphism is a global equivalence 
  by Theorem <ref>.
  At an inner product space V, the second morphism is the map
  
    (^^m_⊕_O(m),^m)(V) =  (^m,V⊕^m)/O(m)     → G r_|V|(V⊕^m) , 
    φ· O(m)     ⟼  φ(^m)^⊥ ,

  the orthogonal complement of the image.
  This is a homeomorphism, so the second morphism is an isomorphism.
  Altogether this shows that γ_m is a global equivalence.
  
  The second claim is also reasonably straightforward from the definitions,
  but it needs one homotopy. 
  We let j:^m→^m+1 denote the linear isometric embedding defined by
  j(x_1,…,x_m)=(x_1,…,x_m,0).
  The tautological class u_O(m),^m is defined as the path component of the point
  
    _^m· O(m)  ∈ ((^m,^m)/O(m))^O(m) ,

  the O(m)-orbit of the identity of ^m.
  So the classes
  
    incl_*( u_m )      and    ^O(m)_O(m+1)(u_m+1)    in π_0^O(m)(_(m+1))

  are represented by the two subspaces
  
    (^m+1-j(^m))⊕ j(^m)      respectively     
      0⊕^m+1

  of ^m+1⊕^m+1. These two representatives are not the same.
  However, the O(m+1)-action on _(m+1)(^m+1),
  and hence also the restricted O(m)-action, 
  is through the first copy of ^m+1
  (and not diagonally!). So there is a path of
  O(m)-invariant subspaces of ^m+1⊕^m+1 connecting the
  two representatives.
  The two points thus represent the same class in π_0^O(m)( _(m+1)),
  and this proves the second claim.



[Commutative versus E_∞-orthogonal monoid spaces]
Non-equivariantly, every E_∞-multiplication on an 
orthogonal monoid space can be rigidified to a strictly commutative multiplication.
We will now see that this is not the case globally,
with power operations being an obstruction.

To illustrate the difference between a strictly commutative multiplication 
and an E_∞-multiplication,
we take a closer look at the orthogonal space .
If we try to define a multiplication on  in a similar way as for , 
we run into the problem that ^∞⊕^∞
is different from ^∞; even worse, although
^∞⊕^∞ and ^∞ are isometrically isomorphic, there is no
preferred isomorphism. The standard way out is to use all isomorphisms
at once, i.e., to parametrize the multiplications by the E_∞-operad
of linear isometric self-embeddings of ^∞. We recall that the n-th
space of the linear isometries operadsubjectlinear isometries operad
is 

    (n)  =  ((^∞)^n,^∞)  ,

with operad structure by direct sum and composition of linear isometric embeddings
(see for example <cit.> for details).
For all n≥ 0 and all inner product spaces V_1,…,V_n
we define a linear isometry

    κ :  (V_1⊕^∞)⊕…⊕(V_n⊕^∞)  
    V_1⊕…⊕ V_n⊕(^∞)^n

by shuffling the summands, i.e.,

    κ(v_1,x_1,…,v_n,x_n) =  (v_1,…,v_n,x_1,…,x_n) .

We can then define a continuous map

    μ_n :  (n)×(V_1)×…×(V_n) → (V_1⊕…⊕ V_n)

by

    μ_n(φ,L_1,…,L_n) =  
    ((V_1⊕…⊕ V_n)⊕φ)∘κ)(L_1⊕…⊕ L_n)  .

For fixed φ these maps form a multi-morphism, so the universal
property of the box product produces a morphism of orthogonal spaces 

    μ_n(φ,-) :  ⊠…⊠ →  .

For varying φ, these maps define a morphism 
of orthogonal spaces 

    μ_n :  (n)× (⊠…⊠) →  .

As n varies, all these morphism together make the orthogonal space 
into an algebra (with respect to the box product)
over the linear isometries operad . Since the linear isometries operad
is an E_∞-operad, we call , endowed with this -action,
an E_∞-orthogonal monoid space.subjectE infty@E_∞-orthogonal monoid space


Now we explain why the E_∞-structure on  cannot
be refined to an ultra-commutative multiplication.
An E_∞-structure gives rise to abelian monoid
structures on the equivariant homotopy sets. In more detail,
we let R be any E_∞-orthogonal monoid space, such as for example .
We obtain binary pairings

    π_0^G(R)×π_0^G(R)  × π_0^G(R⊠ R) π_0^G(μ_2(φ,-)) π_0^G(R) ,

where φ∈(2) is any linear isometric embedding of
(^∞)^2 into ^∞.
The second map (and hence the composite) is independent of φ
because the space (2) is contractible.
In the same way as for strict multiplications in (<ref>),
this binary operation makes π_0^G(R) 
into an abelian monoid for every compact Lie group G,
such that all restriction maps are homomorphisms.
In other words, the E_∞-structure provides a lift of the
Rep-functor _0(R) to an abelian -monoid, i.e., a functor

    _0(R) :  ^ →  Mon  .

This structure is natural for homomorphisms of
E_∞-orthogonal monoid spaces.subjectabelian Rep-monoidsubjectRep-monoid!abelian|seeabelian Rep-monoid

An ultra-commutative monoid can be viewed as
an E_∞-orthogonal monoid space by letting every element of (n)
act as the iterated multiplication. Equivalently: we let the 
linear isometries operad act along the unique homomorphism to the terminal operad
(whose algebras, with respect to the box product, are the ultra-commutative monoids).
For E_∞-orthogonal monoid spaces
arising in this way from  ultra-commutative monoids, 
the products on _0 defined here coincide with those originally defined 
in (<ref>).
For ultra-commutative monoids R, the abelian -monoid _0(R)
is underlying a global power monoid, i.e., it comes with power operations and
transfer maps that satisfy various relations.
We show in Proposition <ref> below 
that the abelian -monoid _0()
cannot be extended to a global power monoid whatsoever;
hence  is not globally equivalent, as an E_∞-orthogonal monoid space,
to any ultra-commutative monoid.
A curious fact, however, is that after global group completion
the E_∞-multiplication of  can be refined to an
ultra-commutative multiplication, compare Remark <ref>.




We compare the E_∞-orthogonal monoid space  to the
ultra-commutative monoid  in the most highly structured way
possible. Every ultra-commuta­tive monoid can be viewed as
an E_∞-orthogonal monoid space, 
and we now define a `weak E_∞-morphism' from  to .
The zigzag of morphisms passes through the orthogonal space ' with values 

    '(V)  =   Gr_|V|(V^2⊕^∞) .

The structure maps of ' are a mixture of those for  and , i.e.,

    '(φ)(L)  =  (φ^2⊕^∞)(L) ⊕ 
    ((W-φ(V))⊕ 0⊕ 0) ,

where now the orthogonal complement of the image of φ
is viewed as sitting in the first summand of W⊕ W⊕^∞.
The linear isometries operad acts on ' in much the same
way as for , making it an  E_∞-orthogonal monoid space. 
Postcomposition with the direct summand embeddings

    V⊕^∞ (v,x)↦(v,0,x) 
    V^2 ⊕^∞ (v,v',0)(v,v') V^2

induces maps of Grassmannians

    (V)   a(V)  '(V)   b(V)  (V)

that form morphisms of E_∞-orthogonal monoid spaces

    a  '   b   .


 
The morphism b:→' is a global equivalence of
orthogonal spaces.


We define an exhaustive filtration 

    =  '_(0) ⊂ '_(1) ⊂ … ⊂ '_(m) ⊂ …

of ' by orthogonal subspaces by setting

    '_(m)(V)  =   G r_|V|(V^2⊕^m) .
 
We denote by =^⊕_ the additive shift functor 
defined in Example <ref>,
and by i_X:X → X the morphism of orthogonal spaces 
given by applying X to the direct summand embeddings V→ V⊕. 
The morphism i_X is a global equivalence 
for every orthogonal space X, by Theorem <ref>.
We define a morphism

    j  :  '_(m+1) →  ('_(m))

at an inner product space V by

    j(V) :  '_(m+1)(V)    =  G r_|V|(V⊕ V⊕^m+1)
       → G r_|V|+1(V⊕⊕ V⊕⊕^m)=  ('_(m))(V)

by applying the linear isometric embedding

    V⊕ V⊕^m+1     →   
    V⊕⊕ V⊕⊕^m
    
    (v,v',(x_1,…, x_m+1))    ⟼ (v,0,v',x_1,(x_2,…, x_m+1))

and adding the first copy of  
(the orthogonal complement of this last embedding).
Then the following diagram commutes:

    @C=15mm'_(m)[r]^-incl[dr]_i_'_(m)   '_(m+1)[d]_j [dr]^-i_'_(m+1)
        ( '_(m)) [r]_-(incl)    ( '_(m+1))

The two diagonal morphisms are global equivalences, hence so is
the inclusion '_(m)→'_(m+1), by the 2-out-of-6 property 
for global equivalences (Proposition <ref> (iii)).
The inclusion of '_(m) into '_(m+1) is also objectwise a closed embedding,
so the inclusion 

    = '_(0) → ⋃_m≥ 0 '_(m) =  '

is a global equivalence, 
by Proposition <ref> (ix).


We recall that (G) denotes the augmentation ideal in the real representation
ring (G) of a compact Lie group G.subjectaugmentation ideal!of the orthogonal representation ring
In the same way as for  we define a monoid homomorphism 

    γ :  π_0^G(')  → (G)

by sending the path component of W∈'(V)^G to the
class [W]-[V].
Then the triangle of monoid homomorphisms
on the right of following diagram commutes:

    π_0^G() [r]^-a_*[dr]    π_0^G(') [d]_γ ^   π_0^G()  [l]^-_-b_*[dl]_(.6)^(<ref>)
       (G)

The two maps on the right are isomorphisms, hence so is the map γ.

The following proposition says that π_0^G() is isomorphic to the
free abelian submonoid of (G) generated by
(λ)· 1-[λ] as λ runs over all
isomorphism classes of non-trivial irreducible G-representations.
We emphasize that for non-trivial groups G,
the monoid π_0^G() does not have inverses,
so  is not group-like.


The composite morphism of abelian -monoids

    _0()   a_*  _0(')   γ

is a monomorphism. For every compact Lie group G the image of
π_0^G() in the augmentation ideal (G) consists of
the submonoid of elements of the form (U)· 1- [U], for G-representations U.
The abelian -monoid _0() 
cannot be extended to a global power monoid.subjectaugmentation ideal!of the orthogonal representation ring


If L⊂ V⊕^∞ is a G-invariant subspace of the same dimension
as V, then

    [L] - [V]      =   
     ((L)-(L^⊥))· 1  + [L^⊥]  - 
     ((V)-(V^⊥))· 1  - [V^⊥]
       =  (V^⊥-L^⊥)· 1 - [V^⊥-L^⊥]

in the group (G).
This show that the following square commutes:

    @C=20mmπ_0^G() [r]^-a_*[d]_(<ref>)   π_0^G(') [d]^γ
    ^♯(G)[r]_[U] ↦ (U)· 1- [U]   (G)

The left vertical map is bijective by Proposition <ref> (ii).
The right vertical map is an isomorphism as explained above. 
This shows the first two claims because the lower horizontal map is injective and
has the desired image.

Now  we argue, by contradiction, that the abelian -monoid _0() 
cannot be extended to a global power monoid.
The additional structure would in particular specify a transfer map

    _A_3^Σ_3 :  π_0^A_3() → π_0^Σ_3()

from the alternating group A_3 to the symmetric group Σ_3.
Since the monoid π_0^e() has only one element,
the double coset formula shows that

    ^Σ_3_(1 2)∘_A_3^Σ_3 = _e^(1 2)∘^A_3_e =  0  :  π_0^A_3() → π_0^(1 2)()   .

The group Σ_3 has two non-trivial irreducible orthogonal representations,
the 1-dimension sign representation σ and
the 2-dimensional reduced natural representation ν.
So π_0^Σ_3() `is' (via γ∘ a_*)
the free abelian submonoid of (Σ_3) generated by

    1 -σ     and     2 - ν .

We abuse notation and also write σ for the 1-dimensional
sign representation of the cyclic subgroup of Σ_3 generated by
the transposition (1 2).
Then

    ^Σ_3_(12)(1 -σ)  =  1 -σ   and  ^Σ_3_(12)(2 -ν)  =  2 - ( 1 +σ) =  1 - σ ;

so the only element of π_0^Σ_3() that restricts to 0 in
π_0^(1 2)() is the zero element. 
Hence the transfer map (<ref>) must be the zero map.
However, another instance of the double coset formula is

    ^Σ_3_A_3∘_A_3^Σ_3 =   +  (c_(1 2))^* =  2·  :  π_0^A_3() → π_0^A_3()   .

The second equality uses that conjugation by the transposition (1 2) 
is the non-trivial automorphism of A_3,
which acts trivially on (A_3). 
Since (A_3) is a non-trivial free abelian monoid, 
this contradicts the relation _A_3^Σ_3=0.
So the abelian -monoid _0() cannot be endowed with transfer maps
that satisfy the double coset formulas.




[Periodic ]
The E_∞-orthogonal monoid space 
also has a straightforward `periodic' variant  that we briefly discuss.
symbol - periodic global BO
For an inner product space V we set

    (V)  =  ∐_n≥ 0  Gr_n(V⊕^∞) ,

the disjoint union of all the Grassmannians in V⊕^∞.
The structure maps of  are defined in exactly the same way as for .
The orthogonal space  is naturally -graded: for m∈ we let

    ^[m](V) ⊂ (V)

be the path component consisting of all subspaces L⊂ V⊕^∞
such that (L)=(V) + m. For fixed m these spaces
form an orthogonal subspace ^[m] of . 
The E_∞-multiplication of  extends naturally
to a -graded E_∞-multiplication on , 
taking  ^[m]⊠^[n] to ^[m+n].
Moreover, =^[0], the homogeneous summand of  of degree 0.



We offer two descriptions of the global homotopy type of the orthogonal space 
in terms of other global homotopy types previously discussed.
As we show in Proposition <ref> below, 
each of the homogeneous summands ^[m]
is in fact globally equivalent to the degree 0 summand , and hence

    ≃ ×

globally as orthogonal spaces.
When combined with Proposition <ref>, this
yields a description of the homotopy types of the fixed point spaces (_G)^G.

To compare the different summands of 
we choose a linear isometric embedding ψ:^∞⊕→^∞
and define an endomorphism ψ_♯:→ of the orthogonal space 
at an inner product space V as the map

    ψ_♯(V) :  (V)  →(V)  ,    L  ⟼(V⊕ψ)( L⊕ ) .

The morphism ψ_♯ increases the dimension of subspaces by 1, so
it takes the summand ^[k] to the summand ^[k+1].
Any two linear isometric embeddings from 
^∞⊕ to ^∞ are homotopic through
linear isometric embeddings, so the homotopy class of ψ_♯
is independent of the choice of ψ.


For every linear isometric embedding ψ:^∞⊕→^∞
the morphism of orthogonal spaces  
ψ_♯:→ is a global equivalence.
Hence for every m∈ the restriction is a global equivalence

    ψ_♯ :  ^[m] →^[m+1] .



We let =^_⊕ denote the additive shift of an orthogonal space
as defined in Example <ref>,subjectshift!of an orthogonal space!additive
and i:→-⊕ is the natural transformation given by the embedding
of the first summand.
The morphism ψ_♯ factors as the composite

    ∘ i   ψ_!    .

The second morphism is defined at V as the map

    ()(V) =  (V⊕) → (V)

that sends a subspace of V⊕⊕^∞ to its image under 
the linear isometric embedding

    V⊕⊕^∞ → V⊕^∞ ,   
    (v,x,y) ⟼ (v,ψ(y,x)) .


The morphism ∘ i: →
is a global equivalence by Theorem <ref>.  
Any two linear isometric embeddings from ^∞⊕ to ^∞
are homotopic through linear isometric embeddings;
in particular, the linear isometric embedding ψ
is homotopic to a linear isometric isomorphism.
Thus ψ_! is homotopic to an isomorphism, hence a global equivalence.
Since both ∘ i and ψ_! are global equivalences, 
so is the composite ψ_♯.


Another description of the global homotopy type of  
is as a global homotopy colimit of a sequence of self-maps of :

    ≃ _m≥ 1  .

The homotopy colimit is taken over iterated instances of a morphism 
→ that classifies the map `add a trivial summand '.
To make this relation rigorous, we use the filtration 

    ∗  _(0) ⊂ _(1) ⊂ … ⊂ _(m) ⊂ …

of  by orthogonal subspaces, 
analogous to the filtration (<ref>) for .
In other words,

    _(m)(V)  =  ∐_n≥ 0  G r_n(V⊕^m) ;
 
as before we consider ^m as the subspace of ^∞
of all vectors of the form (x_1,…,x_m,0,0,…).

We write ^m for ^^m_⊕, the additive shift by ^m
in the sense of Example <ref>.
We define a morphism 
γ_m:^m →_(m) by

    γ_m(V) :  (^m)(V) =  (V⊕^m)     → _(m)(V)
    
      L        ⟼ L^⊥  = (V⊕^m) - L ,

the orthogonal complement of L inside V⊕^m.
We omit the straightforward verification that these maps indeed form
a morphism of orthogonal spaces. Since each map γ_m(V) is a homeomorphism,
the morphism γ_m is in fact an isomorphism of orthogonal spaces.

The morphism

    i_m  :   → ^m

induced by the direct summand embedding V→ V⊕^m
is a global equivalence by Theorem <ref>.
The inclusion of _(m) into _(m+1) is a closed embedding,
so the global invariance property of Proposition <ref> (ix)
entitles us to view the union  as a global homotopy colimit of
the filtration.
This justifies the interpretation of  as a global homotopy colimit
of a sequence of copies of .
For this description to be useful we should
identify the global homotopy classes of the morphisms in the sequence, i.e.,
the weak morphisms

    [∼] γ_m∘ i_m  _(m)  incl _(m+1) [∼] γ_m+1∘ i_m+1   .

As is straightforward from the definition, this weak morphism models
`adding a summand  with trivial action'.

Finally, we compare the E_∞-orthogonal monoid space  to the
ultra-commutative monoid .
In analogy with the non-periodic version in (<ref>),
we introduce the orthogonal space ' with values 

    '(V)  =  ∐_n≥ 0  Gr_n(V^2⊕^∞) .

The structure maps of ' are defined in the same way as for ',
and they mix the structure maps for  and .
Postcomposition with the direct summand embeddings

    V⊕^∞ (v,x)↦(v,0,x) 
    V^2 ⊕^∞ (v,v',0)(v,v') V^2

induces morphisms of E_∞-orthogonal monoid spaces

    a  '   b   .

These morphisms preserve the -grading;
the restrictions to the homogeneous degree 0 summand are precisely
the morphisms with the same names introduced in (<ref>).
The same argument as in Proposition <ref>
also shows that the morphism b:→' is a global equivalence of
orthogonal spaces.

We define a monoid homomorphism 

    γ :  π_0^G(')  → (G)

by sending the path component of W∈'(V)^G to the
class [W]-[V].
Then the triangle of monoid homomorphisms
on the right of following diagram commutes:

    π_0^G() [r]^-a_*[dr]    π_0^G(') [d]_γ ^   π_0^G()  [l]^-_-b_*[dl]_(.6)^(<ref>)
       (G)

The two maps on the right are isomorphisms, hence so is the map γ.
The same argument as in Proposition <ref>
shows that the composite morphism of abelian Rep-monoids
γ∘ a_*:_0() → is a monomorphism, and
for every compact Lie group G the image of
π_0^G() in the representation ring (G) consists of
the submonoid of elements of the form n· 1- [U], for n∈ and U 
any G-representation.




[Complex and quaternionic periodic Grassmannians]
The ultra-commutative monoids  and  and the E_∞-orthogonal monoid
spa­ces  and  have straightforward complex and quaternionic analogs;
we quickly give the relevant definitions for the sake of completeness.
We define the periodic Grassmannians  and  bysymbol - periodic global BUsubjectperiodic global BU
symbol - periodic global BSpsubjectperiodic global BSpsymbol - global BUsymbol - global BSp

    (V)  =  ∐_m≥ 0  Gr_m^(V_^2)   respectively  (V)  =  ∐_m≥ 0  Gr_m^(V_^2) ,

the disjoint union of the respective Grassmannians,
with structure maps  as for . External direct sum
of subspaces defines a -graded ultra-commu­tative multiplication
on  and on , again in much the same way as for .
The homogeneous summands of degree zero are closed under the multiplication
and form ultra-commutative monoids =^[0] 
respectively =^[0]. More explicitly,

    (V)  =    Gr_|V|^(V_^2)     respectively    (V)  =   Gr_|V|^(V_^2) .

The complex and quaternionic analogues of Theorem <ref> 
provide isomorphisms of global power monoids

    _0()        and    _0()    ;

here (G) and (G) are the Grothendieck groups,
under direct sum, of isomorphism classes of unitary respectively symplectic
G-representations. The isomorphisms above match the -grading of
 and  with the grading by virtual dimension of representations,
so they restrict to isomorphisms of global power monoids from
_0() respectively _0() to the augmentation ideal
global power monoids inside  respectively .

Theorem <ref> also generalizes to natural group isomorphisms,
compatible with restrictions,

    - :  _G(A)   → _G(A)     and  - :  _G(A)   → _G(A)

to the equivariant unitary respectively symplectic K-groups,
where G is any compact Lie group and A a compact G-space.
subjectequivariant K-theory

Example <ref> can be modified to define E_∞-orthogonal monoid spaces
symbol - global B U and symbol - global B S p with values

    (V)  =   Gr_|V|^(V_⊕^∞)
       respectively  (V)  =   Gr_|V|^(V_⊕^∞) .

The structure maps and E_∞-multiplication are defined as for .
As orthogonal spaces,  and  are global homotopy colimits
of the sequence of global classifying spaces B_U(m) respectively B_Sp(m).
Periodic versions symbol - periodic global B U 
and symbol - periodic global B S p
are defined by taking the full Grassmannian
inside V_⊕^∞ respectively V_⊕^∞,
as in the real case in Example <ref>.
As orthogonal spaces, the periodic versions  and  
are global homotopy colimits
of iterated instances of the self-maps of ^ respectively ^
that represent `adding a trivial 1-dimensional representation'.






§ GLOBAL GROUP COMPLETION AND UNITS



For every orthogonal monoid space R and every compact Lie group G,
the operation (<ref>)
makes the equivariant homotopy set π_0^G(R) into a monoid,
and this multiplication is natural with respect to restriction maps in G.
If the multiplication of R is commutative,
then so is the multiplication of π_0^G(R).
In this section we look more closely at the group-like ultra-commutative 
monoids, i.e., the ones where all these monoid structures have inverses.
There are two universal ways to make an ultra-commutative monoid group-like:
the `global units' (Construction <ref>)
are universal from the left and
the `global group completion' (Construction <ref>
and Corollary <ref>)
is universal from the right.
In the homotopy category of ultra-commutative monoids, these constructions
are right adjoint respectively left adjoint to the inclusion of group-like objects.
On π_0^G both constructions have the expected effect:
the global units pick out the invertible elements of π_0^G
(see Proposition <ref>),
and the effect of global group completion is group completion of
the abelian monoids π_0^G
(see Proposition <ref>).
Naturally occurring examples of global group completions 
are the morphism i:→ from the additive Grassmannian to
the periodic global version of B O, and its complex and quaternionic
versions, see Theorem <ref>.
At the end of this section we use global group completion to prove
a global, highly structured version of Bott periodicity:
Theorem <ref>
shows that  is globally equivalent, as an
ultra-commutative monoid, to Ω.




The category of ultra-commutative monoids is pointed,
and product respectively box product are the
categorical product respectively coproduct 
in the category of ultra-commutative monoids.
These descend to product and respectively coproduct in the 
homotopy category (umon) of ultra-commutative monoids,
with respect to the global model structure of Theorem <ref>. 
The morphism ρ_R,S:R⊠ S→ R× S is a global
equivalence by Theorem <ref> (i),
so in (umon) the canonical morphism from a coproduct to a product is an isomorphism.
Various features of units and group completions only depend on these
formal properties, and work just as well in any pointed model
category in which coproducts and products coincide up to weak equivalence.
So we develop large parts of the theory in this generality.


 
Let  be a category which has finite products and a zero object.
We write A× B for any product of the objects A and B and
leave the projections A× B→ A and A× B→ B implicit.
Given morphisms f:T→ A and g:T→ B we write
(f,g):T→ A× B for the unique morphism that projects to f
respectively g. We write 0 for any morphism that factors through a zero object.

We call the category  pre-additivesubjectpre-additive category
if `finite products are coproducts';
more precisely, we require that every product A× B
of two objects A and B is also a co-product, with respect to 
the morphisms 

    i_1 = (_A,0) : A →  A× B      and    
    i_2 =  (0,_B) : B → A× B .

In other words, we demand that for every object X the map

    (A× B,X)  → (A,X) ×(B,X)  ,   
    f↦ (f i_1,f i_2)

is bijective. The main example we care about is =(umon),
the homotopy category of ultra-commutative monoids.

In this situation we can define a binary operation on the
morphism set (A,X) for every pair of objects A and X.
Given morphisms a,b:A→ X we let a b:A× A→ X
be the unique morphism such that (a b)i_1=a and (a b)i_2=b.
Then we define 

    a+b =  (a b)Δ : A → X  ,
 
where Δ=(_A,_A):A→ A× A is the diagonal morphism.


The next proposition is well known, but I do not know a convenient reference.


  Let  be a pre-additive category.
    For every pair of objects A and X of 
    the binary operation + makes the set
    (A,X) of  morphisms into an abelian monoid
    with the zero morphism as neutral element.
    Moreover, the monoid structure is natural 
    for all morphisms in both variables, or, equivalently, 
    composition is biadditive.

 The proof is lengthy, but completely formal.
For the associativity of `+' we consider three morphisms a,b,c:A→ X. Then a+(b+c) respectively (a+b)+c are the two 
outer composites around the diagram:

    @R=5mm    A [dl]_Δ[dr]^Δ
    
    A× A [d]_×Δ       A× A[d]^Δ×
    
    A×(A× A) [dr]_a(b c)[rr]^-α       
    (A× A)× A[dl]^(a b) c
        X

Here α is the associativity isomorphism.
The upper part of the diagram commutes because the diagonal morphism
is coassociative. The lower triangle then commutes since
the two morphisms 

    a(b c) ,  ((a b) c)∘α : A×(A× A) → X

have the same `restrictions', namely a, b respectively c.

The commutativity is a consequence of two elementary facts:
first, b a=(a b)τ
where τ:A× A→ A× A
is the automorphism that interchanges the two factors; this follows from
τ i_1=i_2 and  τ i_2=i_1. Second, the diagonal morphism
is cocommutative, i.e., τΔ=Δ:A→ A× A.
Altogether we get 

    a+b = (a b)Δ = (a b)τΔ = (b a)Δ = b+a  .

As before we denote by 0∈(A,X) the unique morphism that factors
through a zero object.
We let p_1:A× A→ A be the projection to the first factor. Then

    (a 0)i_1     =  a  =  a p_1 (,0)  =  a p_1 i_1
       and
    
     (a 0)i_2     =  0  =  a p_1 (0,)  =  a p_1 i_2 .

So we have a 0=a p_1 in (A× A,X).
Hence a+0=(a 0)Δ=a p_1Δ=a; by commutativity we
also have 0+a=a.

Now we verify naturality of the addition.
To check (a+b)c=ac+b c for a,b:A→ X and c:A'→ A we consider
the commutative diagram

    @C=12mm
    A'[d]_Δ[r]^-c    
    A[r]^-Δ[d]^Δ    A× A[d]^a b
    
    A'× A' @/_1pc/[rr]_-ac b c[r]^-c× c    
    A× A [r]^-a b    X

in which the composite through the upper right corner is (a+b)c.
We have 

    (a b)(c× c)i_1 = (a b)(c,0) =  a c = (ac b c)i_1

and similarly for i_2 instead of i_1. 
So (a b)(c× c)=ac b c since both sides
have the same `restrictions' to the two factors of A'× A'.
Since the composite through the lower left corner is
ac+b c, we have shown (a+b)c=ac+b c. Naturality in X is even easier.
For a morphism d:X→ Y we have
d(a b)=d a db:A× A→ Y since both sides 
have the same `restrictions' d a respectively
db to the two factors of A× A. Thus 
d(a+ b)=d a+ d b by the definition of `+'.


Now we introduce the group-like objects in a pre-additive category.

  
  Let  be a pre-additive category.
    For every object A of  the following two conditions are equivalent:
    
    
  * The shearing morphism 
      Δ i_2=(Δ p_1)+i_2 p_2:A× A→ A× A 
      is an isomorphism.
    
  * The identity of A has an inverse in the abelian monoid (A,A). 
  
We call A group-likesubjectgroup-like
if it satisfies (a) and (b).
If A is group-like, then moreover for every object X of  
the abelian monoids (A,X) and (X,A) have inverses. 


(a)⟹(b) 
Since the shearing map is an isomorphism, there is a morphism (k,j):A→ A× A
such that

    (_A, 0) =  (Δ i_2)∘ (k,j)  =  (k,k+j)  .

So k=_A and _A+j=0, i.e., j is an additive inverse of the identity of A.

(b)⟹(a) 
If j∈(A,A) is an inverse of the identity of A, 
then the morphism

    (p_1,j_A) =  (_A,j) i_2  :  A× A  → A× A

is a two-sided inverse to the shearing morphism, which is thus an isomorphism.

If j∈(A,A) is an additive inverse to the identity of A, 
then for all f∈(X,A)

    f +(j f) =  (_A∘ f)+ (j∘ f) =  
    (_A+j)∘ f =  0∘ f  =  0 ;

so j∘ f is inverse to f. Similarly, for every g∈(A,X) 
the morphism g∘ j is additively inverse to g.




In the next definition and in what follows, we denote by M^×
the subgroup of invertible elements in an abelian monoid M.


Let  be a pre-additive category.
A morphism u:R^×→ R is a 
unit morphismsubjectunit morphism!in a pre-additive category
if for every object T the map

    (T,u) :  (T,R^×) → (T,R)

is injective with image the subgroup (T,R)^× of invertible elements.
A morphism i:R→ R^⋆ in  is a 
group completionsubjectgroup completion!in a pre-additive category  
if for every object T the map

    (i,T) :  (R^⋆,T ) → (R,T)

is injective with image the subgroup (R,T)^× of invertible elements.






If u:R^×→ R is a unit morphism 
then the abelian monoid (R^×,R^×) is a group, by the defining property;
so the object R^× is in particular group-like.
Since the pair (R^×,u) represents the functor

    → (sets) ,   
    T  ⟼ (T,R)^× ,

it is unique up to preferred isomorphism.
A formal consequence is that if we choose  
a unit morphism u_R:R^×→ R  for every object R, 
then this extends canonically to a functor

    (-)^× :   →

and a natural transformation u:(-)^×→.
Since the functor (-)^× takes values in group-like objects, 
it is effectively a right adjoint to the inclusion of
the full subcategory of group-like objects.

A category  is pre-additive if and only if its opposite
category ^ is pre-additive. 
Moreover, in that situation (A,X) and ^(X,A)
are not only the same set (by definition), but they also have the
same monoid structure. Thus the concepts of unit morphism and
group completion are `dual' (or `opposite') to each other:
a morphism is a unit morphism in  if and only
if it is a group completion in ^. 
This is why many properties of unit maps have a corresponding `dual'
property for group completions, and why most proofs for unit maps 
have `dual' proofs for group completions.
Since the identity of any object of  is also the identity of
the same object in ^, 
part (b) of Proposition <ref>  
shows that `group-like' is a self-dual property:
an object is group-like in  if and only if it is group-like in ^.

So the above properties of unit morphisms dualize:
if i:R→ R^⋆ is a group completion, then R^⋆
is in particular group-like.
The pair (R^⋆,i) is unique up to preferred isomorphism,
and if we choose a group completion i_R:R→ R^⋆ 
for every object R, then this extends canonically to a functor

    (-)^⋆ :   →

and a natural transformation i:→ (-)^⋆,
producing a left adjoint to the inclusion of group-like objects.





[Unit morphisms and group completion for abelian monoids] 
The category of abelian monoids is the prototypical example of
a pre-additive category, and the general theory of units and group completions
is an abstraction of this special case. 
So we take the time to convince ourselves that the concepts of `unit morphism'
and `group completion' have their familiar meanings in the motivating example.

A basic observation is that in the category of abelian monoids, 
the abstract addition of morphism as in Proposition <ref>
is simply pointwise addition of homomorphisms. So an abelian monoid
is group-like in the abstract sense of Proposition <ref> 
if and only if every element has an inverse; so the group-like objects
are precisely the abelian groups.

A given homomorphism f:M→ N of abelian monoids is invertible 
if and only if it is pointwise invertible in N, 
which is the case if any only if the image of f lies in
the subgroup N^× of invertible elements.
So the inclusion u:N^×→ N of the subgroup of 
invertible elements is a unit morphism in the sense of Definition <ref>.

We recall the Grothendieck groupsubjectGrothendieck group!of an abelian monoid of an abelian monoid M.
An equivalence relation ∼ on M^2 is defined by
declaring (x,y) equivalent to (x',y') if and only if there is an element z∈ M
with 

    x + y' + z  =  x' + y + z  .

The componentwise addition on M^2 is well-defined on equivalence classes,
so the set of equivalence classes

    M^⋆ =  M^2/∼

inherits an abelian monoid structure. We write [x,y]
for the equivalence class in M^⋆ of a pair (x,y).
The pair (x+y,y+x) is equivalent to (0,0), so 

    [x,y] + [y,x]  =   [x+y,y+x] =  0

in the monoid M^⋆. This shows that every element of M^⋆ has an inverse,
and M^⋆ is an abelian group.
We claim that the monoid homomorphismsubjectgroup completion!of an abelian monoid 

    i  :  M  → M^⋆ ,     i(x) =  [x,0]

is a group completion in the sense of 
Definition <ref>. 
Indeed, given a monoid homomorphism h:M→ N that is pointwise invertible,
then we can define f:M^⋆→ N by

    f[x,y] =  h(x) - h(y) .

A routine verification shows that f is indeed a well-defined homomorphism
and that sending h to f is inverse to the restriction map

    Mon(i,N)  :   Mon(M^⋆,N)  →  Mon(M,N)^× .

A slightly different way to summarize the construction of the Grothendieck group
is to say that the group completion of an abelian monoid M is a cokernel,
in the category of commutative monoids, of the diagonal morphism
Δ:M → M× M. 

We observe that

    [x,y] =  [x,0] + [0,y] =  i(x) - i(y)  ,

so every element in M^⋆ is the difference of two elements in the
image of i:M→ M^⋆. Moreover, if x,x'∈ M satisfy i(x)=i(x'),
then the pairs (x,0) and (x',0) are equivalent, which happens if and only if
there is an element z∈ M such that x+z=x'+z.
Conversely, these properties of the Grothendieck construction
characterize group completions of abelian monoids:
a homomorphism j:M→ N of abelian monoids is a group completion
if and only if the following three conditions are satisfied:


  * the monoid N is a group;

  * every element in N is the difference of two elements in the
image of j;

  * if x,x'∈ M satisfy j(x)=j(x'), then 
there is an element z∈ M such that x+z=x'+z.

Indeed, the first condition guarantees that j extends (necessarily uniquely)
to a homomorphism M^⋆→ N; the second and third conditions guarantee that
the extension is surjective respectively injective, hence an isomorphism.





We mostly care about the situation where =() is the homotopy category
of a pointed model category , such as the category of ultra-commutative monoids.
As we shall now proceed to prove, in this situation
units and group completions always exist. 

We consider two composable morphisms
f:A→ B and g:B→ C in a pointed category .
We recall that f is a kernel of g if g f=0 and
for every morphism β:T→ B such that gβ=0,
there is a unique morphism α:T→ A  such that fα=β.
Dually, g is a cokernel of f if g f=0 and
for every morphism β:B→ Y such that β f=0,
there is a unique morphism γ:C→ Y  such that γ g=β.




Let R be an object of a pre-additive category .


  * Let e:R^×→ R× R be a kernel 
of the codiagonal morphism :R× R→ R.
Then the composite

    u  =  ( 0)∘ e  :  R^× → R

is a unit morphism and 

    e  =  (u, - u) : R^× → R× R  .

Conversely, if u:R^×→ R is a unit morphism,
then the  morphism (u,-u): R^×→  R× R
is a kernel of the codiagonal morphism :R× R→ R.

  * Let d:R× R→ R^⋆ be a cokernel 
of the diagonal morphism (,):R→ R× R.
Then the composite

    i  =  d∘(,0) :  R  → R^⋆

is a group completion and 

    d  =  i (- i ) : R× R → R^⋆ .

Conversely, if i:R→ R^⋆ is a group completion, 
then the  morphism i (- i) : R× R →  R^⋆
is a cokernel of the diagonal morphism (,):R→ R× R.



We prove part (ii). Part (i) is dual, i.e., equivalent to
part (ii) in the opposite category ^.
We let T be any object . Then the map

    { (f,g)∈(R,T)^2 |   f +g = 0 } → (R,T)^× ,   
    (f,g)  ⟼ f

is bijective because inverses in abelian monoids, if they exist, are unique.
A cokernel of the diagonal morphism is a morphism d:R× R→ R^⋆ 
that represents the left hand side of this bijection;
a group completion is a morphism i:R→ R^⋆ 
that represents the right hand side of this bijection.
Hence d:R× R→ R^⋆ is a cokernel of the diagonal 
if and only if d∘(,0) is a group completion.

The relation d∘ (,0)=i holds by definition.
The relation

    ( d∘ (0,) ) + i    =   
     d∘  ( (0,)  +   (,0))  =   d∘   (,)    =  0

holds in the monoid (R,R^⋆), and thus 
d∘ (0,) = - i. This shows that d= i (- i).



The previous characterization of unit morphisms and group completions
as certain kernels respectively cokernels formally implies the following corollary.


Let F:→ be a functor between pre-additive categories that
preserves products.


  * If F preserves kernels of splitable epimorphisms, then for every
unit morphism u:R^×→ R in , the morphism
F u:F( R^×)→ F R is a unit morphism.

  * If F preserves cokernels of splitable monomorphisms, then for every
group completion i:R → R^⋆ in , the morphism
F i:F R→ F( R^⋆) is a group completion.





Now we consider a pointed model category  whose homotopy category is
pre-additive. The main example we have in mind is =umon,
the category of ultra-commutative monoids with the global model structure 
of Theorem <ref>. 
The homotopy category () then comes with 
an adjoint functor pair (Σ,Ω) of suspension and loop,
compare <cit.>.

 
Let  be a pointed model category whose homotopy category is
pre-additive. 


  * For every object R of ,
the loop object Ω R  and the suspension Σ R 
are group-like in ().

  * If u:R^×→ R is a unit morphism, then its loop
Ω u:Ω (R^×)→Ω R is an isomorphism in ().

  * If i:R→ R^⋆ is a group completion, then its suspension
Σ i:Σ R→Σ( R^⋆) is an isomorphism in ().



(i) This is a version of the Eckmann-Hilton argument. 
For every object T of , the set [T,Ω R]
has one abelian monoid structure 
via Construction <ref>, coming from the fact that ()
is pre-additive.
A second binary operation on the set [T,Ω R]
arises from the fact that Ω R is a group object in (),
compare <cit.>. This operation makes [T,Ω R] into a group. 
The monoid structure of Construction <ref>
is natural for morphisms in the second variable Ω R, in particular
for the group structure morphism  Ω R×Ω R→Ω R.
This means that the two binary operations satisfy the interchange law.
Since they also share the same neutral element, they coincide.
Since one of the two operations has inverses, so does the other.
 
The argument that Σ R is group-like is dual, using 
that Σ R is the loop object of R in ()^=(^),
and that `group-like' is a self-dual property.

(ii) 
Since the functor Ω:()→() is right adjoint to Σ,
it preserves products and kernels.
So Ω u:Ω(R^×)→Ω R 
is a unit morphism by Corollary <ref>.
Since Ω R is already group-like by part (i), Ω u is an isomorphism.
Part (iii) is dual to part (ii); so it admits the dual proof,
or can be obtained by applying part (ii) to the opposite model category.




Consider a commutative square 

    @C=10mm 
    A [r]^-f [d]     B [d]^g 
    
    C [r]     D

in a pointed model category  such that the object C is weakly contractible.


  * If the square is homotopy cartesian and g admits a section in (),
then the morphism f is a kernel of g in (). 

  * If the square is homotopy cocartesian and f admits a retraction in (),
then the morphism g is a cokernel of f in ().



We prove part (i). Part (ii) can be proved by dualizing the argument or by
applying part (i) to the opposite category with the opposite model structure.
Since the square is homotopy cartesian and C is weakly contractible,
the object A is weakly equivalent to the homotopy fiber, 
in the abstract model category sense, of the morphism g. 
As Quillen explains in Section I.3 of <cit.>,
there is an action map (up to homotopy)

    A × (Ω D)  → A  ,

by an abstract version of `fiber transport'.
For every other object T of , Proposition 4 of <cit.>
provides a sequence of based sets

    [T, Ω B]  [T,Ω g] 
    [T, Ω D]  [T,∂]  [T, A]  [T,f] 
     [T,B]  [T,g]  [T,D]

that is exact in the sense explained in <cit.>,
where [-,-] denotes the morphism sets in the homotopy category of .
In particular, the image of [T,f] is equal to the preimage of
the zero morphism under the map [T,g].

So in order to show that f is a kernel of g it remains to 
check that the map [T,f] is injective. So we consider 
two morphisms α_1,α_2∈[T,A] such that f∘α_1=f∘α_2.
Then by Proposition 4 (ii) of <cit.>, 
there is an element λ∈ [A,Ω D]
such that α_2=α_1·λ. 
Since the morphism g:B→ D has a section, 
so does the morphism Ω g:Ω B→Ω D. So there is a morphism
λ̅∈ [T,Ω B] 
such that λ=(Ω g)∘λ̅.
But all elements in the image of [T,Ω g] act trivially on [T,A],
so 
then

    α_2 =  α_1·λ =  α_1·( (Ω g)∘λ̅)  =  α_1 .





Let  be a pointed model category whose homotopy category is
pre-additive. 


  * Every object of  has a unit morphism and a group completion in (). 

  * If  is right proper, then every object R 
admits a -morphism u:R^×→ R 
that becomes a unit morphism  in the homotopy category (). 

  * If  is left proper, then every object R 
admits a -morphism i:R→ R^⋆ 
that becomes a group completion in the homotopy category (). 



(i) We let R be any object of . 
It suffices to show, by Proposition <ref>,
that the codiagonal morphism :R× R→ R has a kernel in ()
and the diagonal morphism (,):R→ R× R has a cokernel in ().
The arguments are again dual to each other, so we only show the first one.

We can assume without loss of generality that R is cofibrant and fibrant.
Then the fold map ∇:R⨿ R→ R in the model category 
becomes the codiagonal morphism of R in ().
We factor ∇=q∘ j for some weak equivalence j:R⨿ R∼ Q
followed by a fibration q:Q→ R. Then we choose a pullback,
so that we arrive at the homotopy cartesian square:

    P [r]^f[d]     Q[d]^q 
    ∗[r]     R

The morphism q still becomes a codiagonal morphism in (),
and so it has a section. By
Proposition <ref> (i) the morphism f
becomes a kernel of q in (). So the codiagonal morphism
of R has a kernel.  

(ii) We choose a weak equivalence q:R→R̅
to a fibrant object. A unit morphism R̅^×→R̅ exists in () 
by part (i). By replacing the source R̅^×
by a weakly equivalent object, if necessary, we can assume that it
is cofibrant as an object in the model category .
Every morphism in () from a cofibrant to a fibrant object 
is the image of some -morphism under the localization functor, i.e.,
there is a -morphism u̅:R̅^×→R̅ 
that becomes a unit morphism in (). 
By factoring u̅ as a weak equivalence
followed by a fibration we can moreover assume without loss of generality
that u̅ is a fibration. We form a pullback

    R^×[r]^-u [d]_p     R[d]^q_∼
    R̅^×[r]_-u̅   R̅

Since q is a weak equivalence, u̅ a fibration and  is right proper,
the base change p of q is also a weak equivalence.
So u is isomorphic to u̅ in the arrow category in (),
hence u is also a unit morphism of ().
Part (iii) is dual to part (ii).




We claim that unit morphisms and group completions also behave nicely
on derived mapping spaces. We explain this in detail for unit morphisms,
the other case being dual, one more time.
Model categories have derived mapping spaces (i.e., simplicial sets) ^h(-,-), 
giving well-defined homotopy types such that

    π_0 (^h(T,R))   ()(T,R) ,

compare <cit.> or <cit.>.
We let u:R^×→ R be a -morphism that becomes a unit morphism in (),
and T any other object of . Because of the bijection (<ref>)
the map

    u_* :  ^h(T,R^×) → ^h(T,R)

lands in the subspace ^h,×(T,R), defined as the union of those 
path components that represent invertible elements in the monoid ()(T,R).
We claim that u_* is a weak equivalence onto the subspace ^h,×(T,R).
To see this we exploit that both 
^h(T,R^×) and ^h,×(T,R) are group-like H-spaces,
the multiplication arising from the fact T is a comonoid object up to homotopy.
Moreover, the map u_* is an H-map and bijection on path components
(by the universal property of unit morphisms and the bijection (<ref>)).
So it suffices to show that the restriction of u to the identity path components
is a weak equivalence. For this it suffices in turn to show that
the looped map 

    Ω (u_*) :  Ω( ^h(T,R^×)) → Ω(^h,×(T,R))

is a weak equivalence.
But this map is weakly equivalent to 

    ( Ω u)_* :  ^h(T,Ω( R^×)) → ^h(T,Ω R)  .

Since Ω u is an isomorphism in () 
(by Proposition <ref> (ii))
it is a weak equivalence in ,
hence so is the induced map on derived mapping spaces. 


The next proposition will be used to show that loops on the bar construction 
provide functorial global group completions of ultra-commutative monoids.

 
Let  be a pointed model category whose homotopy category is
pre-additive. Suppose that for every group-like object R of  the
adjunction unit η:R→Ω(Σ R) is an isomorphism in ().
Then for every object R of  the
adjunction unit η:R→Ω(Σ R) is a group completion.


We let i:R→ R^⋆ be a group completion, which exists by 
Theorem <ref>.
In the commutative square in ()

    @C=12mm 
    R [r]^-η_R[d]_i    Ω(Σ R) [d]^Ω(Σ i)_
    
    R^⋆[r]_-η_R^⋆^-   Ω(Σ( R^⋆))

the lower horizontal morphism is an isomorphism by hypothesis because R^⋆
is group-like.
The morphism Σ i:Σ R→Σ(R^⋆) is an isomorphism by   
Proposition <ref> (iii),
hence the right vertical morphism
Ω(Σ i) is also an isomorphism.
So η_R is isomorphic, as an object in the comma category R↓(),
to i, and hence also a group completion.




The previous proposition also has a dual statement
(with the dual proof):  
if for every group-like object R of  the
adjunction counit ϵ:Σ(Ω R)→ R is an isomorphism in (),
then ϵ is a unit morphism.
In practice, however, this dual formulation is less useful:
in the important examples that arise `in nature',
for example for ultra-commutative monoids,
the adjunction unit η:R→Ω(Σ R) is an isomorphism 
for all group-like objects R, whereas
the adjunction counit ϵ:Σ(Ω R)→ R is not
always an isomorphism. 



Now we specialize the theory of units and group completions to
ultra-commu­tative monoids. We recall that the category of ultra-commutative
monoids has the trivial monoid as zero object, and
the canonical morphism ρ_R,S:R⊠ S→ R× S
from the coproduct to the product of two ultra-commutative monoids
is a global equivalence by Theorem <ref> (i).
So the homotopy category (umon) is pre-additive.


An ultra-commutative monoid R is 
group-likesubjectgroup-like!ultra-commutative monoid 
if it is group-like as an object of the pre-additive category (umon).
A morphism u:R^×→ R of ultra-commutative monoids is a 
global unit morphismsubjectglobal unit morphismsubjectunits!of an ultra-commutative monoid  
if it is a unit morphism in the pre-additive category (umon).
A morphism i:R→ R^⋆ of ultra-commutative monoids is a 
global group completionsubjectglobal group completionsubjectgroup completion!global|seeglobal group completion  
if it is a group completion in the pre-additive category (umon).



The global model structure on the category of ultra-commutative monoids
is proper, see Theorem <ref>.
Theorem <ref> thus guarantees that every ultra-commutative
monoid R admits 
a global unit morphism u:R^×→ R and
a global group completion i:R→ R^⋆.


As a reality check we show that for ultra-commutative monoids R, 
the abstract definition of `group-like' is equivalent to the requirement
that all the abelian monoids π_0^G(R) are groups.
This part works more generally for all orthogonal monoid spaces,
not necessarily commutative. 
A monoid M (not necessarily abelian) is a group if and only if the shearing map

    χ :  M^2 → M^2  ,    (x,y)  ⟼ (x, x y)

is bijective. Indeed, if M is a group, then the map
(x,z)↦ (x, x^-1 z ) is inverse to χ.
Conversely, if χ is bijective, then for every x∈ M
there is a y∈ M such that χ(x,y)=(x,1), i.e., with x y=1. 
Then χ(x,y x)=(x,x y x)=(x,x)=χ(x,1), so y x=1 
by injectivity of χ. Thus y is a two-sided inverse for x.

For orthogonal monoid spaces R (not necessarily commutative), 
the group-like condition has a
similar characterization as follows. 
The shearing morphismsubjectshearing morphism!of an  orthogonal monoid space
is the morphism of orthogonal spaces

    χ = (ρ_1,μ) :   R⊠ R  → R× R

whose first component is the projection ρ_1 to the first factor
and whose second component is the multiplication morphism μ:R⊠ R→ R.



The multiplication morphism μ:R⊠ R→ R, and hence the
shearing morphism χ, is a homomorphism of orthogonal monoid spaces
only if R is commutative.


Let R be an orthogonal monoid space. Then the following two conditions are equivalent:


  * The shearing morphism χ:R⊠ R→ R× R is a global equivalence
of orthogonal spaces.

  * For every compact Lie group G the monoid π_0^G(R) is a group.

For commutative orthogonal monoid spaces, conditions (i) and (ii) 
are moreover equivalent to being group-like in the pre-additive 
homotopy category of ultra-commutative monoids.


(i)⟹(ii)
The vertical maps in the commutative diagram

    @C=25mmπ_0^G(R⊠ R) [d]_(π_0^G(ρ_1),π_0^G(ρ_2))^[r]^-π_0^G(χ)   π_0^G( R× R) [d]^(π_0^G(p_1),π_0^G(p_2))_
    π_0^G( R ) ×π_0^G( R ) [r]_-(x,y) ⟼ (x, x y)   π_0^G( R ) ×π_0^G( R )

are bijective by Corollary <ref>.
If the shearing morphism is a global equivalence,
then the map π_0^G(χ) is bijective, hence so is the algebraic shearing map
of the monoid π_0^G(R). This monoid is thus a group.

(ii)⟹(i)
Now we assume that all the monoids π_0^G(R) are groups.
We assume first that R is flat as an orthogonal space;
then R⊠ R is also flat, 
by Proposition <ref> (i) for the global family of
all compact Lie groups.
The product R× R is also flat, 
by Proposition <ref>.
Since R⊠ R and R× R are flat, they are also closed as
orthogonal spaces by Proposition <ref> (iii).
We may thus show that for every compact Lie group G the continuous map

    χ^G  =  χ(_G)^G  :  
    (R⊠ R)(_G)^G  → (R× R)(_G)^G =  R(_G)^G × R(_G)^G

is a weak equivalence, compare Proposition <ref>. 
Since the monoid π_0^G(R) has inverses, the shearing morphism
χ:R⊠ R→ R× R induces a bijection on π_0^G,
by the commutative diagram (<ref>) with vertical bijections.
On path components we have

    π_0^G(R⊠ R)   π_0((R⊠ R)(_G)^G)
       and  π_0^G(R× R)   π_0((R× R)(_G)^G) ,

compare Corollary <ref>. 
So we conclude that the map χ^G induces a bijection on path components. 

Now we show that χ^G also induces bijections on homotopy groups
in positive dimensions. We consider a point
x∈ ((R⊠ R)(_G))^G and k≥ 1. 
We let φ:_G^2→_G be any G-equivariant linear isometric embedding.
As we explained in Remark <ref>, 
this map induces an H-space structure
on (R⊠ R)(_G)^G, and hence a continuous map 

    φ_*(-,x)^G :  (R⊠ R)(_G)^G  → 
    (R⊠ R)(_G)^G   .

Since the unit element 1 is a homotopy unit for the H-space structure, 
the element x'=φ_*(1,x) belongs to the same path component as x. 
Since the monoid π_0(R(_G)^G) is isomorphic to the group π_0^G(R),
the H-space structure has inverses.
So the map φ_*(-,x)^G is a homotopy equivalence.
The same argument applies to R× R instead of R⊠ R,
and we obtain a commutative diagram

    @C=20mmπ_k^G((R⊠ R)(_G)^G, 1)
    [r]^-π_k(χ^G,1)[d]^_π_k(φ_*(-,x)^G)   π_k^G( (R× R)(_G)^G, 1) [d]^π_k(φ_*(-,χ^G(x))^G)_
    π_k^G((R⊠ R)(_G)^G, x')
    [r]_-π_k(χ^G,x')   π_k^G( (R× R)(_G)^G, χ^G(x'))

in which both vertical maps are bijective.
So to show that the map χ^G induces bijections of homotopy
groups based at x, it suffices to show this for the special case x=1
of the unit element.

Now the map π_k(χ^G,1) is a group homomorphism such that the composite

    (π_k(R(_G)^G,1))^2  (x,y)↦ x× y   π_k((R⊠ R)(_G)^G,1)  
    π_k(χ^G,1)    π_k((R× R)(_G)^G,1) ((ρ_1)_*,(ρ_2)_*) 
    (π_k(R(_G)^G,1))^2

sends (x,y) to (x,μ_*(x× y)), 
where μ:R⊠ R→ R is the multiplication
map. By the Eckmann-Hilton argument, μ_*(x× y)=x y, the product with respect
to the group structure of π_k(R(_G)^G,1).
The first and third maps are bijective, and so is the composite 
(because π_k(R(_G)^G,1) is a group). So the middle map is bijective.
Altogether this shows that the map χ^G is a weak equivalence.

For general R we choose a global equivalence f:R'→ R of
orthogonal monoid spaces such that R' is flat as an orthogonal space.
One way to arrange this is by cofibrant replacement in the
global model structure of orthogonal monoid spaces
(Corollary <ref> (ii) with R=∗ and =).
Then f⊠ f is a global equivalence by Theorem <ref>
and f× f is a global equivalence 
by Proposition <ref> (vi).
Since χ^R' is a global equivalence by the previous paragraph and
χ^R∘ (f⊠ f)=(f× f)∘χ^R',
the morphism χ^R is also a global equivalence.

Finally, if R is ultra-commutative, then the point-set level shearing morphism χ
becomes the shearing morphism in the sense of 
Proposition <ref> (a) in the pre-additive homotopy
category (umon). So χ is a global equivalence if and only if
the shearing morphism in (umon) is an isomorphism, i.e.,
precisely when R is group-like.



Now we look more closely at global unit morphisms, and we give an explicit,
functorial point set level construction.
For elements in an abelian monoid M, left inverses are automatically
right inverses, and they are unique (if they exist). 
So the subgroup of invertible elements of an abelian monoid
is isomorphic to the kernel of the multiplication map, by

    M^×  (+:M^2→ M)  ,    x  ⟼ (x,-x) .

Proposition <ref> (i) gives an abstract
formulation of this and explains how an abstract kernel of the multiplication map
gives rise to a unit morphism. 
The proof of Theorem <ref>
then shows that the homotopy fiber of the multiplication map,
formed at the model category level, constructs such a kernel.
If we make all this explicit for the model category of
ultra-commutative monoids, we arrive at the following construction.


[Units of an ultra-commutative monoid]subjectunits!of an ultra-commutative monoid
We introduce a functorial pointset level construction
of the global units of an ultra-commutative monoid R.
We define R^× as the homotopy fiber, over the additive unit element 0,
of the multiplication morphism μ:R⊠ R→ R, i.e.,

    R^× =  F(μ)  =  (R⊠ R)×_μ R^[0,1]× _R {0} .

So at an inner product space V, we have

    (R^×)(V) =  (R⊠ R)(V)×_μ(V) R(V)^[0,1]× _R(V){0} ,

the space of pairs (x,ω) consisting of a point x∈ (R⊠ R)(V)
and a path ω:[0,1]→ R(V) such that μ(V)(x)=ω(0)
and ω(1)=0, the unit element in R(V).
Since limits and cotensors with topological spaces of ultra-commutative mo­noids 
are formed on underlying orthogonal spaces, this homotopy fiber 
inherits a preferred structure of ultra-commutative monoid.

We claim that the  composite

    u  :  R^×  p  R⊠ R   ρ_1  R

is a global unit morphism, where p is the projection
onto the first factor. 
Indeed, the commutative square

    R^×[r]^-p [d]_q      R⊠ R [d]^μ
    
    R^[0,1]× _R {0}[r]_-_0    R

is a pullback of ultra-commutative monoids,
by definition, and both horizontal morphisms are strong level fibrations,
where q denotes the projection to the second factor.
So the square is homotopy cartesian.
The multiplication morphism μ has a section, so
Proposition <ref> (i) shows that 
the morphism p:R^×→ R⊠ R becomes a kernel of the multiplication
morphism in the homotopy category (umon).
So u is a unit morphism by Proposition <ref> (i).



We recall from Example <ref>subjectunits!of a global power monoid
that every global power monoid M has a global power submonoid M^× of units;
the value M^×(G) at a compact Lie group G is
the group of invertible elements of M(G).
The next proposition verifies that global unit morphisms have the
expected behavior on _0.

 
Let u:R^×→ R be a unit morphism of ultra-commutative monoids.
  Then the morphism of global power monoids  
  
    _0(u)  :  _0(R^×) → _0(R)

  is an isomorphism onto the global power submonoid (_0(R))^×
  of units of _0(R).


This is a formal consequence of the fact that the functor π_0^G
from the homotopy category of ultra-commutative monoids is representable.
We let G be a compact Lie group and V a non-zero faithful G-representation.
Then the global classifying space B_G=_G,V 
supports the tautological class u_G,V∈π_0^G(B_ G),
compare (<ref>).
We recall that

    u_G^umon = η_*(u_G,V) ∈π_0^G((B_ G))

where η:B_G→(B_ G) is the inclusion of the linear summand.
We claim that evaluation at u_G^umon is an isomorphism of abelian monoids

    (umon)( (B_ G), T )   π_0^G(T)  ,   
    [f] ⟼ f_*(u_G^umon)

for every ultra-commutative monoid T.
Indeed, both sides take global equivalence in T to isomorphisms,
so we may assume that T is fibrant in the global model structure
of ultra-commutative monoids, hence positively static.
Now we consider the composite

    π_0( T(V)^G)     π_0(^umon((B_G),T))   γ_* 
       (umon)( (B_ G), T )  [f]↦ f_*(u_G^umon) π_0^G(T)

with the adjunction bijection and
the map induced by the localization 
functor γ:umon→(umon).
Since V is non-zero and faithful, B_G=_G,V
is positively flat, so (B_G) is cofibrant in the global
model structure of ultra-commutative monoids.
Since (B_G) is cofibrant and T is fibrant,
the middle map is bijective.
The composite is the canonical map π_0(T(V)^G)→π_0^G(T)
to the colimit, which is bijective because T is positively static.

The evaluation isomorphism is natural in the second variable, so we arrive at a 
commutative square of abelian monoids

    (umon)((B_ G),R^×) [r]^-[d]_u_*   π_0^G(R^×) [d]^u_*
    (umon)((B_ G),R) [r]_-   π_0^G(R)

in which both horizontal maps are bijective.
The left vertical map is injective with image the subgroup of
invertible elements; hence the same is true for the right vertical map.



subjectglobal group completion!of an ultra-commutative monoid|(
Now we turn to global group completions of ultra-commutative monoids.
Every ultra-commutative monoid has a group completion in the homotopy category,
by Theorem <ref> (i).
Even better: since the model category of ultra-commutative monoids is left proper,
every ultra-commutative monoid is the source of a global group completion 
in the model category of ultra-commutative monoids,
by  Theorem <ref> (iii).
Now we discuss two functorial pointset level constructions of global group
completions. The first one is dual to Construction <ref> of global units.


[Global group completion of an ultra-commutative mo­noid]
We let R be an ultra-commutative monoid
that is cofibrant in the global model structure of Theorem <ref>. 
We define the cone of R as a pushout in the category of ultra-commutative monoids:

    R{0}[r] [d]_Rincl   ∗[d]
     
    R [0,1][r]     C R

Here  is the tensor of an  ultra-commutative monoid
with a topological space, as explained in Construction <ref>
(not to be confused with the objectwise product of an orthogonal space
with a space).
So the cone is R [0,1],
the tensor of R with the based space ([0,1], 0), 
as defined more generally in (<ref>).
Since R is cofibrant and the global model structure is topological, 
the left vertical morphism
is an acyclic cofibration, and so the cone C R 
is globally equivalent to the zero monoid. 

We can then construct a global group completion as a homotopy cofiber
of the diagonal morphism Δ:R→  R× R, i.e.,
a pushout in the category of ultra-commutative monoids:

    R [r]^-Δ[d]     R× R [d]^d
     
    R [0,1][r]     R^⋆

The left vertical morphism is induced by 1∈ [0,1],
and it is a cofibration since R is cofibrant.
We claim that the  composite

    i  :  R (,0) R× R   d  R^⋆

is a global group completion.
Indeed, the square above is homotopy cocartesian by construction
and the diagonal morphism has a retraction. 
So Proposition <ref> (ii) shows that 
the morphism d:R× R→ R^⋆ becomes a cokernel of the diagonal
morphism in the homotopy category (umon).
So i is a global group completion by Proposition <ref> (ii).

The previous construction of the global group completion can be
rewritten as two-sided bar construction as follows.
The unit interval [0,1]=Δ^1 is also the topological 1-simplex, 
and hence canonically homeomorphic to the geometric
realization of the simplicial 1-simplex Δ[1],symbolΔ[1] - simplicial 1-simplex by

    [0,1] → |Δ[1]| ,    t  ⟼ [ _[1], t]  .

So Proposition <ref> 
shows that C R=R [0,1] is effectively a bar construction
of R with respect to the box product, i.e., the internal realization
of the simplicial object of ultra-commutative monoids 
B(R,Δ[1]), where Δ[1] is pointed by the vertex 0.
Since Δ[1] has (m+1) non-basepoint simplices of dimension m, 
expanding this yields 

    B_m(R,Δ[1])  =  RΔ[1]_m   R^⊠ (m+1) ,

with simplicial structure induced by that of Δ[1].

Since the internal realization of simplicial ultra-commutative monoids
is a coend, it commutes with pushouts. 
So the defining pushout for R^⋆ can be rewritten
as the realization of a simplicial ultra-commutative monoid,
the two-sided bar construction
with respect to the box product:

    R^⋆ =  (R [0,1])⊠_R (R× R)  
    B(R,Δ[1])⊠_R (R× R)  B^⊠(∗, R, R× R) .

In simplicial dimension m,
this bar construction is given by

    B_m^⊠(∗, R, R× R) =   R^⊠ m ⊠  (R× R) ;

the simplicial face morphisms are given by
projections away from the first factor (for d_0),
the multiplication of R on two adjacent factors (for d_1,…,d_m-1)
respectively the action of R on R× R through the diagonal.
The simplicial degeneracy morphisms are inserting the unit of R.
In the context of topological monoids, this bar construction
of a group completion for `sufficiently homotopy commutative' monoids
is sketched by Segal on p. 305 of <cit.>.


The next proposition is a reality check, showing that global group completion
has the expected effect on equivariant homotopy sets.


subjectglobal group completion!of an ultra-commutative monoid
Let i:R→ R^⋆ be a global group completion of ultra-commutative monoids.
Then for every compact Lie group G the map

    π_0^G(i) :  π_0^G(R) → π_0^G( R^⋆)

is an algebraic group completion of abelian monoids and 

    _0(i) :  _0(R) → _0( R^⋆)

is a group completion of global power monoids.


We may assume that R is cofibrant in the global model
structure of ultra-commutative monoids of Theorem <ref>.
As we explained in Construction <ref>, a group completion R^⋆
can then be constructed as the realization of a certain simplicial 
ultra-commutative monoid,
the two-sided bar construction B^⊠(∗, R, R× R), 
where R acts on R× R through the diagonal morphism.
By Proposition <ref>, 
the realization can equivalently be taken internal to the category
of ultra-commutative monoids, or in the underlying category of
orthogonal spaces. The `underlying' realization
is the sequential colimit of partial realizations B^[n],
i.e., `skeleta' in the simplicial direction, defined as the coend

    B^[n] =  ∫^[m]∈_≤ n B^⊠_m(∗, R, R× R)
    ×Δ^m

of the restriction to the full subcategory _≤ n
of  with objects all [m] with m≤ n.
Since _≤ n is contained in _≤ n+1,
there is a canonical morphism 
B^[n]→ B^[n+1] and the realization B^⊠(∗, R, R× R) 
is the colimit of the sequence of orthogonal spaces

    R× R = B^[0] → B^[1] → ⋯ → B^[n] → ⋯ .

The 1-skeleton B^[1] is the pushout of orthogonal spaces:

    @C=12mm 
     R⊠(R× R)×{0,1}[r]^-incl[d]    
     R⊠(R× R)× [0,1][d] 
    
    R× R[r]    B^[1]

The orthogonal space R⊠(R× R)×{0,1}
is the disjoint union of two copies of
R⊠(R× R), and the left vertical map is projection to R× R
on one copy; on the other copy the morphism has the two components

    μ∘(R⊠ p_1) , μ∘(R⊠ p_2) : 
    R⊠(R× R) → R ,

where μ:R⊠ R→ R is the multiplication
and p_1,p_2:R× R→ R are the two projections.
The functor π_0^G takes both the box product and the product
of orthogonal spaces to products of sets, by Corollary <ref>.
So for a compact Lie group G, the set π_0^G(B^[1]) is the coequalizer 
of the two maps

    π_0^G(R)×π_0^G(R)×π_0^G(R) @<.4ex>[r]^-a @<-.4ex>[r]_-b     π_0^G(R)×π_0^G(R)

given by

    a(x,y,z) =  (y,z)     respectively    
     b(x,y,z) =  (x+y,x+z) .

The equivalence relation on the set π_0^G(R)×π_0^G(R)
generated by declaring a(x,y,z) equivalent to b(x,y,z) for all
(x,y,z)∈π_0^G(R) is precisely the equivalence relation
that constructs the Grothendieck group of π_0^G(R),
compare Example <ref>. 
So π_0^G(B^[1]) bijects with the algebraic group completion of π_0^G(R).
For n≥ 1 the passage from B^[n] to B^[n+1] involves
attaching simplices of dimension at least 2 along their boundary,
and this process does not change the path components of G-fixed points
of the values at any G-representation. So the morphism
B^[1]→ B^⊠(∗, R, R× R) induces a bijection on π_0^G.
This proves that the morphism R→ B^⊠(∗, R, R× R)=R^⋆
is a group completion of abelian monoids.

The second claim then follows because group completions 
of global power monoids are calculated `group-wise', compare Example <ref>.



For topological monoids, the loop space of the bar construction
(see Construction <ref>)subjectbar construction!of a topological monoid
provides a functorial group completion. We will now explain that a similar construction
also provides global group completion for ultra-commutative monoids,
before passing to the homotopy category.
Part of this works for arbitrary orthogonal monoid spaces, not necessarily commutative.
We refer to Construction <ref> for generalities
about realization of simplicial objects, in particular simplicial orthogonal spaces.

If R is an orthogonal monoid space, 
then the bar constructionsubjectbar construction!of an orthogonal monoid space
is the simplicial object of orthogonal spaces

    B_∙(R) =  ([n]↦ R^⊠ n)  .

The simplicial face morphisms are induced by the multiplication in R, and
the degeneracy morphisms are induced by the unit morphism of R,
much like for the bar construction with respect to cartesian product
(as opposed to box product) in Construction <ref>.
The geometric realization in the category of orthogonal spaces
is then the orthogonal space

    B(R)  =  | B_∙(R) |  .

Geometric realization of orthogonal spaces is `objectwise',
i.e., for an inner product space V we have

    B(R)(V)   =  |B_∙(R) (V) | ,

the realization of the simplicial space [n]↦ R^⊠ n(V).

The next proposition shows that the bar construction of orthogonal
monoid spaces preserves global equivalences under a mild non-degeneracy
condition on the unit.


  An orthogonal monoid space R has a flat unitsubjectflat unit!of an orthogonal monoid space
  if the unit morphism ∗→ R is a flat cofibration of orthogonal spaces.


The condition that the unit morphism ∗→ R is a flat cofibration
is equivalent to the requirements that the underlying orthogonal space of R
is flat and the unit map ∗→ R(0) is a cofibration of spaces.

We recall from Definition <ref>
that a simplicial orthogonal space X is Reedy flatsubjectReedy flat!orthogonal space
if the latching morphism l_n^Δ:L_n^Δ(X)→ X_n in the simplicial
direction is a flat cofibration of orthogonal spaces for every n≥ 0.


  
  
  * For every orthogonal monoid space R with flat unit
    the simplicial orthogonal space B_∙(R) is Reedy flat.
  
  * Let f:R→ S be a morphism of orthogonal monoid spaces with flat units.
    If f is a global equivalence, so is the morphism B(f):B(R)→ B(S).  
  

  (i) The n-th latching morphism L_n^Δ(B_∙(R))→ B_n(R) in the simplicial
  direction is the iterated pushout product
  
    i^ n :  Q^n(i) → R^⊠ n

  with respect to the unit morphism ∗→ R. Since this unit morphism
  is a flat cofibration, the pushout product property of the flat cofibrations
  shows that i^ n, and hence the latching morphism,
  is a flat cofibration for all n≥ 0.

  (ii) Since R and S have flat units, the simplicial orthogonal spaces
  B_∙(R) and B_∙(S) are Reedy flat by part (i).
  Moreover, the morphism B_n(f)=f^⊠ n:R^⊠ n→ S^⊠ n
  is a global equivalence since f is and because
  the box product is homotopical for global equivalences 
  (by Theorem <ref>).
  So the claim follows from the global invariance of realizations
  between Reedy flat simplicial orthogonal spaces 
  (Proposition <ref> (ii)). 




[Comparing bar constructions]
We let M be a monoid valued orthogonal space 
in the sense of Definition <ref>.subjectorthogonal space!monoid valued 
Then we have two bar constructions available: on the one hand we can
take the bar construction 
objectwise as in Example <ref>,
resulting in the orthogonal space 𝐁^∘ M.subjectbar construction!of a monoid valued orthogonal space
On the other hand, we can first pass to the associated orthogonal monoid space
as in (<ref>), and then perform the bar construction
with respect to the ⊠-multiplication as in (<ref>). 
There is a natural comparison map: the symmetric monoidal transformation
ρ_X,Y : X⊠ Y→ X× Y   
defined in (<ref>) has an analog 
for any finite number of factors, and for varying n, the morphisms

    ρ_M,…,M :  M^⊠ n → M^n

form a morphism of simplicial orthogonal spaces

    ρ_∙ :  B_∙^⊠(M) → B^×_∙(M)

from the ⊠-bar construction to the ×-bar construction.
If M has a flat unit, then the simplicial orthogonal spaces
B_∙^⊠(M) and B_∙^×(M) are both Reedy flat. 
Indeed, for the former, this is 
Proposition <ref> (i),
and for the latter the same proof works because the categorical
product of orthogonal spaces also satisfies the pushout product property
with respect to flat cofibrations 
(Proposition <ref>).
So ρ_∙ is a morphism between Reedy flat simplicial orthogonal spaces that 
is a global equivalence
in every simplicial degree (by Theorem <ref> (i)).
The induced morphism |ρ_∙|:B(M)→𝐁^∘ M 
between the realizations is then a global equivalence 
by Proposition <ref> (ii).



The canonical morphism

    R× [0,1]  =   B_1(R)×Δ^1  → 
     | B_∙(R) | =  B(R)

takes R×{0,1} to the basepoint, so it factors over a morphism
of orthogonal spaces

    R ([0,1]/{0,1})  →  B(R) .

We let u :S^1 →[0,1]/{0,1} be the composite homeomorphism

    S^1   c    U(1) log [0,1]/{0,1} ,

of the Cayley transformsubjectCayley transform 

    c  :  S^1  → U(1)  ,    x  ⟼ (x+i)(x-i)^-1 ,

and the logarithm, i.e., the inverse of the exponential homeomorphism

    [0,1]/{0,1}  U(1) ,    t  ⟼ e^2π i t .

This yields a composite morphism

    R S^1  []R u  R ([0,1]/{0,1})  →  B(R)

which is adjoint to a morphism of orthogonal spaces

    η_R  :   R  → Ω B(R) .





  Let R be an orthogonal monoid space with flat unit.
  If for every compact Lie group G the monoid π_0^G(R) is a group, 
  then the morphism η_R : R→Ω B(R) is a global equivalence.


  Since R has a flat unit, the simplicial orthogonal space B_∙(R)
  is Reedy flat by Proposition <ref> (i),
  so the underlying orthogonal space of B(R) is flat by 
  Proposition <ref> (i). 
  As a flat orthogonal space, B(R) is in particular closed.
  The loop space functor preserves closed inclusions 
  by <cit.>,
  so the pointwise loop space Ω B(R) is also closed as an orthogonal space.
  Since R and Ω B(R) are both closed orthogonal spaces,
  we can detect global equivalences on G-fixed points,
  see Proposition <ref>.

  So we let G be a compact Lie group. Then
  ((Ω B(R))(_G))^G is homeomorphic to
  Ω( ( B(R)(_G))^G ), which is in turn homeomorphic to
  the loop space of the geometric realization of the simplicial space
  
    [n] ⟼  (R^⊠ n(_G))^G  .

  We define i_k:[1]→ [n] by i_k(0)=k-1 and i_k(1)=k.
  Then the morphism 
  
    (i_1^*,…,i_n^*) :  R^⊠ n =  B_n(R) →  (B_1(R))^n  =   R^n

  is precisely the morphism ρ_R,…,R:R^⊠ n→ R^n,
  and hence a global equivalence by Theorem <ref> (i). 
  Since π_0^G(R) is a group, 
  the simplicial space (<ref>)
  satisfies the hypotheses of Segal's theorem <cit.>;
  so the adjoint of the canonical map
  
    R(_G)^G S^1  → 
      |[n]↦ ((R^⊠ n)(_G))^G|     ( B(R)(_G) )^G

  is a weak equivalence. Here we have used again that fixed points commute
  with geometric realization, 
  see Proposition <ref> (iv).
  This adjoint is precisely the underlying map 
  of G-fixed points of the morphism η_R:R→Ω B(R).





The previous proposition works for general orthogonal monoid spaces,
not necessarily commutative; in that generality the bar construction
B(R) is an orthogonal space, but it does not have any natural multiplication. 
When we apply the bar construction to ultra-commutative monoids,
then something special happens: since the multiplication morphism
μ:R⊠ R→ R is then a homomorphism of ultra-commutative monoids,
B_∙(R) is a simplicial object in the category of
ultra-commutative monoids, i.e., a simplicial ultra-commutative monoid.
The geometric realization B(R) is then canonically an
ultra-commu­tative monoid, and it coincides with the realization of 
B_∙(R) internal to the category of ultra-commutative monoids,
compare Proposition <ref>.

Moreover, we claim that for ultra-commutative monoids,
the bar construction B(R) is naturally isomorphic to R S^1,
the `suspension' of R internal to the category of
ultra-commutative monoids.
To see this we consider the `simplicial circle' 𝐒^1,subjectsimplicial circle
the simplicial set given by

    (𝐒^1)_n  =  {0,1,…,n} ,

with face maps d_i:(𝐒^1)_n→ (𝐒^1)_n-1 given by

    d_i(j)  = 
    j-1     for i <j, and
       j     for i ≥ j and j n,
       0     for i=j=n,

and degeneracy maps s_i:(𝐒^1)_n→(𝐒^1)_n+1 given by

    s_i(j)  = 
    j+1     for i  <j, and
       j     for i≥ j.

The simplicial set 𝐒^1 is based by 0; it is isomorphic to 
the simplicial 1-simplex modulo its boundary,
and its realization is homeomorphic to a circle, whence the name.

The  `obvious' isomorphisms

    p_n  :  R^⊠ n     
     R{0,1,…,n} =  R (𝐒^1)_n  =  B_n(R,𝐒^1) ,

are compatible with the simplicial structure maps as n varies,
so they define an isomorphism of simplicial ultra-commutative monoids

    p_∙ :  B_∙(R)  B_∙(R,𝐒^1)  .

When we specialize Proposition <ref> 
to A=𝐒^1, we obtain an isomorphism of ultra-commutative monoids

    R |𝐒^1|  B(R)   .

The homeomorphism u:S^1→[0,1]/{0,1} from (<ref>) 
and the homeomorphism

    [0,1] / {0,1}     |𝐒^1|   ,   
    t  ⟼ [1,t]

turn this into an isomorphism of ultra-commutative monoids

    R S^1 []R u R ( [0,1] / {0,1} )    R |𝐒^1|    B(R)

whose adjoint R→Ω B(R) is the morphism η_R
of (<ref>).



For every ultra-commutative monoid with flat unit R the adjunction unit

    η_R  :  R → Ω(R S^1)

is a global group completion.subjectflat unit!of an ultra-commutative monoid


We let R be a group-like cofibrant ultra-commutative monoid.
Then R has a flat unit by Theorem <ref> (ii a).
Since ultra-commutative monoids form a topological model category,
R S^1 is an abstract suspension of R.
The isomorphism (<ref>)
transforms the adjunction unit R→Ω(R S^1)
into the morphism η_R:R→Ω B(R) defined in (<ref>). 
Proposition <ref> shows
that this adjunction unit is a global equivalence 
for every cofibrant group-like ultra-commutative monoid R.
In the homotopy category (umon) this implies that for
every group-like ultra-commutative monoid R
the derived adjunction unit η:R→Ω(Σ R) 
is an isomorphism.
So Proposition <ref>
shows that for every ultra-commutative monoid R
the derived adjunction unit η:R→Ω(Σ R) 
is a group completion in the pre-additive category (umon).
For cofibrant R the pointset level adjunction unit
R→Ω(R S^1) realizes the derived unit,
hence the claim follows for every ultra-commutative monoid that is
cofibrant in the global model structure of Theorem <ref>.

In the general case we choose a cofibrant replacement q:R^c→ R 
in the global model structure of Theorem <ref>,
i.e., a global equivalence of ultra-commutative monoids with cofibrant source.
Since R^c and R have flat units, the induced morphism of bar constructions
B(q):B(R^c)→ B(R) is a global equivalence
by Proposition <ref> (ii).
Hence the morphism q S^1:R^c S^1→ R S^1 is a global equivalence, and so 
is Ω(q S^1):Ω(R^c S^1)→Ω(R S^1).
The morphism η_R^c:R^c→Ω(R^c S^1) is a global group
completion by the previous paragraph. Since η_R:R→Ω(R S^1) 
is isomorphic to the morphism η_R^c in the homotopy category (umon),
the morphism η_R is also a global group completion.




An example of a global group completion that comes up naturally 
is the morphism i:→ introduced in Example <ref>. 
The verification of the group completion property
will be done through a homological criterion.
For that purpose we define the homology groups of an orthogonal space Y as

    H_*( Y^G;)  =  _V∈ s(_G) H_*( Y(V)^G;)  .

Every global equivalence induces isomorphisms on H_*((-)^G;)
for all compact Lie groups G. Indeed, the functor H_*((-)^G;)
takes strong level equivalences to isomorphisms, which reduces the claim
(by cofibrant approximation in the strong level model structure)
to global equivalences f:X→ Y between flat orthogonal spaces. Flat orthogonal
spaces are closed, so the global equivalence induces weak equivalences
f(_G)^G:X(_G)^G→ Y(_G)^G on G-fixed points.
The poset s(_G) is filtered, so homology commutes with this colimit, i.e.,

    H_*( Y^G;)    H_*( Y(_G)^G;)  .

Thus the morphism f also induces an isomorphism on H_*((-)^G;). 

The multiplication of an orthogonal monoid space R
induces a graded multiplication on the homology groups
H_*(R^G;), by simultaneous passage to colimits in both variables
of the maps

    H_m( R(V)^G;)  H_n( R(W)^G;)   ×     H_m+n( R(V)^G×  R(W)^G;)  
    ( (μ_V,W)^G)_*    H_m+n( R(V⊕ W)^G;)  .

Assigning to a path component its homology class is a map

    π_0(R(V)^G)  → H_0(R(V)^G;)

compatible with increasing V. On colimits over s(_G)
this provides a map

    π_0^G(R)  → H_0(R^G;) .

This map takes the addition in R to the multiplication in H_0(R^G;),
so its image is a multiplicative subset of H_0(R^G;).
If the multiplication of R is commutative, then the  
product of H_*( R^G;) is commutative in the graded sense.
In particular, the multiplicative subset of π_0^G(R)
is then automatically central.


A morphism i:R→ R^⋆ of ultra-commutative monoids is a 
global group completion if and only if the following two conditions
are satisfied.


  * The ultra-commutative monoid R^⋆ is group-like, and

  * for every compact Lie group G the map of graded commutative
rings

    H_*(i^G;) :  H_*( R^G;)  → H_*( (R^⋆)^G;)

is a localization at the multiplicative subset π_0^G(R) of
H_0(R^G;).



We start by showing that a global group completion satisfies properties (i)
and (ii). Property (i) holds by definition of `group completion'. 
We give two alternative proofs for why a global group completion
satisfies property (ii), based on the two different bar construction models
in Construction <ref> respectively 
Corollary <ref>.

The first argument uses the loop space of the bar construction B(R),
which is isomorphic to internal suspension R S^1.
By Corollary <ref> 
it suffices to show that for every cofibrant
ultra-commutative monoid R the morphism η_R:R→Ω(R S^1)=Ω(B R)
has property (ii). 
Since R is cofibrant and the global model structure
of ultra-commutative monoids is topological 
(Theorem <ref>), 
the basepoint inclusion of S^1 induces a cofibration

    Rincl :   R{∞} → R S^1 .

The cobase change is the unique morphism ∗→ R S^1 from the terminal
ultra-commutative monoid to the reduced tensor, and this is thus a cofibration.
In other words, B(R)=R S^1 is again cofibrant as an ultra-commutative monoid.
Since R and B(R) are cofibrant as ultra-commutative monoids,
Theorem <ref> (ii) shows that their underlying orthogonal spaces
are flat, hence closed.
The loop space functor preserves closed inclusions by <cit.>,
so the pointwise loop space Ω B (R) is also closed as an orthogonal space.

Since R and Ω B(R) are closed as orthogonal spaces,
it suffices to show that for every compact Lie group G the map

    H_*( R(_G)^G;)  → H_*( (Ω B (R))(_G)^G;)

is a localization at the multiplicative subset π_0(R(_G)^G) of
the source. Since the H-space structure of R(_G)^G comes from
the action of an E_∞-operad, the graded ring 
H_*( R(_G)^G;) is graded commutative.
We can thus apply Quillen's group completion theorem from the unpublished, 
but widely circulated preprint `On the group completion of a simplicial monoid'.
Quillen's manuscript was later published as
Appendix Q of the Friedlander-Mazur paper <cit.>,
where the relevant theorem appears on page 104 in Section Q.9.

The second, alternative, argument first reduces to cofibrant ultra-commuta­tive monoids
by cofibrant approximation in the global model of Theorem <ref>.
As explained in Construction <ref>, 
a group completion R^⋆ can then be constructed
as the homotopy cofiber of the diagonal Δ:R→ R× R,
which is concretely given by the geometric realization of 
the simplicial ultra-commutative monoid B^⊠(∗, R, R× R), 
the bar construction with respect to the box product.
Segal <cit.> sketches an argument why the 
resulting morphism R→ B^⊠(∗, R, R× R) is localization on
homology with field coefficients. The argument is reproduced in more
detail in the proof of <cit.>.

Now we prove the reverse implication.
We let i:R→ R^⋆ be a morphism of ultra-commutative monoids that 
satisfies properties (i) and (ii); we need to show that i is a global group 
completion.
We assume first that both R and R^⋆ are cofibrant 
in the global model structure of ultra-commutative monoids 
of Theorem <ref>.
Then the unit morphisms of R and R^⋆ are flat cofibrations
of underlying orthogonal spaces by Theorem <ref> (ii).
So the morphism η_R:R→Ω(R S^1)=Ω B(R)
is a global group completion by Corollary <ref>.
Since R^⋆ is group-like, the morphism η_R^⋆
is a global equivalence by Proposition <ref>. 

We claim that the morphism B (i):B(R)→ B( R^⋆) is a global equivalence.
For every coefficient system L on (B (R^⋆))(_G)^G 
we compare the spectral sequence

    E^2_p,q =  _p^H_*(R^G;k)(k,L)  ⟹H_*( (B (R))(_G)^G;L)

(obtained by filtering the bar construction by simplicial skeleta)
with the analogous one for the homology of (B (R^⋆))(_G)^G.
The localization hypothesis implies that the map of Tor groups

    _p^H_*(R^G;k)(k,L)  → _p^H_*((R^⋆)^G;k)(k,L)

is an isomorphism, 
see for example <cit.>
or <cit.>.
So we have a morphism of first quadrant spectral sequences
that is an isomorphism of E^2-terms; the map on abutments
is then an isomorphism as well.
This shows that B(i):B (R)→ B (R^⋆) is a global equivalence.
Since looping preserves global equivalences, the morphism 
Ω B (i) is a global equivalence.
Now we contemplate the commutative square

    @C=12mm
    R [r]^-i[d]_η_R    R^⋆[d]^η_R^⋆
    Ω B (R) [r]_-Ω B(i)   Ω B (R^⋆)

The left vertical morphism is a global group completion,
and the lower horizontal and right vertical morphisms are
global equivalences. So the upper horizontal morphism i is global group completion.

Now we reduce the general case to the special case by cofibrant
replacement.
We choose a cofibrant replacement q:R^c→ R in the global model structure of
ultra-commutative monoids of Theorem <ref>,
and then factor the morphism i q:R^c→ R^⋆ as a cofibration
i^c:R^c→ R^† followed by a global equivalence φ:R^†→ R^⋆.
Properties (i) and (ii) are invariant under global equivalences of pairs,
so the morphism i^c:R^c→ R^† satisfies (i) and (ii).
Since R^c and R^† are both cofibrant, the morphism i^c
is a global group completion by the special case above.
So the morphism i is also a global group completion.





We showed in Theorem <ref> that the ultra-commutative 
monoid  is group-like and that its equivariant homotopy sets
_0() realize the orthogonal representation rings additively.
In Example <ref> we introduced a morphism i:→
of ultra-commutative monoids from the additive Grassmannian
and showed in Proposition <ref>
that for every compact Lie group G and every G-space A, the homomorphism

    [A,i]^G  :  [A,]^G  → [A,]^G

is a group completion of abelian monoids.
In particular, the map π_0^G(i):π_0^G()→π_0^G()
is an algebraic group completion.
In much the same way we can define morphisms of ultra-commutative monoids

    i : ^ →      and    i : ^ →

by replacing -subspaces in V by -subspaces in V_, 
respectively -subspaces in V_.


The morphisms i:→, i:^→
and i:^→ are global group completions  
of ultra-commutative monoids.


We prove the real case in detail and leave the complex and quaternionic cases
to the reader.
We verify the localization criterion of Proposition <ref>.
To this end we define a bi-orthogonal space, i.e., a functor

    ^♯ :  × →

on objects by

    ^♯(U,V) =  (U⊕ V)  .

For linear isometric embeddings φ:U→U̅ and ψ:V→V̅, 
the induced map is

    ^♯(φ,ψ) :  ^♯(U,V) → ^♯(U̅,V̅)  ,   
    L  ⟼ (φ⊕ψ)(L)  +  ((U̅-φ(U))⊕ 0) .

We emphasize that the behavior on objects is not symmetric in the
two variables, and in the first variable it is not just applying φ.

Now we fix a compact Lie group G and consider the colimit
of the bi-orthogonal space ^♯  over the poset s(_G)× s(_G).
Since the diagonal is cofinal in the poset s(_G)× s(_G),
this `double colimit' is also a colimit over the restriction to the diagonal.
But the diagonal of ^♯ is precisely the orthogonal space , and so

    _(U,V)∈ s(_G)^2^♯(U,V) =  _W∈ s(_G)(W)  =  (_G) .

On the other hand, if we fix an inner product space U as the
first variable, then ^♯(U,-) is isomorphic to
the additive U-shift 
(in the sense of Example <ref>)
of the Grassmannian . Hence for fixed U,

    _V∈ s(_G)^♯(U,V) =  (U⊕_G) .

A colimit over s(_G)× s(_G) can be calculated in two steps,
first in one variable and then in the other, so we conclude that

    (_G) =  ^♯_U∈ s(_G) (U⊕_G) ;

under this identification, the map i(_G):(_G)→(_G) 
becomes the canonical morphism 

    i^♯ :  (_G)  → ^♯_U∈ s(_G) (U⊕_G)

to the colimit, for U=0.

The decoration `♯' is meant to emphasize that the structure maps 
in this colimit system come from the functoriality of ^♯ 
in the first variable, so they are not
the maps obtained by applying (-⊕_G) to an inclusion U⊂U̅.
For example, the maps in the colimit (<ref>)
do not preserve the -grading by dimension.
So one should not confuse the colimit (<ref>)
with the space (_G⊕_G), 
which is G-homeomorphic to (_G) by a choice
is equivariant linear isometry _G⊕_G_G.



We claim that the map

    H_*((i^♯)^G)  :  H_*((_G)^G) → 
    H_*( ^♯_U∈ s(_G)(U⊕_G)^G )

is a localization at the multiplicative subset π_0^G(),
where homology stands for singular homology with integer coefficients.
To see this we observe that all the maps in the colimit system are
closed embeddings; so singular homology commutes with this particular colimit.

For U∈ s(_G) we denote by j_U:(_G)^G→(U⊕_G)^G
the map induced by applying the direct summand inclusion _G→ U⊕_G.
The map j_U is a homotopy equivalence because _G is a complete G-universe.
For all U⊂ V in s(_G) the following square commutes

    @C=15mm
    H_*((_G)^G) [r]^-· [V-U][d]^_H_*(j_U)   
    H_*((_G)^G)[d]_^H_*(j_V)
    
    H_*((U⊕_G)^G) [r]_-H_*(i_U,V^♯)    H_*((V⊕_G)^G)

and the vertical maps are isomorphisms.
So the target of (<ref>) is the colimit 
of the functor on s(_G) that takes all objects to the
ring H_*((_G)^G;) and an inclusion U⊂ V
to multiplication by the class [V-U] in the multiplicative subset
under consideration. Hence the map (<ref>)
is indeed a localization as claimed. Since the ultra-commutative monoid 
is group-like, the criteria of Proposition <ref>
are satisfied, and so the morphism i:→ is a global group
completion.



We had earlier defined an E_∞-orthogonal monoid space ' as a mixture of
 and : the value at an inner product space V is

    '(V)  =  ∐_m≥ 0  Gr_m(V^2⊕^∞) .

The structure maps and an E_∞-multiplication can be defined 
in essentially the same way as for ',
which was defined in (<ref>).
So ' becomes the -graded periodic analog
of the orthogonal space '.
In the same way as for the homogeneous degree 0 summands in (<ref>),
we defined two morphisms
of E_∞-orthogonal monoid spaces

    a  '  [≃] b   .

The same arguments as in Proposition <ref>
show that the morphism b is a global equivalence.
A very similar argument as in Proposition <ref>
shows that the morphism a
is a global group completion in the homotopy category of
E_∞-orthogonal monoid spaces. Strictly speaking we would first have
to justify that the homotopy category is pre-additive (which we won't do),
so that the formalism of group completions applies.

As we argued in Proposition <ref>, the E_∞-structure
on  cannot be refined to an ultra-commutative multiplication.
The argument was based on an algebraic obstruction that exists 
in the same way in _0(), so  
cannot be refined to an ultra-commutative monoid either.
The fact that  has an ultra-commutative group completion 
can be interpreted as saying that in this particular case 
`global group completion kills to obstruction to ultra-commutativity'.




subjectBott periodicity!global|(

Bott periodicity is traditionally seen as a homotopy equivalence between
the space × B U and the loop space of the infinite unitary group U.
Since a loop space only sees the basepoint component, 
and the loop space of B U is weakly equivalent to U, 
the 2-fold periodicity then takes the form of  a chain of weak equivalences:

    Ω^2(× B U)  ≃ Ω(Ω(B U))  ≃ Ω U  ≃ × B U .

We are going to prove a highly structured version of complex
Bott periodicity, in the form of a global equivalence of ultra-commutative monoids
between  and Ω.subjectunitary group ultra-commutative monoid 
Bott periodicity has been elucidated from many different angles,
and before we start, we put our approach into perspective.
Since Bott's original geometric argument <cit.>
a large number of different proofs became available,
see for example <cit.> for an overview.
In essence, our proof of global Bott periodicity is an adaptation 
of Harris' proof <cit.> of complex Bott periodicity.
The reviewer for Math Reviews praises Harris proof
as `a beautiful well-motivated proof of the complex Bott periodicity theorem 
using only two essential properties of the complex numbers'.
Suslin <cit.> calls this the `trivial proof'
of Bott periodicity and extends it to a `Real' (i.e., C_2-equivariant)
context. I also think that for readers with a homotopy theory background, Harris'
proof may be particularly accessible and appealing.

Harris' argument uses two main ingredients.
On the one hand, the group completion theorem is used
to identify the loop space of the bar construction of ⨿_n≥ 0  G r_n
(under the monoid structure induced by orthogonal direct sum)
with × B U. On the other hand, Harris exhibits an explicit homeomorphism
between the bar construction of ⨿_n≥ 0  Gr _n
and the infinite unitary group, essentially the inverse to the eigenspace decomposition
of a unitary matrix.
Together these two ingredients provide a chain of weak equivalences

    × B U  ≃ Ω( B( ⨿_n≥ 0  G r_n))  ≃Ω U  .

Our global proof is analogous:
Theorem <ref> above
shows that  is a global group completion of ^,
essentially by applying the group completion theorem to
all fixed point spaces. This part of the argument works just
as well for the real and symplectic versions of ^ and . 
Theorem <ref> below
shows that Ω is also a  global group completion of ^,
by globally identifying the bar construction of ^ 
(with respect to the box product of orthogonal spaces) with ,
using Harris' homeomorphism between 
the realization |^_∙(W)| and U(W_).
Two global group completions of the same ultra-commutative monoid
are necessarily globally equivalent, which yields the global
version of complex Bott periodicity of 
Theorem <ref>.

[Global Bott periodicity]
After this outline, we now provide the necessary details.
The ultra-commutative monoid  of unitary groups
was defined in Example <ref>.subjectunitary group ultra-commutative monoid
The orthogonal space Ω inherits an ultra-commutative multiplication by
pointwise multiplication of loops,
where Ω means objectwise continuous based maps from S^1.
We define a morphism of ultra-commutative monoids

    β :  ^ → Ω

at an inner product space V by

    β(V)(L)(x) =  c(x)· p_L + p_L^⊥ .

Here L is a complex subspace of V_, x∈ S^1,

    c  :  S^1  → U(1)  ,    x  ⟼ x+i/x-i ,

is the Cayley transform,subjectCayley transform
and p_L and p_L^⊥ denote the orthogonal projections to L 
respectively its orthogonal complement.
In other words, L and L^⊥ are the eigenspaces of β(V)(L)(x),
for the eigenvalues c(x) respectively 1.
Then 

    β(V)(0)(x) =   p_V_ =  _V_ ;

so β(V)(0) is the constant loop at the identity, which is the
unit element of Ω(V).
Now we consider subspaces L∈^(V) and L'∈^(W). Then

    β(V⊕ W)(L⊕ L')(x)     =  
     (c(x)· p_L⊕ L') + p_L^⊥⊕(L')^⊥
       = 
    ( (c(x)· p_L) + p_L^⊥) ⊕
    ( (c(x)· p_L') + p_(L')^⊥) 
       =  β(V)(L)(x)⊕β(W)(L')(x)   .

In other words, the square

    @C=20mm^(V)×^(W)[r]^-β(V)×β(W)[d]_⊕   Ω(V) ×Ω(W) [d]^μ^Ω_V,W
    ^(V⊕ W)[r]_-β(V⊕ W)   Ω(V⊕ W)

commutes, i.e., β is compatible with the multiplications
on both side. Since β respects multiplication and unit,
it also respects the structure maps. The upshot is that
β is a morphism of ultra-commutative monoids.

The category of ultra-commutative monoids is tensored and cotensored
over based spaces, so the functor of taking objectwise loops
is right adjoint to the functor - S^1 defined in (<ref>).
For an ultra-commutative monoid R, the based tensor R S^1
is isomorphic to the bar construction B(R), compare (<ref>).



The adjoint β^♭:B(^)=^ S^1→ 
of the morphism β:^→Ω is a global equivalence
of ultra-commutative monoids.
The morphism β:^→Ω is a global group completion of 
ultra-commu­tative monoids.


We factor β^♭ as a composite
of two morphisms of ultra-commutative monoids

    B(^) =  |B_∙(^)| [≃] ζ  
     | ^_∙ |  [] ϵ   ;

then we show that the morphism ζ is a global equivalence and the
morphism ϵ is an isomorphism. Together this shows the first claim.

The middle object is the realization 
of a simplicial ultra-commutative monoid ^_∙, 
and the first morphism is the realization of a simplicial morphism.
The object of n-simplices ^_n is the ultra-commutative monoid
of n-tuples of pairwise orthogonal complex subspaces, i.e.,

    ^_n(V) =  { (L_1,…,L_n)∈ ( Gr^(V_))^n :  L_i is orthogonal to L_j for i j} .

For varying n, the ultra-commutative monoids ^_n
assemble into a simplicial ultra-commutative monoid: the face morphisms

    d_i  :  ^_n → ^_n-1

are given by

    d_i(L_1,…,L_n)  = {[                          (L_2,…,L_n)                                     ; (L_1,…,L_i-1,L_i⊕ L_i+1,L_i+2,…,L_n)                                     ;                        (L_1,…,L_n-1)                                      ].

For n≥ 1 and 0≤ i ≤ n-1 the degeneracy morphisms are given by

    s_i(L_1,…, L_n-1)  =  (L_1,…,L_i,0,L_i+1,…,L_n-1)  .

The direct sum maps

    ^(V_1)×…×^(V_n)    → ^_n(V_1⊕…⊕ V_n)   
    
    (L_1,…,L_n)       ⟼   (i_1(L_1),…,i_n(L_n))

form a multi-morphism, where i_k:(V_k)_→ (V_1⊕…⊕ V_n)_
is the embedding as the k-th summand. 
The universal property of the box product
turns this multi-morphism into a morphism of orthogonal spaces

    ζ_n  :   B_n(^) =   (^)^⊠ n → ^_n .

The morphisms ζ_n are compatible with the simplicial face and
degeneracy maps, since these are given by orthogonal direct sum respectively
insertion of 0 on both sides. So for varying n, they form a morphism of
simplicial ultra-commutative monoids

    ζ_∙ :  B_∙(^)  → ^_∙ .


We claim that ζ_n is a global equivalence for every n≥ 0. 
Since ^=∐_j≥ 0^,[j] and 
the box product distributes over disjoint unions,
(^)^⊠ n is the disjoint union of the orthogonal spaces

    ^,[j_1]⊠…⊠^,[j_n]

indexed over all tuples (j_1,…,j_n)∈^n.
The orthogonal space ^_n has an analogous decomposition, where
^,[j_1,…,j_n]_n consists of those tuples
(L_1,…,L_n) with (L_i)=j_i. The morphism ζ_n respects
the decomposition, i.e., it matches the two summands indexed by the same tuple
(j_1,…,j_n). A disjoint union of global equivalences is a global equivalence
(Proposition <ref> (v)),
so we are reduced to showing that each of the  morphisms

    ζ_j_1,…,j_n :  ^,[j_1]⊠…⊠^,[j_n] →^,[j_1,…,j_n]_n

is a global equivalence.
This is in fact a restatement of an earlier result about box products
of orthogonal spaces `represented' by unitary representations.
In Construction <ref> we defined
an orthogonal space ^_G,W 
from a unitary representation W of a compact Lie group G.
The value at a euclidean inner product space V is

    ^_G,W(V)  =   ^(W,V_) / G .

Here ^ is the space of -linear maps that preserve
the hermitian inner products.
In the special case of the tautological U(n)-representation on ^n,
the homeomorphisms

    ^(^n,V_) / U(n)  → G r^_n(V_) =  ^,[n](V) ,  φ· U(n) ⟼ φ(^n)

form an isomorphism of orthogonal spaces ^_U(n),^n^,[n].
Similarly, passage to images provides 
an isomorphism of orthogonal spaces

    ^_U(j_1)×…× U(j_n),^j_1⊕…⊕^j_n  ^,[j_1,…,j_n]_n .

Under these identifications, the morphism ζ_j_1,…,j_n becomes the morphism

    ζ_U(j_1),…,U(j_n);^j_1,…,^j_n :  ^_U(j_1),^j_1⊠…⊠^_U(j_n),^j_n →^_U(j_1)×…× U(j_n),^j_1⊕…⊕^j_n ,

the iterate of the morphism
discussed in Proposition <ref>.
Proposition <ref>
thus shows that the morphism ζ_j_1,…,j_n is a global equivalence.
This completes the proof that 
the morphism ζ_n:(^)^⊠ n→^_n is a global equivalence. 

Now we observe that the underlying simplicial orthogonal spaces of source and target
of ζ_∙ are Reedy flat in the sense of Definition <ref>, 
i.e., all latching morphisms (in the simplicial direction) are
flat cofibrations of orthogonal spaces.
Indeed, the unit of the ultra-commutative monoid ^
is the inclusion of the summand ^,[0]
into the disjoint union of all ^,[n].
Since the orthogonal space ^,[n] is isomorphic to ^_U(n),^n,
it is flat by Proposition <ref> (ii).
So ^ has a flat unit, and its bar construction is Reedy flat by 
Proposition <ref> (i).
Since ^_∙ is not the bar construction of any
orthogonal monoid space, we must show Reedy flatness directly.
Each simplicial degeneracy morphism of ^_∙ 
inserts the zero vector space in one slot; so the degeneracy morphisms
are embeddings of summands in a disjoint union.
The latching morphism

    L_m^(^_∙)  → ^_m

is then also the inclusions of certain summands, namely those 
^,[j_1,…,j_n]_n for which j_i=0 for at least one i.
Since the summand ^,[j_1,…,j_n]_n is isomorphic to
^_U(j_1)×…× U(j_n),^j_1⊕…⊕^j_n,
it is flat by Proposition <ref> (ii).
This verifies the Reedy flatness condition for ^_∙. 
Since source and target of the morphism ζ_∙
are Reedy flat as simplicial orthogonal spaces, and ζ_∙
is a global equivalence in every simplicial dimension,
the induced morphism of realizations

    ζ =   |ζ_∙| :   B(^) =  
    | B_∙(^)| → |^_∙|

is a global equivalence by Proposition <ref> (ii).

The isomorphism of ultra-commutative monoids
ϵ:|^_∙|
is taken from Harris <cit.>,
and we recall it in some detail for the convenience of the reader.
We let V be an inner product space and consider the continuous map

    ϵ_n :  ^_n(V)×Δ^n     →       U(V_) 
      
    (L_1,…,L_n;  t_1,…,t_n)    ⟼ ∏_j=1^n  exp(2π i t_j· p_L_j) ,

where p_L:V_→ V_ is the orthogonal projection onto L.
In other words, ϵ_n(L_1,…,L_n;  t_1,…,t_n)
is the unitary automorphism of V_ that has L_j as
eigen­space with eigenvalue exp(2π i t_j),
and is the identity on the orthogonal complement of all L_j's.
We have the relations

    ϵ_n(L_1,…,L_n;  0, t_1,…, t_n-1)
      =  ϵ_n-1(L_2,…, L_n;  t_1,…,t_n-1)

and

    ϵ_n(L_1,…,L_n;  t_1,…, t_n-1,1) =  ϵ_n-1(L_1,…, L_n-1;  t_1,…,t_n-1)

because exp(0)=exp(2π i · p_L)=_V_; moreover,

    ϵ_n(L_1,…,L_n;  t_1,…,t_i,t_i…,t_n-1)
     =  ϵ_n-1(L_1,…,L_i⊕ L_i+1,…, L_n;  t_1,…,t_n-1)

for all 0<i<n, because exp(2π i t· p_L_i)·exp(2π i t· p_L_i+1)=exp(2π i t· p_L_i⊕ L_i+1);
and finally

    ϵ_n+1(L_1,…,L_i,0,L_i+1,…,    L_n;  t_1,…,t_n+1)
       =  ϵ_n(L_1,…,L_n;  t_1,…,t_i+1,…,t_n+1)

for all 0≤ i≤ n.
So the maps ϵ_n are compatible with the equivalence relation
defining geometric realization, and they induce a continuous map

    ϵ(V) :  |^_n(V)| → U(V_)  .

The map ϵ(V) is bijective because every unitary automorphism is diagonalizable
with pairwise orthogonal eigenspaces and eigenvalues in U(1).
As a continuous bijection from a compact space
to a Hausdorff space, ϵ(V) is a homeomorphism.
The homeomorphisms ϵ(V) are compatible
with linear isometric embeddings in V and the ultra-commutative multiplications
on both sides, i.e., they define an isomorphism of ultra-commutative monoids
ϵ:|^_∙|.

Now we can conclude the proof.
Unraveling all definitions shows that the composite

    ^ S^1 → B(^)   ζ  |^_∙|
       ϵ

is given at an inner product space V by the map

    G r^(V_)   S^1 →  (V_) 
    
    L x     ⟼ exp(2π i·log(c(x))· p_L) =c(x)· p_L  +  p_L^⊥ =  β(V)(L)(x) .

This means that the original morphism β is the composite

    ^  η_^  Ω B(^) Ω(ζ) Ω|^_∙(V)|Ω(ϵ) Ω ,

where η_^ was defined in (<ref>).
Thus the adjoint β^♭ factors as
ϵ∘ζ, a global equivalence followed by an isomorphism.
So β^♭ is  a global equivalence of ultra-commutative monoids.

We showed above that ^ has a flat unit, so the 
adjunction unit η_^ 
is a global group completion by Corollary <ref>.
Since β^♭ is a global equivalence, so is Ω(β^♭).
Hence the composite β:^→Ω is a global group completion.



Theorem <ref>
and Theorem <ref>
show that the morphisms 

    i : ^ →      respectively    β : ^ → Ω

are both global group completions.
The universal property of group completions 
already implies that  is isomorphic to Ω
in the homotopy category of ultra-commutative monoids;
the two can thus be linked by a chain of global equivalences of ultra-commutative monoids.
In a sense we could stop here, and call this `complex global Bott periodicity'.
However, we elaborate a bit more and exhibit an explicit chain
of two global equivalences between  and Ω,
see Theorem <ref> below.

We define a morphism of ultra-commutative 
monoidssubjectadditive Grassmanniansymbolβ@β̅ - Bott morphism from  to Ω

    β̅ :   → Ω( _)  .

Here _=^^2_ is the multiplicative shift by ^2 
defined in Example <ref>.
The orthogonal space  has a commutative multiplication by
direct sum of unitary automorphisms; 
thus Ω(_) inherits a commutative multiplication by
pointwise multiplication of loops.
The target of β̅ is globally equivalent, as an ultra-commutative monoid,
to Ω, the objectwise loops of the unitary group monoid.
The definition of the map

    β̅(V) : (V) → Ω (_) (V)  =  (S^1, ( V_^2)) ,

for an inner product space V, is similar to, but slightly more elaborate than
the definition of β(V) in (<ref>) above.
An element of (V) is a complex subspace L of V_^2;
as before we denote by p_L and p_L^⊥ the orthogonal projections to L 
respectively to its orthogonal complement.
We define the loop

    β̅(V)(L) :  S^1  → U(V_^2)

by

    β̅(V)(L)(x) =  
    ( (c(x)· p_L) + p_L^⊥) ∘
    ( (c(-x)· p_V_⊕ 0) + p_0⊕ V_) .

As before c:S^1 → U(1) is the Cayley transform.
The map β̅(V) is continuous in L.


For every inner product space V we have

    β̅(V)(V_⊕ 0)(x) =   
    ( (c(x)· p_V_⊕ 0) + p_0⊕ V_) ∘
    ( (c(-x)· p_V_⊕ 0) + p_0⊕ V_) = _V_ ;

so β̅(V)(V_⊕ 0) is the constant loop at the identity, which is the
unit element of Ω (_)(V).
Now we consider subspaces L∈(V) and L'∈(W). 
We recall that κ^V,W:V_^2⊕ W_^2 (V⊕ W)_^2
is the preferred natural isometry, which enters into the definition of
the multiplication of .
The argument for the additivity relation

    β̅(V⊕ W)(L⊕ L')(x) =  κ^V,W_* ( β̅(V)(L)(x)⊕β̅(W)(L')(x))

is straightforward and similar to (but somewhat longer than)
the argument for the map β in (<ref>), and we omit it. 
Hence β̅ is compatible with the multiplications
on both sides. Since β̅ respects multiplication and unit,
it also respects the structure maps. The upshot is that
β̅ is a morphism of ultra-commutative monoids.
We also have

    ( β̅(V)(L)(x) )     =  ( (c(x)· p_L)+p_L^⊥)
    ·( (c(-x)· p_V_⊕ 0)+p_0⊕ V_)
       =  c(x)^(L)-(V) ,

exploiting that c(-x)=c(x)=c(x)^-1.
So the map β̅(V):(V) →Ω U(V_^2)
sends the subspace (V)=^[0](V) to Ω (S U(V_^2)).
Hence the morphism β̅ restricts to a morphism of 
ultra-commutative monoids

    β̅^[0] :   → Ω ( _)  .



Now we can properly state our global version of complex Bott periodicity.
The embeddings j:V_→ V_^2 as the first summand
induce a morphism of ultra-commutative monoids ∘ j:→_.


 
The morphisms of ultra-commutative monoids 

    β̅  Ω (_) Ω(∘ j) Ω

are global equivalences.
The morphism  β̅^[0]:→Ω (_) 
is a global equivalence.


The morphism ∘ j is a global equivalence 
by Theorem <ref>,
hence so is Ω(∘ j).
The following diagram of homomorphisms of ultra-commutative monoids
commutes by direct inspection:

    @C=15mm^[r]^-β[d]_i     Ω[d]^Ω(∘ j)
    [r]_-β̅   Ω(_)

The morphism β:^→Ω is a global group completion
by Theorem <ref>.
So the composite Ω(∘ j)∘β:^→Ω(_) 
with a global equivalence is also a global group completion.
The morphism i:^→ is a global group completion by
Theorem <ref>.
The universal property of group completions then shows that 
β̅:→Ω(_) becomes an
isomorphism in the homotopy category of ultra-commutative monoids.
Since the global equivalences are part of a model structure,
this implies that β̅:→Ω(_) 
is a global equivalence.
Since the morphism β̅^[0]:→Ω(_) is
a retract of the global equivalence β̅, it is a global equivalence itself.








For every compact Lie group G and every finite G-CW-complex A, the map

    [A,β]^G :  [A,^]^G → [A,Ω]^G

is a group completion of abelian monoids.


We contemplate the following commutative square of abelian monoid homomorphisms:

    @C=15mm
    [A,^]^G [d]_[A,i]^G[r]^-[A,β]^G   [A,Ω]^G 
    [d]^[A,Ω(∘ j)]^G_
    
    [A,]^G [r]^-_-[A,β̅]^G    [A,Ω(_) ]^G

Since β̅ and Ω(∘ j)
are global equivalences by Theorem <ref>,
the two homomorphisms [A,β̅]^G and [A,Ω(∘ j)]^G 
are isomorphisms, by Proposition <ref> (ii).
The morphism [A,i]^G is a group completion of abelian monoids by
Proposition <ref>
(or rather its complex analog, which is proved analogously).
So [A,β]^G:[A,^]^G→[A,Ω]^G 
is also a group completion of abelian monoids.



subjectBott periodicity!global|)
subjectglobal group completion!of an ultra-commutative monoid|)
subjectultra-commutative monoid|)









CHAPTER: EQUIVARIANT STABLE HOMOTOPY THEORY




In this chapter we give a largely self-contained 
exposition of many basics about equivariant stable homotopy theory
for a fixed compact Lie group; our model is the category of orthogonal G-spectra.
In Section <ref>
we review orthogonal spectra and orthogonal G-spectra;
we define equivariant stable homotopy groups and prove their basic properties,
such as the suspension isomorphism and long exact sequences of
mapping cones and homotopy fibers, and the additivity
of equivariant homotopy groups on sums and products.
Section <ref> discusses the Wirthmüller isomorphism
that relates the equivariant homotopy groups 
of a spectrum over a subgroup to the equivariant homotopy groups 
of the induced spectrum; intimately related to the 
Wirthmüller isomorphism are various transfers that we also recall.
In Section <ref> we introduce and study
geometric fixed point homotopy groups.
We establish the isotropy separation sequence that facilitates
inductive arguments, and show that equivariant equivalences can
also be detected by geometric fixed points.
We use geometric fixed points to derive a functorial description of the
0-th equivariant stable homotopy group of a G-space Y in terms
of the path components of the fixed point spaces Y^H.
Section <ref> gives a self-contained proof
of the double coset formula for the composite of a transfer 
followed by a restriction to a closed subgroup.
We also discuss various examples and end with a discussion of Mackey functors for
finite groups.
After inverting the group order, the category of G-Mackey functors
splits as a product, indexed by conjugacy classes of subgroups,
of module categories over the Weyl groups, 
see Theorem <ref>.
We show that rationally and for finite groups, geometric fixed point homotopy groups
can be obtained from equivariant homotopy groups by dividing out transfers
from proper subgroups.
Section <ref> is devoted to multiplicative aspects of
equivariant stable homotopy theory. In our model,
all multiplicative features can be phrased in terms of the
smash product of orthogonal spectra (or orthogonal G-spectra),
another example of a Day type convolution product.
The smash product gives rise to pairings of equivariant homotopy groups;
when specialized to equivariant ring spectra, these pairings turn 
the equivariant stable homotopy into graded rings.

We do not discuss model category structures for orthogonal G-spectra;
the interested reader can find different ones in the memoir of
Mandell and May <cit.>,
in the thesis of Stolz <cit.>,
the article by Brun, Dundas and Stolz <cit.> 
and (for finite groups) in the paper of Hill, Hopkins and Ravenel <cit.>.





§ EQUIVARIANT ORTHOGONAL SPECTRA



In this section we begin to develop some of the basic 
features of equivariant stable homotopy theory for compact Lie groups
in the context of equivariant orthogonal spectra. 
After introducing orthogonal G-spectra
and equivariant stable homotopy groups, 
we discuss shifts by a representation and show that they are
equivariantly equivalent to smashing with the representation sphere
(Proposition <ref>).
We establish the loop and suspension isomorphisms 
(Proposition <ref>)
and the long exact homotopy
group sequences of homotopy fibers and mapping cones
(Proposition <ref>).
We prove that equivariant homotopy groups take wedges to sums and
preserve finite products (Corollary <ref>).
We end by showing that the equivariant homotopy group functor π_0^H,
for a closed subgroup H of a compact Lie group G, is represented
by the unreduced suspension spectrum of the homogeneous space G/H
(Proposition <ref>).  



We recall orthogonal spectra. 
These objects are used, at least implicitly, already in <cit.>;
the term `orthogonal spectrum' was introduced by
Mandell, May, Shipley and the author in <cit.>, where the
(non-equivariant) stable model structure for orthogonal spectra
was constructed.
Orthogonal spectra are stable versions of orthogonal spaces, and before recalling
the formal definition we try to motivate it – already with a view
towards the global perspective.
An orthogonal space Y assigns values 
to all finite-dimensional inner product spaces.
The global homotopy type is encoded in the G-spaces Y(_G), 
where _G is a complete G-universe, which we can informally think of 
as `the homotopy colimit of Y(V) over all G-representations V'.
So besides the values Y(V), an orthogonal space uses the information
about the O(V)-action (which is turned into a G-action when G acts on V)
and the information about inclusions of inner product spaces
(in order to be able to stabilize to the colimit _G).
The information about the O(V)-actions and how to stabilize are conveniently
encoded together as a continuous functor 
from the category  of linear isometric embeddings.

An orthogonal spectrum X is a stable analog of this: it assigns
a based space X(V) to every inner product space, and it keeps
track of an O(V)-action on X(V) (to get G-homotopy types when G acts on V)
and of a way to stabilize by suspensions (needed when exhausting a complete universe
by its finite-dimensional subrepresentations). 
When doing this in a coordinate free way, the stabilization data
assigns to a linear isometric embedding φ:V→ W a 
continuous based map

    φ_⋆ :  S^W-φ(V)  X(V) → X(W)

where W-φ(V) is the orthogonal complement of the image of φ.
This structure map should `vary continuously with φ',
but this phrase has no literal meaning because the source of φ_⋆
depends on φ. The way to make the continuous dependence
rigorous is to exploit that the complements W-φ(V) vary in 
a locally trivial way, i.e., they are the fibers of a distinguished vector bundle,
the `orthogonal complement bundle', 
over the space of (V,W) of linear isometric embeddings.
All the structure maps φ_⋆ together define a map on
the smash product of X(V) with the Thom space of this complement bundle,
and the continuity of the dependence on φ is formalized by
requiring continuity of that map.
All these Thom spaces together form the morphism spaces of a based
topological category, and the data of an orthogonal spectrum 
can conveniently be packaged as a continuous based
functor on this category.


We let V and W be inner product spaces. Over the space
(V,W) of linear isometric embeddings sits a certain `orthogonal complement'
vector bundle with total space

    ξ(V,W)  =  {  (w,φ) ∈ W×(V,W)  |  w⊥φ(V) } .

The structure map ξ(V,W)→(V,W)symbolξ(V,W) - orthogonal complement vector bundle 
is the projection to the second factor.
The vector bundle structure of ξ(V,W) is 
as a vector subbundle of the trivial vector bundle W×(V,W),
and the fiber over φ:V→ W is the orthogonal complement W-φ(V) 
of the image of φ.


We let (V,W)symbol(V,W) - Thom space of orthogonal complement bundle be the Thom space of the bundle ξ(V,W),
i.e., the one-point compactification of the total space of ξ(V,W).
Up to non-canonical homeomorphism, we can describe the space (V,W) 
differently as follows.
If the dimension of W is smaller than the dimension of V,
then the space (V,W) is empty and (V,W) consists
of a single point at infinity. Otherwise we can choose a linear isometric embedding
φ:V→ W, and then the maps

    O(W)/O(W-φ(V) )        → (V,W) ,   
    A· O(W-φ(V) )  ⟼ Aφ   and
    
    O(W)⋉_O(W-φ(V)) S^W-φ(V)     → (V,W) ,   
    [A,w]  ⟼ (A w, A φ)

are homeomorphisms.
Here, and in the following, we writesymbolG⋉_H A - induced based G-space

    G⋉_H A  =  (G_+)_H A  =  (G_+ A) / ∼

for a closed subgroup H of G and a based G-space A;
the equivalence relation is g h a∼ g  h a
for all (g,h,a)∈ G× H× A.
Put yet another way: if V=n and W=n+m,
then (V,W) is homeomorphic to the homogeneous space O(n+m)/O(m)
and (V,W) is homeomorphic to O(n+m)⋉_O(m)S^m.
The vector bundle ξ(V,W) becomes trivial upon product with
the trivial bundle V, via the trivialization

    ξ(V,W)× V   W×(V,W) ,   
     ((w,φ), v)  ⟼ (w+φ(v),φ)  .

When we pass to Thom spaces on both sides this becomes the
untwisting homeomorphism:subjectuntwisting homeomorphism

    (V,W) S^V   S^W(V,W)_+ .




The Thom spaces (V,W) are the morphism spaces of a based topological category.
Given a third inner product space U, the bundle map 

    ξ(V,W) ×ξ(U,V)  → ξ(U,W)  ,   
    ((w,φ), (v,ψ))  ⟼ (w+φ(v), φψ)

covers the composition map (V,W)×(U,V)→(U,W).
Passage to Thom spaces gives a based map

    ∘ :  (V,W) (U,V)  → (U,W)

which is clearly associative, and is the composition in the category . 
The identity of V is (0,_V) in (V,V).



An orthogonal spectrumsubjectorthogonal spectrum 
subjectspectrum!orthogonal|seeorthogonal spectrum 
is a based continuous functor from  to the category _* of based spaces.
A morphism of orthogonal spectrasubjectmorphism!of orthogonal spectra 
is a natural transformation of functors.
We denote the category of orthogonal spectra by .symbol   - category of orthogonal spectra


Given two inner product spaces V and W we define a continuous based map

    i_V  :  S^V  → (W,V⊕ W)     by     
    v  ⟼ ((v,0), (0,-))  ,

where (0,-):W→ V⊕ W is the embedding of the second summand.
We define the structure mapsubjectstructure map!of an orthogonal spectrum 
σ_V,W:S^V X(W)→ X(V⊕ W) of the orthogonal spectrum X
as the compositesymbolσ_V,W - structure map of an orthogonal spectrum

    S^V    X(W)  i_V X(W) (W,V⊕ W) X(W)  X   X(V⊕ W)  .

Often it will be convenient to use the opposite structure mapsubjectstructure map!of an orthogonal spectrum!oppositesymbolσ_V,W^ - opposite structure map of an orthogonal spectrum

    σ^_V,W :     X(V) S^W  →  X(V⊕ W)

which we define as the following composite:

    X(V) S^W  twistS^W    X(V)  σ_W,V  X(W⊕ V)  X(τ_V,W)  X(V⊕ W)


[Coordinatized orthogonal spectra]
Every inner product space is isometrically isomorphic to ^n
with standard inner product, for some n≥ 0.
So the topological category  has a small skeleton, 
and the functor category of orthogonal spectra has `small' morphism sets.
This also leads to the following
more explicit coordinatized description of orthogonal spectra
in a way that resembles a presentation by generators and relations. 

Up to isomorphism, an orthogonal spectrum X is determined by the values X_n=X(^n) 
and the following additional data relating these spaces:


  * a based continuous left O(n)-action on X_n for each n≥ 0, 

  * based maps σ_n:S^1 X_n→ X_1+n for n≥ 0.

This data is subject to the following condition:
for all m, n≥ 0, the iterated structure map
S^m X_n →  X_m+n defined as the composition

    @C=15mm
    S^m X_n ^-S^m-1σ_n[r]    
    S^m-1 X_1+n^-S^m-2σ_1+n[r]      ⋯  ^-σ_m-1+n[r]     X_m+n

is (O(m)× O(n))-equivariant. Here the group O(m)× O(n) 
acts on the target by restriction, along orthogonal sum, of the O(m+n)-action.
Indeed, the map 

    O(n)_+ → (^n,^n)  ,    A ⟼ (0,A)

is a homeomorphism, so O(n) `is' the endomorphism monoid of ^n
as an object of the category ; via this map, O(n) acts on the
value at ^n of any functor on .
The map σ_n=σ_,^n 
is just one of the structure maps (<ref>).




Let G be a compact Lie group.
An orthogonal G-spectrumsubjectorthogonal G-spectrum 
is a based continuous functor from  to the category G_* of based G-spaces.
A morphism 
of orthogonal G-spectrasubjectmorphism!of orthogonal G-spectra 
is a natural transformation of functors.
We write Gsymbol  G - category of orthogonal G-spectra
for the category of orthogonal G-spectra and G-equivariant morphisms.
  


A continuous functor to based G-spaces is the same data as a G-object
of continuous functors. So orthogonal G-spectra could equivalently
be defined as orthogonal spectra equipped with a continuous G-action.
An orthogonal G-spectrum X can be evaluated on a G-representation V,
and then X(V) is a (G× G)-space by the `external' G-action on X 
and the `internal' G-action from the G-action on V
and the O(V)-functoriality of X.
We consider X(V) as a G-space via the diagonal G-action.
If V and W are G-representations, then the 
structure map (<ref>) 
and the opposite structure map (<ref>)  
are G-equivariant where 
the group G also acts on the representation spheres.




Our definition of orthogonal G-spectra is not the same as the one used
by Mandell and May <cit.> and Hill, Hopkins and Ravenel <cit.>,
who define orthogonal G-spectra as G-functors on a G-enriched extension
of the category  that contains all G-representations as objects.
However, our category of orthogonal G-spectra is equivalent to theirs by
<cit.>. The substance of this equivalence is
the fact that for every orthogonal G-spectrum in the sense of Mandell and May,
the values at arbitrary G-representations are in fact determined
by the values at trivial representations.



Next we recall the equivariant stable homotopy groups π_*^G(X)
(indexed by the complete G-universe) of an orthogonal G-spectrum X.
We introduce a convenient piece of notation.
If φ:V→ W is a linear isometric embedding
and f:S^V→ X(V) a continuous based map, 
we define φ_*f:S^W→ X(W) as the composite

    S^W  S^W-φ(V) S^V    S^W-φ(V) f 
    S^W-φ(V) X(V) 
       σ_W-φ(V),V X((W-φ(V))⊕ V)     X(W)

where two unnamed homeomorphisms use the linear isometry

    (W-φ(V))⊕ V   W   ,    (w,v) ⟼ w+ φ(v)  .

For example, if φ is bijective (i.e., an equivariant isometry),
then φ_* f becomes the φ-conjugate of f, i.e., the composite

    S^W   S^φ^-1  S^V   f  X(V)  X(φ)  X(W)  .

The construction is continuous in both variables, i.e., the map

    (V,W)×_*(S^V,X(V))  → _*(S^W,X(W))  ,   
    (φ,f) ⟼ φ_* f

is continuous.


As before we let s(_G) denote the poset, under inclusion,
of finite-dimensional G-subrepresentations 
of the chosen complete G-universe _G.
We obtain a functor from s(_G) to sets by sending V∈ s(_G) to 

    [S^V, X(V)]^G  ,

the set of G-equivariant homotopy classes of based G-maps
from S^V to X(V).
For V⊆ W in s(_G) the inclusion i:V→ W is sent to the map

    i_*  :  [S^V, X(V)]^G  → [S^W,X(W)]^G  ,   [f] ⟼ [i_*f]  .

The 0-th equivariant homotopy group π_0^G(X) is then defined 
assubjecthomotopy group!equivariantsubjectequivariant homotopy group!of an orthogonal G-spectrum

    π_0^G(X)  =  _V∈ s(_G)  [S^V, X(V)]^G   ,

the colimit of this functor over the poset s(_G).

The sets π_0^G(X) have a lot of extra structure;
we start with the abelian group structure.
We consider a finite-dimensional G-subrepresentation V of the 
universe _G with non-zero fixed points.
We choose a G-fixed unit vector v_0∈ V, 
and we let V^⊥ denote the orthogonal complement of v_0 in V.
This induces a decomposition

    ⊕ V^⊥  V  ,    (t,v) ⟼ t v_0 +v

that extends to a G-equivariant homeomorphism
S^1 S^V^⊥ S^V on one-point compactifications.
From this we obtain a bijection

    [S^V,X(V)]^G  [S^1,_*^G(S^V^⊥,X(V))]_*   
     = π_1(_*^G(S^V^⊥,X(V))) ,

natural in the orthogonal G-spectrum X. 
We use the bijection (<ref>)
to transfer the group structure on the fundamental group
into a group structure on the set [S^V,X(V)]^G.

Now we suppose that the dimension of the fixed point space V^G is at least 2.
Then the space of G-fixed unit vectors in V is connected and
similar arguments as for the commutativity of higher homotopy groups show:


  * the group structure on the set [S^V,X(V)]^G defined by
the bijection (<ref>)
is commutative and independent of the choice of G-fixed unit vector;

  * if W is another finite-dimensional G-subrepresentation of
_G containing V, then the map

    i_*  :  [S^V,X(V)]^G → [S^W,X(W)]^G

is a group homomorphism.

The G-subrepresentations V of _G with (V^G)≥ 2
are cofinal in the poset s(_G), so the two properties above
show that the abelian group structures on [S^V,X(V)]^G
for  (V^G)≥ 2 assemble into a well-defined and
natural abelian group structure on the colimit π_0^G(X).

We generalize the definition of π_0^G(X) to
integer graded equivariant homotopy groups of an orthogonal G-spectrum X.
If k is a positive integer, then we set

    π_k^G (X)     =  _V∈ s(_G)  [S^V⊕^k, X(V)]^G  
         and
    π_-k^G (X)     =  _V∈ s(_G)  [S^V, X(V⊕^k)]^G   .

The colimits are taken over the analogous stabilization maps as for π_0^G,
and they come with abelian group structures by the same reasoning as for π_0^G(X).


A morphism f:X→ Y of orthogonal G-spectra 
is a _*-isomorphismsubjectpi star isomorphism@_*-isomorphism!of orthogonal G-spectra
if the induced map π_k^H(f):π_k^H(X) →π_k^H(Y) is an isomorphism
for all closed subgroups H of G and all integers k.



While the definition of π_k^G(X) involves a case distinction
in positive and negative dimensions k, every class in π_k^G( X)
can be represented by a G-map

    f  :  S^V⊕^n+k → X(V⊕^n)

for suitable n∈ such that n+k≥ 0.
Moreover, V can be any finite-dimensional G-representation,
not necessarily a subrepresentation of the chosen complete G-universe.
Since we will frequently use this way to represent elements of π_k^G(X),
we make the construction explicit here.

We start with the case k≥ 0. We choose a G-equivariant linear isometry 
j:V⊕^n→V̅ onto a G-subrepresentation V̅ of _G.
Then the composite

    S^V̅⊕^k [](S^j⊕^k)^-1 S^V⊕^n+k  f   
    X(V⊕^n)  []X(j) X(V̅)

represents a class f∈π_k^G(X).
For k≤ 0, we choose a G-equivariant linear isometry
j:V⊕^n+k→V̅ onto a G-subrepresentation V̅ of _G.
Then the composite

    S^V̅ [](S^j)^-1 S^V⊕^n+k  f   
    X(V⊕^n)  []X(j⊕^-k) X(V̅⊕^-k )

represents a class f∈π_k^G(X). 

We also need a way to recognize that `stabilization along a linear isometric embedding'
does not change the class in π_k^G(X).
For this we let φ:V→ W be a G-equivariant linear isometric embedding
and f:S^V⊕^n+k→ X(V⊕^n) a continuous based G-map as above. 
We define φ_*f:S^W⊕^n+k→ X(W⊕^n) as the composite

    S^W⊕^n+k  S^W-φ(V)   S^V⊕^n+k S^W-φ(V) f 
    S^W-φ(V) X(V⊕^n) 
       σ_W-φ(V),V⊕^n  X((W-φ(V))⊕ V⊕^n)  
       X(W⊕^n)   ;

the two unnamed homeomorphisms use the linear isometry

    (W-φ(V))⊕ V⊕^m  W⊕^m   ,    (w,v,x) ⟼ (w+ φ(v), x)

for m=n+k respectively m=n.
In the special case n=k=0, this construction reduces 
to (<ref>).


The same reasoning as in the unstable situation 
in Proposition <ref>
shows the following stable analog:




Let G be a compact Lie group and X an orthogonal G-spectrum.
Let V be a G-representation and f:S^V⊕^n+k→ X(V⊕^n) a 
based continuous G-map, where n∈ and k∈ are such that n+k≥ 0.


  * The class f in π_k^G(X) is independent of the choice 
of linear isometry onto a subrepresentation of _G.

  * For every G-equivariant linear isometric embedding 
φ:V→ W the relation

    φ_*f =  f     holds in  π_k^G(X).




Now we let K and G be two compact Lie groups.
Every continuous based functor F:G_∗→ K_∗
from based G-spaces to based K-spaces gives rise to a functor

    F∘-  :  G → K

from orthogonal G-spectra to orthogonal K-spectra
by postcomposition: if X is a G-orthogonal spectrum,
then the composite

    X   G_*  F  K_∗ .

is an orthogonal K-spectrum.
The next construction is an example of this.

[Restriction maps]subjectrestriction homomorphism!for equivariant stable homotopy groups
We let α:K→ G be a continuous homomorphism between compact Lie groups.
Given an orthogonal G-spectrum X, we apply restriction of scalars levelwise
and obtain an orthogonal K-spectrum α^* X.
We define the restriction homomorphism

    α^*  :  π^G_0(X) → π^K_0(α^* X)      by    α^*[f] =  α^*(f) .

In other words, the class represented by a based G-map 
f:S^V→ X(V) is sent to the class represented by the K-map

    α^*(f) :  S^α^*(V) = α^*(S^V)  →α^*(X(V)) =  (α^* X)(α^* V)  ,

appealing to Construction <ref>.
The restriction  maps α^* are clearly transitive 
(contravariantly functorial) for composition of group homomorphisms.


For g∈ G the conjugation homomorphism is defined as

    c_g  :  G  → G  ,    c_g(h)  =  g^-1 h g .

For every G-space A, left multiplication by g is then a
G-equivariant homeomorphism l^A_g:c_g^*(A)→ A.
For an orthogonal G-spectrum X the maps 
l_g^X(V):(c_g^* X)(V)=c_g^*(X(V))→ X(V)
assemble into an isomorphism of orthogonal G-spectra l_g^X:c_g^*X→ X,
as V runs over all inner product spaces (with trivial G-action).



Let G be a compact Lie group, X an orthogonal G-spectrum and g∈ G.
Then the two isomorphisms

    c_g^*  :  π_0^G(X)  → π_0^G(c_g^* X)      and    
    (l_g^X)_* :  π_0^G(c_g^* X)  → π_0^G( X)

are inverse to each other.  


We let V be a G-representation,
and we recall that the G-action on X(V) is diagonally,
from the external G-action on X and the internal G-action on V.   
Hence the map l_g^X(V):c_g^*(X(V))→ X(V)
is the composite of the map l_g^X(c_g^* V):(c_g^*X)(c_g^* V)→ X(c_g^* V)
and the map X(l_g^V):X(c_g^* V)→ X(V).
Now we let f:S^V→ X(V) be a G-map representing a class in π_0^G(X).
The following diagram of G-spaces and G-maps commutes
because f is G-equivariant:

    @C=20mm 
    S^c_g^*V[r]^-c_g^* f[d]_l_g^V    
    c_g^*(X(V)) =(c_g^* X)(c_g^* V)[r]^-l_g^X(c_g^*V)@/^1pc/[dr]_(.6)l_g^X(V)   
    X(c_g^* V)[d]^-X(l_g^V)
    
    S^V [rr]_-f        X(V)

The upper horizontal composite represents the class (l_g^X)_*(c_g^*[f]).
Since it differs from f by conjugation with an equivariant isometry,
the upper composite represents the same class as f,
by Proposition <ref> (ii).
Thus we conclude that (l_g)_*(c_g^*[f])=[f].


[Weyl group action on equivariant homotopy groups]
We consider a closed subgroup H of a compact Lie group G 
and an orthogonal G-spectrum X.
Every g∈ G 
gives rise to a conjugation homomorphism
c_g:H→ H^g by c_g(h)=g^-1h g,
where H^g={g^-1 h g | h∈ H} is the conjugate subgroup.
One should beware that while c_g^*(^G_H^g(X)) and ^G_H(X)
have the same underlying orthogonal spectrum, they come with different H-actions. 
However, left translation by g is an isomorphism of orthogonal H-spectra
l_g^X:c_g^*X→ X. So combining the restriction map along c_g
with the effect of l_g^X 
gives an isomorphismsubjectconjugation homomorphism!on equivariant homotopy groups  

    g_⋆ :  π_0^H^g(X)  (c_g)^*  π_0^H(c_g^*X)  (l_g^X)_* π_0^H(X) .

Moreover, 

    g_⋆∘ g'_⋆    =  (l_g^X)_*∘ (c_g)^* ∘ (l_g'^X)_*∘ (c_g')^*   
    =  (l_g^X)_*∘ ( (c_g)^*(l_g'^X))_*∘ (c_g)^* ∘  (c_g')^*  
       = (l_g^X∘ (c_g)^*(l_g'^X))_*∘ (c_g'∘ c_g)^*=  (l_g g'^X)_*∘ (c_g g')^* =  (g g')_⋆

by naturality of (c_g)^*.
If g normalizes H, then g_⋆ is a self-map of the group π_0^H(X).
If moreover g belongs to H, then g_⋆ is the identity
by Proposition <ref>;
so the maps g_⋆ define an action of the Weyl group
W_G H=N_G H/H on the equivariant homotopy group π_0^H(X).

If H has finite index in its normalizer, this is the end of the story
concerning Weyl group actions on π_0^H(X).
In general, however the group H need not have finite index in its normalizer N_G H,
and the Weyl group W_G H may have positive dimension, and hence a non-trivial
identity path component (W_G H)^∘. We will now show that the entire
identity path component acts trivially on π_0^H(X)
for any orthogonal G-spectrum X. This is a consequence of the
fact that every element of (W_G H)^∘
has the form z H for an element z in (C_G H)^∘,
the identity component of the centralizer of H in G,
compare Proposition <ref>.
But then c_z:H→ H is the identity because z centralizes H.
On the other hand, any path from z and 1 in C_G H
induces a homotopy of morphisms of orthogonal H-spectra
from l_z:X→ X to the identity of X.
So

    z_⋆ =  (l_z^X)_*∘ (c_z)^*  =  _π_0^H(X) .

This shows that the identity component of the Weyl group W_G H
acts trivially on π_0^H(X). So the Weyl group action factors over
an action of the discrete group

    π_0(W_G H) =  (W_G H) / (W_G H)^∘ .






If A is a pointed G-space, then smashing with A and taking based maps
out of A are two continuous based endofunctors
on the category of based G-spaces. So for every orthogonal G-spectrum X,
we can define two new orthogonal G-spectra X A and _*(A,X)
by smashing with A (and letting G act diagonally)
or taking based maps from A levelwise (and letting G act by conjugation).
More explicitly, we have

    (X A)(V) = X(V) A     respectively  _*(A,X)(V)  =  _*(A,X(V))

for an inner product space V. 
The structure maps and actions of the orthogonal groups 
do not interact with A: the group O(V) acts through its action on X(V), 
and the structure maps are given by the composite

    S^V (X A)(W)
      = S^V X(W) A  σ_V,W AX(V⊕ W) A =  (X A)(V⊕ W)

respectively by the composite

    S^V_*(A,X(W))  → _*(A, S^V X(W))  _*(A,σ_V,W) _*(A, X(V⊕ W))

where the first is an assembly map that sends v f
to the map sending a∈ A to v f(a).

Just as the functors - A and _*(A,-) are adjoint
on the level of based G-spaces,
the two functors just introduced are an adjoint pair on the level 
of orthogonal G-spectra. The adjunction

    G(X,_*(A,Y) )        G(X A,Y)

takes a morphism f:X→_*(A,Y) to the morphism f^♭:X A→ Y 
whose V-th level f^♭(V):X(V) A→ Y(V) is f^♭(V)(x a)=f(V)(x)(a).

An important special case of this construction is
when A=S^W is a representation sphere, i.e., the one-point compactification
of an orthogonal G-representation.
The W-th suspensionsubjectsuspension!of an orthogonal spectrum
X S^W is defined by

    (X S^W)(V)  =  X(V) S^W  ,

the smash product of the V-th level of X with the sphere S^W.
The W-th loop spectrumsubjectloop spectrum
Ω^W X=_*(S^W,X), defined by

    (Ω^W X)(V)  =  Ω^W X(V)  =  _*(S^W,X(V))  ,

the based mapping space from  S^W to the V-th level of X.
We obtain an adjunction between - S^W and Ω^W
as the special case A=S^W of (<ref>).




[Shift of an orthogonal spectrum]
We introduce a spectrum analog of the additive shift
of orthogonal spaces defined 
in Example <ref>. 
We let V be an inner product space and denote by

    -⊕ V :   →

the continuous functor given on objects by
orthogonal direct sum with V, and on morphism spaces by

    (U,W)  → (U⊕ V, W⊕ V) ,   
    (w,φ) ⟼ ((w,0),φ⊕ V)  .

The V-th shiftsubjectshift!of an orthogonal spectrumsymbol^V X - V-th shift of the orthogonal spectrum X 
of an orthogonal spectrum X is the composite

    ^V X  =  X∘ (-⊕ V) .

In other words, the value of ^V X at an inner product space U is

    (^V X)(U)  =  X(U⊕ V)  .

The orthogonal group O(U) acts through the monomorphism
-⊕ V:O(U)→ O(U⊕ V).
The structure map σ_U,W^^V X of ^V X is 
the structure map σ^X_U,W⊕ V of X.


Since composition of functors is associative, the shift construction commutes 
on the nose with all constructions on orthogonal spectra that are given by
postcomposition with a continuous based functor as in 
Construction <ref>. This applies in particular to smashing
with and taking mapping space from a based space A, i.e., 

    (^V X) A =  ^V(X A)   and  _*(A, ^V X)  =  ^V(_*(A, X)) .

So we can – and will – omit the parentheses in expressions such as ^V X A.

The shift construction is also transitive in the following sense.
The values of ^V(^W X) and ^V⊕ WX at an inner product space U
are given by

    (^V(^W X))(U) =  X((U⊕ V)⊕ W)

respectively

    (^V⊕ WX)(U) =  X(U⊕ (V⊕ W))  .

We use the effect of X on the associativity isomorphism

    (U⊕ V)⊕ W  U⊕ (V⊕ W)  ,   
    ((u,v),w) ⟼ (u,(v,w))

to identify these two spaces; then we abuse notation and write

    ^V(^W X) =  ^V⊕ WX .


The suspension and the shift of an orthogonal spectrum X are related by
a natural morphismsymbolλ^V_X - natural _*.isomorphism of orthogonal G-spectra X S^V→^V X

    λ_X^V : X S^V → ^V X .

In level U, this is defined as λ_X^V(U)=σ_U,V^,
the opposite structure map (<ref>),
i.e., the composite

    X(U) S^V twist S^V X(U)  σ_V,U  X(V⊕ U) X(τ_V,U) X(U⊕ V) = (^V X)(U)  .

In the special case V= we abbreviate λ_X^
to λ_X:X S^1→ X.symbolλ_X - natural global equivalence of orthogonal spectra X S^1→ X
The λ-maps are transitive in the sense that for another inner product space W, 
the morphism λ^V⊕ W_X coincides with the two composites
in the commutative diagram: 

    X S^V⊕ W[r]^-    
    X S^V S^W [rr]^-λ^W_X S^V[d]_λ^V_X S^W      ^W X S^V [d]^λ^V_^W X
       ^V X S^W [rr]_-^V(λ^W_X)      ^V(^W X)@=[r]    ^V⊕ WX


Now we let G be a compact Lie group, V a G-representation and X
an orthogonal G-spectrum. Then the orthogonal spectra X S^V and ^V X
become orthogonal G-spectra by letting G act diagonally on X and V.
With respect to these diagonal actions, the morphism λ^V_X:X S^V→^V X
is a morphism of orthogonal G-spectra.
Our next aim is to show that λ^V_X is in fact a _*-isomorphism.
We define a homomorphism

    ψ^V_X  :  π_k^G(^V X)  → π_k^G(X S^V)

by sending the class represented by a G-map

    f : S^U⊕^n+k → X(U⊕^n⊕ V) = (^V X)(U⊕^n)

to the class represented by the composite

    S^U⊕ V⊕^n+k S^Uτ_V,^n+k S^U⊕^n+k⊕ Vf S^V    X(U⊕^n⊕ V)  S^V 
    X(U⊕τ_^n,V) S^V    X(U⊕ V⊕^n)  S^V  .

We omit the straightforward verification that this assignment
is compatible with stabilization, and hence well-defined.
The map ψ^V_X is natural for morphisms of orthogonal G-spectra in X.
Finally, we define

    ε_V :  π_k^G(X S^V)  → π_k^G(X S^V)

as the effect of the involution

    X S^-_V :  X S^V  → X S^V

induced by the antipodal map of S^V.



Let G be a compact Lie group, X an orthogonal G-spectrum 
and V a G-representation.


  * For every integer k, each of the three composites around the triangle  

    π_k^G(X S^V) [rr]^-(λ^V_X)_*      π_k^G(^V X) [dl]^-ψ^V_X
       π_k^G(X S^V) [ul]^-ε_V

is the respective identity.

  * The morphism

    λ^V_X :  X S^V → ^V X   , 
         its adjoint    λ̃^V_X  :  X  → Ω^V ^V X  ,

the adjunction unit η^V_X:X→Ω^V(X S^V) 
and the adjunction counit ϵ^V_X:(Ω^V X) S^V→ X 
are _*-isomorphisms of orthogonal G-spectra.



We introduce an auxiliary functor π^G(A;-)
from orthogonal G-spectra to abelian groups that generalizes 
equivariant homotopy groups and depends on a based G-space A.
We set

    π^G(A;X)  =  _U∈ s(_G)  [S^U A, X(U)]^G  ,

where the colimit is taken over the analogous stabilization maps as for π_0^G;
the set π^G(A;X) comes with a natural abelian group structure
by the same reasoning as for π_0^G(X).
Then π^G(S^k;X) is naturally isomorphic to π_k^G(X);
more generally, for a G-representation W,
the adjunction bijections

    [S^U S^^k⊕ W, X(U)]^G    [S^U⊕^k, Ω^W X(U)]^G

assemble into a natural isomorphism of abelian groups
between π^G(S^^k⊕ W;X) and π_k^G(Ω^W X).

The definition of the map ψ^V_X has a straightforward generalization
to a natural homomorphism

    ψ^V_X  :  π^G(A; ^V X)  → π^G(A;X S^V)

by sending the class represented by a G-map

    f : S^U A → X(U⊕ V) = (^V X)(U)

to the class represented by the composite

    S^U⊕ V A S^Uτ_S^V,A   S^U A S^V f S^V  X(U⊕ V)  S^V  .

We omit the straightforward verification that this assignment
is compatible with stabilization, and hence well-defined.
We claim that each of the three composites around the triangle  

    π^G(A;X S^V) [rr]^-(λ^V_X)_*      π^G(A;^V X) [dl]^-ψ^V_X
       π^G(A;X S^V) [ul]^-ε_V

is the respective identity. 
We consider a based continuous G-map f:S^U A→ X(U) S^V
that represents a class in π^G(A;X S^V).
Then the class ε_V(ψ^V_X((λ^V_X)_*f))
is represented by the composite

    S^U⊕ V A    S^U A S^V f S^V X(U) S^V S^V σ^_U,V S^-_V X(U⊕ V) S^V  .

The map V⊕(-_V):V⊕ V→ V⊕ V
is homotopic, through G-equivariant linear isometries, to the
twist map τ_V,V:V⊕ V→ V⊕ V that interchanges the two summands.
So ε_V(ψ^V_X((λ^V_X)_*f)) is also represented by
the left vertical composite in the following diagram of based continuous G-maps:

    @C=22mm@R=6mm 
    S^U S^V A[d]_S^Uτ_A,S^V[r]^-τ_U,V A    S^V S^U A[ddd]^(.4)S^V f
    
    S^U A S^V[d]_f S^V   
    
    X(U) S^V S^V[d]_X(U)τ_V,V[dr]^-τ_X(U) S^V,S^V   
    
    X(U) S^V S^V[r]_-τ_X(U),S^V S^V[d]_σ^_U,V S^V   
    S^V X(U) S^V[d]^σ_V,U S^V
    
    X(U⊕ V) S^V[r]_-X(τ_U,V) S^V   
    X(V⊕ U) S^V

The right vertical composite (σ_V,U S^V)∘(S^V f)
is the stabilization of f, so it
represents the same class in π^G(A;X S^V).
Since the left and right vertical composites differ by conjugation with
an equivariant isometry, they also represent the same class in π^G(A;X S^V),
by Proposition <ref> (ii).
Altogether this shows that the composite
ε_V∘ψ^V_X∘(λ^V_X)_* is the identity.
Since ε_V^2 is the identity, this also implies
that the composite ψ^V_X∘(λ^V_X)_*∘ε_V is the identity.

The remaining case is similar.
We consider a based continuous G-map g:S^U A→ X(U⊕ V)=(^V X)(U)
that represents a class in π^G(A;^V X).
Then the class (λ^V_X)_*(ε_V(ψ^V_Xg))
is represented by the composite

    S^U S^V A    S^Uτ_S^V,A   S^U A S^V g S^V X(U⊕ V) S^V 
       X(U⊕ V) S^-_V X(U⊕ V) S^V
       σ^_U⊕ V,V X(U⊕ V⊕ V)  =  (^V X)(U⊕ V)   .

Since 

    σ^_U⊕ V,V∘ (X(U⊕ V) S^-)  =  
    X(U⊕ V⊕ (-)) ∘σ^_U⊕ V,V

and V⊕(-_V):V⊕ V→ V⊕ V
is G-homotopic to the twist τ_V,V,
the class (λ^V_X)_*(ε_V(ψ^V_Xg))
is also represented by the left vertical composite in the following diagram:

    @C=25mm@R=6mm 
    S^U S^V A[d]_S^Uτ_S^V,A[r]^-τ_U,V A    
    S^V S^U A[ddd]^S^V g
    
    S^U A S^V[d]_g S^V   
    
    X(U⊕ V) S^V[d]_σ^_U⊕ V,V[dr]^-τ_X(U) S^V,S^V   
    
    X(U⊕ V⊕ V)[dr]^-X(τ_U⊕ V,V)[d]_X(U⊕τ_V,V)   
    S^V X(U⊕ V)[d]^σ_V,U⊕ V
    
    X(U⊕ V⊕ V)[r]_-X(τ_U,V⊕ V)@=[d]    
    X(U⊕ V⊕ V)  @=[d] 
    
    (^V X)(U⊕ V)[r]_-(^V X)(τ_U,V)   
    (^V X)(U⊕ V)

The right vertical composite σ_V,U⊕ V∘(S^V g)
is the stabilization of g, 
so it represents the same class in π^G(A;^V X).
Since the left and right vertical composites differ by conjugation with
an equivariant isometry, they represent the same class,
so the composite (λ^V_X)_* ∘ε_V∘ψ^V_X is the identity.


Now we prove claim (i) of the proposition. 
For k≥ 0, it is the special case A=S^k of the discussion above.
To deduce the claim for negative dimensional homotopy groups we use  
 the isomorphism of orthogonal G-spectra 
 
    τ_k,V : ^k(^V X)  ^V(^k X)

whose value at an inner product space U is the map

    X(U⊕τ_^k,V) :  X(U⊕^k⊕ V)    
    X(U⊕ V⊕^k) .

Then the following diagram commutes:

    @C=15mm@R=5mmπ_-k^G(X S^V)[r]^-(λ_X^V)_*@=[dd]    π_-k^G(^V X)[r]^-ε_V∘ψ_X^V@=[d]    π_-k^G(X S^V)@=[dd]
       π_0^G(^k(^V X))[d]_(τ_k,V)_*^   
    π_0^G(^k X S^V)[r]_-(λ_^k X^V)_*   π_0^G(^V (^k X))[r]_-ε_V∘ψ_^k X^V   π_0^G(^k X S^V)

So the claim in dimension -k for the orthogonal G-spectrum X
is a consequence of the previously established claim
in dimension 0 for the orthogonal G-spectrum ^k X.

(ii) 
We start with the morphism λ̃^V_X, which can be treated fairly
directly. We discuss the case k≥ 0 and leave the analogous argument
for k<0 to the reader.
We define a map in the opposite direction

    κ :  π_k^G(Ω^V^V X) → π_k^G(X) .

We let g:S^U⊕^k→Ω^V X(U⊕ V)= (Ω^V^V X)(U)
represent a class of the left hand side. The map κ sends [g]
to the class represented by the composite

    S^U⊕ V⊕^k S^Uτ_V,^kS^U⊕^k⊕ V  g^♭  X(U⊕ V)  ,

where g^♭ is the adjoint of g.
This is compatible with stabilization.

We claim that the map κ is injective.
Indeed, if g:S^U⊕^k→Ω^V X(U⊕ V)
represents an element in the kernel of κ,
then after increasing U, if necessary, the composite
g^♭∘(S^Uτ_V,^k) is G-equivariantly null-homotopic.
But then g^♭, and hence also its adjoint g, are
equivariantly null-homotopic. So κ is injective.

The composite κ∘(λ̃^V_X)_* sends the class of a G-map
f:S^U⊕^k→ X(U) to the class of the composite

    S^U⊕ V⊕^k S^Uτ_V,^kS^U⊕^k⊕ V  (η_X^V(U)∘ f)^♭  X(U⊕ V)  .

The adjoint (η_X^V(U)∘ f)^♭ coincides with the composite

    S^U⊕^k⊕ V  f S^V  X(U) S^Vσ^_U,V X(U⊕ V) ,

so the composite (<ref>)
represents the same class as f.
This proves that κ∘(λ̃^V_X)_* is the identity.
Since κ is also injective, the map (λ̃^V_X)_*
is bijective. 

If H is a closed subgroup of G we apply the previous argument
to the underlying H-representation of V to conclude that
λ̃^V_X induces an isomorphism on π_*^H.
So  (λ̃^V_X)_* is a _*-isomorphism of orthogonal G-spectra.



Now we treat the morphism λ^V_X.
Again we show a more general statement, namely that
for every pair of G-representations V and W the morphism

    Ω^W(λ^V_X) : Ω^W(X S^V)  → Ω^W(^V X)

is a _*-isomorphism of orthogonal G-spectra.
We start with the effect on G-equivariant homotopy groups.
For k≥ 0 this follows by applying part (i) with A=S^^k⊕ W 
and exploiting the natural isomorphism π_k^G(Ω^W Y)π^G(S^^k⊕ W;Y).
To get the same conclusion
for negative dimensional homotopy groups we exploit that
π_-k^G(Y)=π_0^G(^k Y), by definition.
Moreover, the following diagram commutes

    @R=5mm@C=25mmπ_-k^G(Ω^W(X S^V)) @=[r] [dd]_π_-k^G(Ω^W(λ^V_X))   π_0^G(Ω^W (^k X S^V))  [d]^π_0^G(Ω^W(λ^V_^k X))
       π_0^G(Ω^W(^V(^k X)))  
    π_-k^G(Ω^W(^V X)) @=[r]    π_0^G(Ω^W(^k(^V X)))
    [u]^_π_0^G(Ω^W τ_k,V)

where the isomorphism τ_k,V was defined in (<ref>).
So the previous argument applied to the spectrum ^k X shows
that the morphism Ω^W(λ^V_X) also induces isomorphisms 
on G-equivariant homotopy groups in negative dimensions.
If H is any closed subgroup of G, then we consider the underlying H-representations
of V and W and conclude that the morphism Ω^W(λ^V_X) 
induces isomorphisms on π_*^H.
This proves that Ω^W(λ^V_X) is a _*-isomorphism 
of orthogonal G-spectra.
The special case W=0 proves that λ^V_X is a _*-isomorphism.

The morphism λ̃^V_X factors as the composite

    X   η^V_X   Ω^V(X S^V)  Ω^V(λ^V_X) Ω^V^V X .

Since both λ̃^V_X and Ω^V( λ^V_X) are
_*-isomorphisms of orthogonal G-spectra,
so is the adjunction unit η^V_X.

Finally, we treat the adjunction counit ϵ^V_X.
The two homomorphisms of orthogonal G-spectra

    Ω^V (λ̃^V_X) ,  λ̃^V_Ω^V X :  Ω^V X  → Ω^V(Ω^V^V X)
 
are not the same; they differ by the involution on the target
that interchanges the two V-loop coordinates.
An equivariant homotopy of linear isometries from τ_V:V⊕ V→ V⊕ V
to (-)⊕ thus induces an equivariant homotopy between the morphism
Ω^V (λ̃^V_X) and the composite

    Ω^V X  λ̃^V_Ω^V XΩ^V(Ω^V^V X)  Ω^V _*(S^-,^V X)Ω^V(Ω^V^V X)  .

Passing to adjoints shows that the square of morphisms of orthogonal G-spectra

    @C=25mm 
    (Ω^V X) S^V[r]^-ϵ_X^V[d]_λ^V_Ω^V X    
    X[d]^λ̃^V_X
    ^V Ω^V X[r]^-_-^V_*(S^-,X)   ^V Ω^V X

commutes up to G-equivariant homotopy. Since the two vertical morphisms
are _*-isomorphisms, and the lower horizontal one is even an isomorphism,
we conclude that ϵ^V_X is a _*-isomorphism.



Now we recall some important properties of equivariant homotopy groups,
such as stability under suspension
and looping, and the long exact sequences associated to
mapping cones and homotopy fibers.
We define the loop isomorphismsubjectloop isomorphism

    α :  π^G_k(Ω X) → π^G_k+1 (X)   .

We represent a given class in π_k^G(Ω X) by a based G-map 
f:S^V⊕^n+k→Ω X(V⊕^n) and 
let f^♭ :S^V⊕^n+k+1→ X(V⊕^n)
denote the adjoint of f, which represents an element of
π_k+1^G(X). Then we set α[f]=[f^♭].

Next we define the suspension isomorphismsubjectsuspension isomorphism

    - S^1  :  π^G_k (X) → π^G_k+1(X S^1)  .

We represent a given class in π_k^G(X) by a based G-map 
f:S^V⊕^n+k→ X(V⊕^n);
then f S^1 : S^V⊕^n+k+1→  X(V⊕^n) S^1
represents a class in π_k+1^G(X S^1), and we set [f] S^1= [f S^1].




Let G be a compact Lie group, X an orthogonal G-spectrum
and k an integer.
Then the loop isomorphism (<ref>) 
and the suspension isomorphism (<ref>) 
are isomorphisms of abelian groups.


The inverse to the loop map (<ref>) is given by
sending the class of a G-map S^V⊕^n+k+1→ X(V⊕^n)
to the class of its adjoint S^V⊕^n+k→Ω X(V⊕^n).
The suspension homomorphism is the composite of the two maps

    π_k^G(X)  (η_X)_*  π_k^G(Ω(X S^1))  [] α π_k+1^G(X S^1) ;

since (η_X)_* is an isomorphism by Proposition <ref> (ii),
so is the suspension isomorphism.




A key feature that distinguishes stable from unstable
equivariant homotopy theory – and at the same time an important calculational tool –
is the fact that mapping cones give rise to long exact sequences of equivariant
homotopy groups. Our next aim is to establish
this long exact sequence, see Proposition <ref>.

[Mapping cone and homotopy fiber]
The reduced mapping conesubjectmapping cone
C f of a morphism of based spaces f:A→ B is defined by

    C f  =  (A [0,1])∪_f B  .

Here the unit interval [0,1] is pointed by 0,
so that A [0,1] is the reduced cone of A. 
The mapping cone comes with an embedding i:B→ C f
and a projection p:C f→ A S^1; the projection
sends B to the basepoint and is given on A [0,1] 
by p(a,x)=a t(x), where

    t : [0,1] →  S^1     is defined as    
    t(x) = 2x-1/x(1-x) .

What is relevant about the map t is not the precise formula, 
but that it passes to a homeomorphism between the quotient space 
[0,1]/{0,1} and S^1=∪{∞}, and that it satisfies t(1-x)=-t(x).

The mapping conesymbolC f - mapping cone of f 
C f of a morphism f:X→ Y of orthogonal G-spectra
is now defined levelwise by

    (C f)(V)  =  C(f(V))  =  (X(V) [0,1])∪_f(V) Y(V)  ,

the reduced mapping cone of f(V):X(V)→ Y(V). 
The groups G and O(V) act on (C f)(V)
through the given action on X(V) and Y(V) and trivially on the interval.
The embeddings i(V):Y(V)→ C(f(V))
and projections p(V):C(f(V))→ X(V) S^1 assemble into
morphisms of orthogonal G-spectra 

    i : Y → C f      and     p : C f →  X S^1 .

We define a connecting homomorphismsubjectconnecting homomorphism!for equivariant homotopy groups 
∂:π^G_k+1(C f)→π^G_k(X) as the composite

    π^G_k+1(C f)  π^G_k+1(X S^1) - S^-1 π^G_k (X)  ,

the effect of the projection p:C f→ X S^1 on homotopy groups, 
followed by the inverse of the suspension isomorphism (<ref>).

The homotopy fibersubjecthomotopy fibersymbolF f - homotopy fiber of f
is the construction `dual' to the mapping cone.
The homotopy fiber of a continuous map f:A→ B  of based spaces
is the fiber product

    F(f) =  *×_B B^[0,1]×_B A 
     =  {(λ,a)∈ B^[0,1]× A |  λ(0)=*, λ(1)=f(a)} ,

i.e., the space of paths in B starting at the basepoint 
and equipped with a lift of the endpoint to A.
As basepoint of the homotopy fiber we take the pair consisting
of the constant path at the basepoint of B and the basepoint of A.
The homotopy fiber comes with maps

    Ω B  i  F(f)  p  A   .

The map p is the projection to the second factor;
the value of the map i on a based loop ω:S^1→ B is

    i(ω) = (ω∘ t,*)   ,

where t:[0,1]→ S^1 was defined in (<ref>).

The homotopy fiber F(f) of a morphism f:X→ Y of orthogonal G-spectra
is the orthogonal G-spectrum defined by

    F(f)(V)  =  F(f(V))  ,

the homotopy fiber of f(V):X(V)→ Y(V). 
The groups G and O(V) act on F(f)(V)
through the given action on X(V) and Y(V) and trivially on the interval.
Put another way, the homotopy fiber is the pullback in the cartesian square 
of orthogonal G-spectra:

    @C=25mm F(f) [d] [r]^-p    
     X [d]^-(*,f)
    
      Y^[0,1][r]_-λ ↦ (λ(0),λ(1))    Y× Y

The inclusions i(V):Ω Y(V)→ F(f(V))
and projections p(V):F(f(V))→ X(V) assemble into
morphisms of orthogonal G-spectra 

    i : Ω Y →  F(f)      and     p : F(f) →  X .

We define a connecting homomorphism
subjectconnecting homomorphism 
∂:π^G_k+1(Y)→π^G_k (F(f)) as the composite

    π^G_k+1(Y)   α^-1 π^G_k(Ω Y)
     π^G_k (F(f)) ,

where the first map is the inverse of the loop isomorphism (<ref>).


The proof of exactness for the mapping cone sequence will 
need some elementary homotopies that we spell out in the next proposition.

 
Let G be a topological group.


  * For every continuous based map f:A→ B 
of based G-spaces the composites

    A  f  B  i  C f      and     
    F(f)  p  A  f   B

are naturally based G-null-homotopic. Moreover, the diagram

    @C=5mm    CA∪_f CB [dl]_p_A∪ *[dr]^*∪ p_B
    
    A  S^1 [rr]_-fτ       B S^1

commutes up to natural, based G-homotopy, where τ 
is the sign involution of S^1 given by x↦ -x.

  * For every based G-space Z the map
p_Z∪ *:CZ∪_Z× 1CZ→ Z S^1
which collapses the second cone is a based G-homotopy equivalence.

  * Let f:A→ B and β:Z→ B be morphisms of based G-spaces 
such that the composite iβ:Z→ C f is equivariantly null-homotopic. 
Then there exists a based G-map h:Z S^1→ A S^1 such that 
(f S^1)∘ h:Z S^1→ B S^1 
is equivariantly homotopic to β S^1.



(i)
We specify natural G-homotopies by explicit formulas.
The map i f:A→ C f is null-homotopic by
A× [0,1]→ C f, (a,s)↦(a,s), 
i.e., the composite of the canonical maps A× [0,1]→ A [0,1]
and A [0,1]→ C f. The map f p:F(f)→ B is null-homotopic by
F(f)× [0,1]→ B, (λ,a,s)↦λ(s).

The homotopy for the triangle will be glued together from
two pieces. We define a based homotopy 
H : C A× [0,1] → B S^1  by the formula

    H(a,s,u)  =  f(a) t(2-s-u)

which is to be interpreted as the basepoint if 2-s-u≥ 1.
Another based homotopy H':C B× [0,1] → B S^1 
is given by the formula

    H'(b,s,u)  =  b t(s-u)  ,

where t:[0,1]→ S^1 was defined in (<ref>).
This formula is to be interpreted as the basepoint if s≤ u.
The two homotopies are compatible in the sense that

    H(a,1,u)  =  f(a) t(1-u)  =  H'(f(a),1,u)  ,

for all a∈ A and u∈[0,1]. So H and H' glue and yield a homotopy

    ( C A∪_f C B) × [0,1]
    (C A× [0,1])∪_f× [0,1] (C B× [0,1])  H∪ H' B S^1  .

For u=0 this homotopy starts with the map *∪ p_B,
and it ends for u=1 with the map (fτ)∘(p_A∪ *).

(ii) 
Since the functor Z- is a left adjoint and Z{0,1} Z× 1,
the space C Z∪_1× Z C Z is homeomorphic to the smash product
of Z and the pushout [0,1]∪_{0,1} [0,1].
This identification

    C Z∪_Z× 1 C Z  Z ([0,1]∪_{0,1} [0,1] )

turns the map p_Z into the map

    Z (t∪∗) :  Z ([0,1]∪_{0,1} [0,1] ) → Z S^1  .

So the claim follows from the fact that the map
t∪∗:[0,1]∪_{0,1} [0,1] →  S^1 is a based homotopy equivalence.

(iii)
Let H:C Z= Z [0,1]→ C f be a based, equivariant null-homotopy of
the composite iβ:Z→ C f, 
i.e., H(z,1)=i(β(z)) for all z∈ Z. 
The composite p_AH: C Z→ A S^1 then factors 
as p_A H=h p_Z for a unique G-map h:Z S^1→ A S^1.
We claim that h has the required property.


To prove the claim we need the G-homotopy equivalence
p_Z∪ *:CZ∪_Z× 1CZ→ Z S^1
which collapses the second cone.
We obtain a sequence of equalities and G-homotopies

    (f S^1)∘ h∘(p_Z∪ *)    =  
    (f S^1)∘ (p_A∪ *)∘ (H∪ C(β)) 
       = (Bτ)∘(fτ)∘ (p_A∪ *)∘ (H∪ C(β)) 
       ≃ (Bτ)∘(*∪ p_B)∘ (H∪ C(β)) 
       =  (Bτ)∘(β S^1)∘(*∪ p_Z) 
       =  (β S^1)∘(Zτ)∘(*∪ p_Z)  ≃ (β S^1)∘(p_Z∪ *)

Here H∪ C(β):CZ∪_1× ZCZ→ 
C f∪_B CB  CA∪_f CB
and τ is the sign involution of S^1.
The two homotopies result from part (i)  
applied to f respectively the
identity of Z, and we used the naturality of various constructions.
Since the map p_Z∪ * is a G-homotopy equivalence by part (ii), this proves
that (f S^1)∘ h is G-homotopic to β S^1.


Now we are ready to prove the long exact homotopy group sequences for
mapping cones and homotopy fibers.



subjectlong exact sequence!of equivariant homotopy groups
For  every compact Lie group G and
every morphism f:X→ Y of orthogonal G-spectra
the long sequences of equivariant homotopy groups

    ⋯ → π^G_k+1 (C f)  π^G_k(X)   π^G_k(Y)  π^G_k (Cf)  → ⋯

and

    ⋯ → π^G_k+1(Y)  π_k^G ( F(f))  π^G_k(X)   π^G_k(Y)  → ⋯

are exact.


We start with exactness of the first sequence at π^G_k(Y). 
The composite of f:X→ Y and the
inclusion i:Y→ C f is equivariantly null-homotopic, 
so it induces the trivial map on π^G_k. 
It remains to show that every element in the
kernel of π_k^G(i):π^G_k ( Y ) →π^G_k (C f)
is in the image of π_k^G(f). 
We show this for k≥ 0; for k<0 we can either use a similar argument 
or exploit that π_k^G(X)=π_0^G(^^-kX)
and shifting commutes with the formation of mapping cones.
We let β:S^V⊕^k→ Y(V) represent an element in the kernel
of π_k^G(i).
By increasing V, if necessary, we can assume that  
the composite of β with the inclusion
i:Y(V)→ (C f)(V)=C(f(V)) is equivariantly null-homotopic.
Proposition <ref> (iii) provides a G-map
h:S^V⊕^k S^1→ X(V) S^1 such that 
(f(V) S^1)∘ h is G-homotopic to β S^1. 
The composite

    S^V⊕^k+1  h  X(V) S^1  σ^_V,  X(V⊕)
 
represents an equivariant homotopy class in π_k^G(X) and we have

    π_k^G(f)σ_V,^∘ h    =  f(V⊕)∘σ_V,^∘ h
     
    =     σ^_V,∘(f(V) S^1)∘ h= σ_V,^∘(β S^1) =  (-1)^k·β .

To justify the last equation we let φ:V→ V⊕ 
denote the embedding of the first summand. 
Then the maps

    σ_V,^∘(β S^1) :  S^V⊕^k⊕ → Y(V⊕)
       and  φ_*β : S^V⊕⊕^k → Y(V⊕)

differ by the permutation of the source that moves one sphere coordinate
past k other sphere coordinates; this permutation has degree k,
so Proposition <ref> (ii) establishes 
the last equation.
Altogether this shows that the class represented by β 
is the image under π_k^G(f) 
of the class (-1)^k·σ^_V,∘ h.

We now deduce the exactness at π_k^G (C f) and π^G_k-1 ( X )
by comparing the mapping cone sequence for f:X→ Y 
to the mapping cone sequence for the morphism i:Y→ Cf 
(shifted to the left).
We observe that the collapse map

    *∪ p :  C i  C Y∪_f C X  → X S^1

is an equivariant homotopy equivalence, and thus induces an isomorphism 
of equivariant homotopy groups.
Indeed, a homotopy inverse

    r  :  X S^1  → C Y∪_f C X

is defined by the formula

    r(x s)  =       (x,2s)   ∈ C X     for 0≤ s≤ 1/2, and
    
    (f(x),2-2s) ∈ C Y     for 1/2≤ s≤ 1,

which is to be interpreted levelwise.
We omit the G-homotopies r(*∪ p)≃ 
and (*∪ p)r≃; explicit formulas can be found in the
proof of <cit.>. 
Now we consider the diagram 

    C f [r]^-i_i[dr]_p     Ci [r]^-p_i[d]_(.4)*∪ p    Y S^1
        X S^1 [ur]_-f S^1

whose upper row is part of the mapping cone sequence
for the morphism i:Y→ C f.
The left triangle commutes on the nose and the right triangle commutes
up to G-homotopy. We get a commutative diagram

    @C=17mmπ^G_k ( Y ) @=[d] [r]^-π_k^G(i)   π^G_k ( C f) [r]^-π_k^G(i_i)@=[d]    π^G_k ( C i) [r]^-(.4)∂[d]_(- S^-1)∘(*∪ p)_*^   π^G_k-1(Y) @=[d] 
    π^G_k ( Y ) [r]_-π_k^G(i)   π^G_k(C f)[r]_-∂   π^G_k-1 (X) [r]_-π_k^G(f)   π^G_k-1(Y)

(using, for the right square, the naturality of the suspension isomorphism).
By the previous paragraph, applied to i:Y→ Cf instead of f,
the upper row is exact at π^G_k(C f). Since all vertical maps
are isomorphisms, the original lower row is exact at π^G_k(C f).
But the morphism f was arbitrary, so when applied to 
i:Y→ Cf instead of f, we obtain that the upper row is
exact at π^G_k( C i).  Since all vertical maps
are isomorphisms, the original lower row is exact at π^G_k-1 (X).
This finishes the proof of exactness of the first sequence.


Now we come to why the second sequence is exact. 
We show the claim for k≥ 0, the other case being similar.
For every V∈ s(_G)
the sequence F(f)(V)=F(f(V))→ X(V)→ Y(V) 
is an equivariant homotopy fiber sequence. 
So for every based G-CW-complex A, the long sequence of based sets

    ⋯ → [A,Ω Y(V)]^G     [A,i(V)] [A,F(f(V))]^G 
       [A,p(V)]^G  [A,X(V)]^G   [A,f(V)]^G  [A,Y(V)]^G

is exact. We take A=S^V⊕^k and form the colimit over the poset s(_G).
Since filtered colimits are exact, the resulting sequence of colimits
is again exact, and that proves the second claim.



 


subjectequivariant homotopy group!of a wedge
Let G be a compact Lie group and k an integer.


  * For every family of orthogonal G-spectra {X_i}_i∈ I the canonical map

    ⊕_i∈ I π^G_k ( X_i ) → π^G_k( ⋁_i∈ I   X_i )

is an isomorphism.

  * For every finite index set I and 
every family {X_i}_i∈ I of orthogonal G-spectra the canonical map

    π^G_k( ∏_i∈ I  X_i )   →∏_i∈ I π^G_k ( X_i )

is an isomorphism.subjectequivariant homotopy group!of a product

  * For every finite family of orthogonal G-spectra
the canonical morphism from the wedge to the product 
induces isomorphisms of G-equivariant homotopy groups.



(i) We first show the special case of two summands.
If X and Y are two orthogonal G-spectra, 
then the wedge inclusion j:X→ X∨ Y has a retraction. 
So the associated long exact homotopy group sequence of
Proposition <ref>
splits into short exact sequences

    0  → π^G_k(X)  π^G_k(j) π^G_k(X∨ Y)  π_k^G(i) π^G_k(C j)  → 0  .

The mapping cone C j is isomorphic to (C X)∨ Y
and thus G-homotopy equivalent to Y. So we can replace 
π^G_k(C j) by π^G_k(Y) and conclude that π^G_k(X∨ Y)
splits as the sum of π^G_k(X) and π^G_k(Y), via the canonical map.
The case of a finite index set I now follows by induction.

In the general case we consider the composite

    ⊕_i∈ I π^G_k ( X_i ) → π^G_k( ⋁_i∈ I  X_i )  → ∏_i∈ I π^G_k (  X_i )  ,

where the second map is induced by the projections to the wedge summands.
This composite is the canonical map from a direct sum to a product
of abelian groups, hence injective.
So the first map is injective as well. For surjectivity we consider
a G-map f:S^V⊕^n+k→⋁_i∈ I X_i(V⊕^n)
that represents an element in the k-th G-homotopy group of
⋁_i∈ I X_i. 
Since the source of f is compact, there is a finite subset J
of I such that f has image in ⋁_j∈ J X_j(V⊕^n),
compare Proposition <ref>.
Then the given class is in the image of 
π^G_k( ⋁_j∈ J X_j );
since J is finite, the class is in the image of the canonical map,
by the previous paragraph.

(ii) The functor X↦ [S^V⊕^n+k, X(V⊕^n)]^G commutes with products. 
Finite products commute with filtered colimits, such as the one defining π_k^G,
so passage to colimits gives the claim.

(iii) For a finite indexing set I and every integer k the composite map

    ⊕_i∈ I π^G_k( X_i)  → π^G_k(⋁_i∈ I  X_i ) 
     → π^G_k(∏_i∈ I  X_i ) → ∏_i∈ I π^G_k(X_i)

is an isomorphism, where the first and last maps are the canonical ones.
These canonical maps are isomorphisms by parts (i) respectively (ii),
hence so is the middle map.


The equivariant homotopy group functor π_k^G
does not in general commute with infinite products. The issue is that
π_k^G involves a filtered colimit, and these do not always commute
with infinite products.
However, this defect is cured when we pass to the equivariant or global stable
homotopy category, i.e., π_k^G  takes `derived' infinite products
to products. We refer to Remark <ref>
below for more details.



We recall that a morphism f:A→ B of orthogonal G-spectra
is an h-cofibrationsubjecth-cofibration!of orthogonal spectra 
if it has the homotopy extension property, 
i.e., given a morphism of orthogonal G-spectra φ:B→ X and a homotopy
H:A [0,1]_+→ X starting with φ f,
there is a homotopy H̅:B [0,1]_+→ X starting with φ
such that H̅∘(f [0,1]_+)=H.
For every such h-cofibration f:A→ B
the collapse map c:C f→ B/A from the mapping cone to the cokernel
of f is a G-homotopy equivalence. Indeed, since the square

    A[r]^-f [d]_-  1    B[d] 
     A [0,1] [r]_-g     C f

is a pushout, the cobase change g:C A=A [0,1]→ C f
also has the homotopy extension property.
The cone C A is contractible, so the claim follows from the following
more general statement: for every h-cofibration i:C→ D such that C is contractible,
the quotient map D→ D/C is a based homotopy equivalence.
The standard proof of this fact in the category of topological spaces
(see for example <cit.>
or <cit.> with B=∗) 
only uses formal properties of homotopies,
and carries over to the category of orthogonal G-spectra.

So for every h-cofibration f:A→ B of orthogonal G-spectra,
the collapse map c:C f→ B/A is an equivariant homotopy equivalence, 
hence it induces isomorphisms of all equivariant homotopy groups.
We can thus define a modified connecting homomorphism
∂:π^G_k+1(B/A)→π^G_k(A) as the composite

    π^G_k+1(B/A)  π^G_k+1( C f)  ∂  π^G_k (A)  .






We call a morphism f:X→ Y of orthogonal G-spectra 
a strong level fibrationsubjectstrong level fibration!of orthogonal G-spectra 
if for every closed subgroup H of G and every H-representation V 
the map f(V)^H:X(V)^H→ Y(V)^H is a Serre fibration.
For every such strong level fibration,
the embedding j:F→ F(f) of the strict fiber into the homotopy fiber
then induces isomorphisms on π_*^G.
We can thus define a modified connecting homomorphism
∂:π^G_k+1(Y)→π^G_k (F) as the composite

    π^G_k+1(Y)   ∂  π^G_k (F(f))  (π_k^G(j))^-1 π^G_k (F) .

So we deduce:


Let G be a compact Lie group.


  * Let f:A→ B be an h-cofibration of orthogonal G-spectra.
Then the long sequence of equivariant homotopy groups

    ⋯ → π_k+1^G (B/A)   π_k^G (A)  π_k^G(f)  π_k^G(B)   π_k^G(q)   π_k^G (B/A)  → ⋯

is exact.

  * Let f:X→ Y be a strong level fibration of orthogonal G-spectra
and j:F→ X the inclusion of the pointset level fiber of f.
Then the long sequence of equivariant homotopy groups

    ⋯ → π_k+1^G (Y)   ∂   π_k^G (F)
     π_k^G(X)   π_k^G(f)  π_k^G(Y) 
      → ⋯

is exact.





  Let G be a compact Lie group and
  
    A [r]^-f [d]_g     B [d]^h
    
        C [r]_-k     D

  a commutative square of orthogonal G-spectra.

 
  * Suppose that the square is a pushout and
   the map π_*^G(f):π_*^G(A)→π_*^G(B) 
   of G-equivariant homotopy groups is an isomorphism.
   If in addition f or g is an h-cofibration, then 
   the map π_*^G(k):π_*^G(C)→π_*^G(D) is also an isomorphism.
 
  * Suppose that the square is a pullback and
   the map π_*^G(k):π_*^G(C)→π_*^G(D) 
   of G-equivariant homotopy groups is an isomorphism.
   If in addition k or h is a strong level fibration, then 
   the map π_*^G(f):π_*^G(A)→π_*^G(B) is also an isomorphism.
 


(i) If f is an h-cofibration, then its long exact homotopy group
sequence (Corollary <ref>)
shows that all G-equivariant homotopy groups of the cokernel B/A
are trivial. Since the square is a pushout, the induced morphism from
B/A to any cokernel D/C of k is an isomorphism, so the groups
π_*^G(D/C) are all trivial. As a cobase change of the
h-cofibration f, the morphism k is again an h-cofibration,
so its long exact homotopy group sequence shows that π_*^G(k)
is an isomorphism.

If g is an h-cofibration, then so is its cobase change h.
Moreover, any cokernel C/A of g maps by an isomorphism to 
any cokernel D/B of h, since the square is a pushout.
The square induces compatible maps between 
the two long exact homotopy group sequences of g and h,
and the five lemma then shows that π_*^G(k) is an isomorphism.
The argument for part (ii) is dual to (i).




Since the group π_k^G(Ω^m X) is naturally isomorphic
to π_k+m^G(X), looping preserves equivariant stable equivalences. 
The next proposition generalizes this.


Let G be a compact Lie group and A a finite based G-CW-complex.


  * Let f:X→ Y be a morphism of orthogonal G-spectra with the
following property: for every closed subgroup H of G that 
fixes some non-basepoint of A,
the map π_*^H(f):π_*^H(X)→π_*^H(Y) is an isomorphism.
Then the morphism _*(A,f):_*(A,X)→_*(A,Y) is a
_*-isomorphism of orthogonal G-spectra.

  * The functor _*(A,-) preserves _*-isomorphisms of orthogonal G-spec­tra.


(i) We start with a special case and let X be an orthogonal G-spectrum
whose equivariant homotopy groups π_*^H(X) vanish for all closed subgroups
H of G that fix some non-basepoint of A. 
We show first that then the G-equivariant homotopy groups
of the orthogonal G-spectrum _*(A,X) vanish.

We argue by induction over the number of equivariant cells in a CW-structure of A.
The induction starts when A consists only of the basepoint,
in which case _*(A,X) is a trivial spectrum and there is nothing to show.
For the inductive step we assume that the groups π_*^G(_*(B,X)) 
vanish and A is obtained from B by attaching an equivariant n-cell 
G/H× D^n along its boundary G/H×∂ D^n, 
for some closed subgroup H of G.
Then the restriction map _*(A,X)→_*(B,X)
is a strong level fibration of orthogonal G-spectra whose fiber is
isomorphic to

    _*(A/B,X)  _*(G/H_+ S^n,X)  _*(G/H_+,Ω^n X) .

The G-equivariant stable homotopy groups of this spectrum
are isomorphic to the H-equivariant homotopy groups of Ω^n X,
and hence to the shifted H-homotopy groups of X.
Since H occurs as a stabilizer of a cell of A, 
the latter groups vanish by assumption.
The long exact sequence of Corollary <ref> (ii)
and the inductive hypothesis for B then show that the groups 
π_*^G(_*(A,X)) vanish.

When H is a proper closed subgroup of G we exploit
that the underlying H-space of A is H-equivariantly homotopy equivalent
to a finite H-CW-complex, see <cit.>.
We can thus apply the previous paragraph to the underlying H-spectrum of X
and the underlying H-space of A and conclude that
the H-equivariant homotopy groups of _*(A,X) vanish.
Altogether this proves the special case of the proposition.

The functor _*(A,-) commutes with homotopy fibers;
so two applications of the long exact homotopy group sequence
of a homotopy fiber (Proposition <ref>)
reduce the general case of the first claim to the special case.
Part (ii) is a special case of (i).





  Let G be a compact Lie group.
  
  
  * Let e_n:X_n→ X_n+1 be morphisms 
    of orthogonal G-spectra that are levelwise closed embeddings, for n≥ 0. 
    Let X_∞ be a colimit of the sequence {e_n}_n≥ 0.
    Then for every integer k the canonical map
    
    _n≥ 0 π_k^G(X_n) → π_k^G(X_∞)

    is an isomorphism.
  
  * Let e_n:X_n→ X_n+1 and f_n:Y_n→ Y_n+1 be morphisms 
    of orthogonal G-spectra that are levelwise closed embeddings, for n≥ 0. 
    Let ψ_n:X_n→ Y_n be _*-isomorphisms of orthogonal G-spectra 
    that satisfy ψ_n+1∘ e_n=f_n∘ψ_n for all n≥ 0.
    Then the induced morphism ψ_∞:X_∞→ Y_∞ 
    between the colimits of the sequences is a _*-isomorphism.
  
  * Let f_n:Y_n→ Y_n+1 be _*-isomorphisms of orthogonal G-spectra
   that are levelwise closed embeddings, for n≥ 0. 
   Then the canonical morphism 
   f_∞:Y_0→ Y_∞ to a colimit of the sequence {f_n}_n≥ 0
   is a _*-isomorphism.
  


 (i) We let V be a G-representation, and m≥ 0 such that m+k≥ 0.
 Since the sphere S^V⊕^m+k is compact and X_∞(V⊕^m) 
 is a colimit of the sequence of closed embeddings
 X_n(V⊕^m)→ X_n+1(V⊕^m),
 any based G-map f:S^V⊕^m+k→ X_∞(V⊕^m)
 factors through a continuous map
 
    f̅ : S^V⊕^m+k → X_n(V⊕^m)

 for some n≥ 0, see for example <cit.>
 or Proposition <ref> (i).
 Since the canonical map X_n(V⊕^m)→ X_∞(V⊕^m) is injective,
 f̅ is again based and G-equivariant. The same applies to homotopies,
 so the canonical map
 
    _n≥ 0  [S^V⊕^m+k,X_n(V⊕^m)]^G  → 
     [S^V⊕^m+k,X_∞(V⊕^m)]^G

 is bijective. Passing to colimits over m and over the poset s(_G) 
 proves the claim.

 Part (ii) is a direct consequence of (i), applied to G and all its closed subgroups.
 Part (iii) is a special case of part (ii) where we set X_n=Y_0,
 e_n=_Y_0 and ψ_n=f_n-1∘…∘ f_0:Y_0→ Y_n.
 The morphism ψ_n is then a _*-isomorphism
 and Y_0 is a colimit of the constant first sequence.
 Since the morphism ψ_∞ induced on the colimits 
 is the canonical morphism Y_0→ Y_∞, part (ii) specializes to claim (iii).



[Equivariant suspension spectrum]
Every based G-space A gives rise to a suspension spectrumsubjectsuspension spectrum!of a G-space
Σ^∞ A. This is the orthogonal G-spectrum with V-term

    (Σ^∞ A)(V)  =  S^V A ,

with O(V)-action only on S^V, with G-action only on A, 
and with structure map σ_U,V given by the canonical homeomorphism
S^U S^V A S^U⊕ V A.
For an unbased G-space Y, we obtain an unreduced suspension spectrum
Σ^∞_+ Y by first adding a disjoint basepoint to Y
and then forming the suspension spectrum as above.


The suspension spectrum functor is homotopical on a large class
of G-spaces; however, since the reduced suspension S^1 - does not preserve
weak equivalences in complete generality, a non-degeneracy condition
on the basepoint is often needed.


A based G-space is well-pointedsubjectwell-pointed
if the basepoint inclusion has the homotopy extension property
in the category of unbased G-spaces.  


Equivalently, a based G-space (A,a_0) is well-pointed
if and only if the inclusion of A×{0}∪{a_0}×[0,1]
into A×[0,1] has a continuous G-equivariant retraction.
Cofibrant based G-spaces are always well-pointed.
Also, if we add a disjoint basepoint to any unbased G-space,
the result is always well-pointed.
The reduced suspension of a well-pointed space is again well-pointed.
Indeed, if (A,a_0) is well-pointed, then the subspace inclusion

    A×{0,1}∪{a_0}× [0,1] → A×[0,1]

is an h-cofibration, see for example
<cit.> or <cit.>.
Since  h-cofibrations are stable under cobase change
(see Corollary <ref> (ii)), 
the inclusion of the basepoint into the quotient space

    A×[0,1]/ (A×{0,1}∪{a_0}× [0,1])

is an h-cofibration.
Since S^1 A is homeomorphic to this quotient space, this proves the claim.

We recall that a continuous map f:X→ Y is m-connected,
for some m≥ 0,subjectconnectivity!of a continuous map
if the following condition holds: for 0≤ k≤ m 
and all continuous maps α:∂ D^k→ X
and β:D^k→ Y such that β|_∂ D^k=f∘α,
there is a continuous map λ:D^k→ X
such that λ|_∂ D^k=α and
such that f∘λ
is homotopic, relative to ∂ D^k, to β.
An equivalent condition is that the map π_0(f):π_0(X)→π_0(Y) is surjective,
and for all x∈ X the map
π_k(f):π_k(X,x)→π_k(Y,f(x)) is bijective for
all 1≤ k<m and surjective for k=m.
The map f is a weak homotopy equivalence if and only if it is m-connected
for every m≥ 0.



Let G be a compact Lie group.


  * Let f:X→ Y be a continuous based G-map 
between well-pointed based G-spaces, and m≥ 0.
Suppose that for every closed subgroup H of G 
the fixed point map f^H:X^H→ Y^H is (m-(W_G H))-connected.
Then the induced map of G-equivariant stable homotopy groups

    π_k^G(Σ^∞ f)  : π_k^G(Σ^∞ X)  → π_k^G(Σ^∞ Y)

is bijective for every integer k with k<m and surjective for k=m.

  * For every well-pointed based G-space X and every negative integer k,
the G-equivariant homotopy group π_k^G(Σ^∞ X) is trivial. 

In particular, the suspension spectrum functor Σ^∞ takes
G-weak equivalences between well-pointed G-spaces to _*-isomorphisms,
and the unreduced suspension spectrum functor Σ^∞_+ takes
G-weak equivalences between arbitrary unbased G-spaces to _*-isomorphisms.


(i)
We let V be a G-representation and we let n≥ 0 be such that n+k≥ 0. 
Reduced suspension increases the connectivity of continuous maps between
well-pointed spaces, compare <cit.>,
and it preserves well-pointedness.
So the natural homeomorphism 

    (S^V⊕^n X)^H   S^V^H⊕^n X^H

shows that the map

    (S^V⊕^n f)^H   : (S^V⊕^n X)^H  → (S^V⊕^n Y)^H

is ((V^H)+m+n-(W_G H))-connected.
On the other hand, we claim that the cellular dimension of S^V⊕^n+k
at H,
in the sense of <cit.>,
is at most (V^H)+n+k-(W_G H). 
Indeed, if any G-CW-structure for S^V⊕^n+k contains
an equivariant cell of the form G/H× D^j, then 
(G/H)^H×D^j embeds into S^V^H⊕^n+k, and hence

    (W_G H)+ j  =  ((G/H)^H)+ j  ≤ (V^H)+ n+ k  .

The cellular dimension at H is the maximal j that occurs in this way, so 
this cellular dimension is at most (V^H)+n+k-(W_G H). 
We conclude that for k<m the cellular dimension of
S^V⊕^n+k at H is smaller than the connectivity of (S^V⊕^n f)^H,
and for k=m the former is less than or equal to the latter.
So the induced map

    [S^V⊕^n+k,S^V⊕^n f]^G  :  
    [S^V⊕^n+k,S^V⊕^n X]^G  →[S^V⊕^n+k,S^V⊕^n X]^G

is bijective for k<m and surjective for k=m, 
by <cit.>.
Passage to the colimit over V∈ s(_G) and n then proves the claim.  

(ii) The unique map f:X→∗ to a one-point G-space has the property 
that f^H is 0-connected for every closed subgroup H of G. 
So for every negative integer k the induced map from
π_k^G(Σ^∞ X) to π_k^G(Σ^∞∗) is an isomorphism
by part (i). Since the latter group is trivial, this proves the claim.




We end this section with a useful representability result
for the functor π_0^H on the category of orthogonal G-spectra,
where H is any closed subgroup of a compact Lie group G.
We define a tautological classsubjecttautological class!for π_0^H

    e_H ∈ π_0^H(Σ^∞_+ G/H)

as the class represented by the H-fixed point e H of G/H. 
symbole_H - stable tautological class in π_0^H(Σ^∞_+ G/H)

  
Let H be a closed subgroup of a compact Lie group G and 
Φ:G→ (sets) a set valued functor 
on the category of orthogonal G-spectra that takes _*-isomorphisms
to bijections. 
Then evaluation at the tautological class is a bijection

    ^G(π_0^H,Φ) 
     → Φ(Σ^∞_+ G/H)  ,   τ ⟼ τ(e_H) .



To show that the evaluation map is injective 
we show that every natural transformation τ:π_0^H→Φ 
is determined by the element τ(e_H).
We let X be any orthogonal G-spectrum and x∈π_0^H(X)
an H-equivariant homotopy class.
The class x is represented by a continuous based H-map

    f  :  S^U → X(U)

for some H-representation U. By increasing U, if necessary,
we can assume that U is underlying a G-representation.
We can then view f as an H-fixed point 
in (Ω^U^U X)(0)=Ω^U X(U).
There is thus a unique morphism of orthogonal G-spectra

    f^♯ :  Σ^∞_+ G/H  → Ω^U^U X

such that f^♯(0):G/H_+→Ω^U X(U) sends the distinguished coset e H to f. 
This morphism then satisfies

    π_0^H(f^♯)(e_H)  =  π_0^H(λ̃^U_X)(x) 
         in  π_0^H(Ω^U^U X)  ,

where λ̃^U_X:X→Ω^U^U X is the _*-isomorphism
discussed in Proposition <ref> (ii).
The diagram

    @C=20mmπ_0^H(Σ^∞_+ G/H)[d]_τ[r]^-π_0^H(f^♯)   π_0^H(Ω^U^U X)[d]_τ   π_0^H(X)[d]^τ[l]^-_-π_0^H(λ̃^U_X)
    Φ(Σ^∞_+ G/H) [r]_-Φ(f^♯)   Φ(Ω^U^U X) 
        Φ(X)[l]^-Φ(λ̃^U_X)_-

commutes and the two right horizontal maps are bijective.
So

    Φ(f^♯)(τ(e_H))  =  τ(π_0^H(f^♯)(e_H))  =  τ(π_0^H(λ̃^U_X)(x)) 
     =  Φ(λ̃^U_X)(τ(x))  .

Since Φ(λ̃^U_X) is bijective, this proves that τ
is determined by its value on the tautological class e_H.

It remains to construct, for every element 
y∈Φ(Σ^∞_+ G/H), a natural transformation
τ:π_0^H→Φ with τ(e_H)=y.
The previous paragraph dictates what to do: 
we represent a given class x∈π_0^H(X)  by
a continuous based H-map f:S^U→ X(U),
where U is underlying a G-representation.
Then we  set

    τ(x)  =  Φ(λ̃^U_X)^-1(Φ(f^♯)(y))  .

We verify that the element τ(x) only depends on the class x.
To this end we need to show that τ(x) does not change if we either
replace f by a homotopic H-map, or increase it by stabilization along
a G-equivariant linear isometric embedding.
If f̅:S^U→ X(U) is H-equivariantly homotopic to f, then the morphism
f̅^♯ is homotopic to f^♯
via a homotopy of morphisms of orthogonal G-spectra

    K  :  (Σ^∞_+ G/H) [0,1]_+ →Ω^U^U X .

The morphism q:(Σ^∞_+ G/H) [0,1]_+→Σ^∞_+ G/H
that maps [0,1] to a single point is a homotopy equivalence, hence
a _*-isomorphism, of orthogonal G-spectra.
So Φ(q) is a bijection. The two embeddings
i_0,i_1:(Σ^∞_+ G/H)→Σ^∞_+ G/H [0,1]_+
as the endpoints of the interval are right inverse to q,
so Φ(q)∘Φ(i_0)=Φ(q)∘Φ(i_1)=.
Since Φ(q) is bijective, Φ(i_0)=Φ(i_1).
Hence

    Φ(f̅^♯) =  Φ(K∘ i_0) =  Φ(K)∘Φ(i_0) =  Φ(K)∘Φ(i_1) =  Φ(K∘ i_1) =  Φ(f^♯) .

This shows that τ(x) does not change if we modify f by an H-homotopy.

Now we let V be another G-representation and φ:U→ V 
a G-equivariant linear isometric embedding. Then φ_*f:S^V→ X(V)
is another representative of the class x.
A morphism of orthogonal G-spectra

    φ_♯ :  Ω^U^U X  → Ω^V^V X

is defined at an inner product space W as the stabilization map

    φ_* :   (Ω^U^U X)(W)    =  _*(S^U,X(W⊕ U)) 
       → _*(S^V,X(W⊕ V))  =  (Ω^V^V X)(W)  .

This morphism makes the following diagram commute:

    @C=15mm@R=7mmΣ^∞_+ G/H[r]^-f^♯[dr]_(.4) (φ_* f)^♯   Ω^U^U X [d]_φ_♯   
    X[l]_-λ̃^U_X[dl]^(.4)λ̃^V_X
       Ω^V^V X

So

    Φ(λ̃^U_X)^-1∘Φ(f^♯) = Φ(λ̃^V_X)^-1∘Φ(φ_♯)∘Φ(f^♯) 
     =  Φ(λ̃^V_X)^-1∘Φ( (φ_* f)^♯)  ,

and hence the class τ(x) remains unchanged upon stabilization of f
along φ.

Now we know that τ(x) is independent of the choice of representative
for the class x, and it remains to show that τ is natural.
But this is straightforward: if ψ:X→ Y is a morphism of orthogonal G-spectra
and f:S^U→ X(U) a representative for x∈π_0^H(X) as above,
then ψ(U)∘ f:S^U→ Y(U) represents the class π_0^H(ψ)(x).
Moreover, the following diagram of orthogonal G-spectra commutes:

    @C=15mmΣ^∞_+ G/H[r]^-f^♯[dr]_(.4) (ψ(U)∘ f)^♯   Ω^U^U X [d]^Ω^U^U ψ   
    X[l]_-λ̃^U_X[d]^ψ
       Ω^U^U Y     Y [l]^-λ̃^U_Y

So naturality follows:

    τ(π_0^H(ψ)(x))    =  
    (Φ(λ̃^U_Y)^-1∘Φ( (ψ(U)∘ f)^♯))(y) 
       =  (Φ(λ̃^U_Y)^-1∘Φ(Ω^U^U ψ)∘Φ(f^♯))(y) 
       =  (Φ(ψ)∘Φ(λ̃^U_X)^-1∘Φ(f^♯))(y) 
     =  Φ(ψ)(τ(x))  .

Finally, the class e_H is represented by the H-map - e H:S^0→ S^0 G/H_+,
which is adjoint to the identity of Σ^∞_+ G/H. Hence
τ(e_H)= Φ()(y) =y.






§ THE WIRTHMÜLLER ISOMORPHISM AND TRANSFERS




This section establishes the Wirthmüller isomorphism
that relates the equivariant homotopy groups 
of a spectrum over a subgroup to the equivariant homotopy groups 
of the induced spectrum, see Theorem <ref>.
Intimately related to the Wirthmüller isomorphism are various transfers 
that we discuss in Constructions <ref> and <ref>.
We show that transfers are transitive (Proposition <ref>)
and commute with inflations (Proposition <ref>).
Theorem <ref> below establishes the `double coset formula'
that expresses the composite of a transfer followed by a restriction 
as a linear combination of restrictions followed by transfers. 





We let H be a closed subgroup of a compact Lie group G. 
We write i^*:G_*→ H_* for the restriction functor
from based G-spaces to based H-spaces.
We write

    G⋉_H -  =  (G_+)_H -  :  H_* → G_*

for the left adjoint induction functor.
For a based H-space A and a based G-space B,
the shearing isomorphismsubjectshearing isomorphism
is the G-equivariant homeomorphism

    B (G⋉_H A)   G⋉_H(i^*B A)  ,   
    b [g, a] ⟼ [g, (g^-1 b) a] .







As before we consider a closed subgroup H of a compact Lie group G.
To define the Wirthmüller morphism, 
we need a specific H-equivariant map

    l_A  :  G⋉_H A  → A S^L

that is natural for continuous based H-maps in A, see (<ref>) below.
Here L=T_e H(G/H) is the tangent H-representation,
i.e., the tangent space at the preferred coset e H of the smooth
manifold G/H. Since the left translation by H on G/H fixes e H,
the tangent space L inherits an H-action, and S^L 
is its one-point compactification.
When H has finite index in G, then L is trivial
and the map l_A:G⋉_H A → A
is simply the projection onto the wedge summand indexed by the preferred H-coset e H.
If the dimension of G is bigger than the dimension of H,
then L is non-zero and the definition of the map l_A is substantially more involved.

We consider the left H^2-action on G given by 

    H^2× G  → G  ,    (h',h)· g  =  h' g h^-1 .

There is another action of the group Σ_2 on G by inversion,
i.e., the non-identity element of Σ_2 acts by τ· g= g^-1.
These two actions combine into an action of the wreath product group
Σ_2≀ H=Σ_2⋉ H^2 on G.
The subgroup H is the (Σ_2⋉ H^2)-orbit of 1∈ G, whose stabilizer
is the subgroup Σ_2×Δ, where Δ= { (h,h)  |  h∈ H}
is the diagonal subgroup of H^2.
The differential of the projection G→ G/H identifies

    ν =  ( T_1 G) / ( T_1 H)  ,

the normal space at 1∈ H of the inclusion H→ G, 
with the tangent representation L =T_e H (G/H).
The representation ν is a representation 
of the stabilizer group Σ_2×Δ of 1, 
and if we identify H with Δ via h↦(h,h), 
then this identification takes the Δ-action on ν  
to the tangent H-action on L. 
Moreover, the differential at 1 of the inversion map g↦ g^-1
is multiplication by -1 on the tangent space; 
so the above identification of ν with L takes the Σ_2-action
to the sign action on L.

We choose an H-invariant inner product on the vector space L.
Since H is the H^2-orbit of 1 inside G, there is a slice,
i.e., a Σ_2×Δ-equivariant smooth embedding

    s :  D( L )  → G

of the unit disc of L with s(0)=1 and such that 
the differential at 0∈ D(L) of the composite

    D(L)  G   G/H

is the identity of L.
The property that s is (Σ_2×Δ)-equivariant means in more concrete terms
that the relations

    s(h· l) =  h· s(l)· h^-1     and    
     s(- l) =   s(l)^-1

hold in G for all (h,l)∈ H× D(L).
After scaling the slice, if necessary, the map

    D(L)× H  → G  ,    (l,h) ⟼ s(l)· h

is an embedding whose image is a tubular neighborhood of H in G.
For a proof, see for example <cit.>.
This embedding is equivariant for the action of H^2 on the source by

    (h_1,h_2)· (l, h) =  (h_1 l,  h_1  h h_2^-1 )

and for the action of Σ_2 on the source by

    τ· (l, h) =  (-h^-1 l,  h^-1 )   .

The map 

    l_H^G  :  G  → S^L  H_+

is then defined as the H^2-equivariant collapse map 
with respect to the above tubular neighborhood.subjectThom-Pontryagin construction
So explicitly,

    l_H^G(g)  =  
     ( l / (1-|l|) )  h     if g=s(l)· h with (l,h)∈ D(L)× H, and 
        ∗    if g is not of this form.

Given any based H-space A, we can now form l_H A,
where -_H - refers to the action of the second H-factor in H^2.
We obtain an H-equivariant based map

    l_A  =  l_H^G_H A  :  G⋉_H A → 
    (S^L H_+)_H A   A S^L

that is natural in A. 
Here H acts by left translation on the source, and diagonally on the target.
This map is thus given by

    l_A[g,a]  =  
     h a ( l / (1-|l|) )      if g=s(l)· h with (l,h)∈ D(L)× H, and 
      ∗    if g is not of this form.





If H has finite index in G, then L=0 and the triangle 
of the following proposition commutes on the nose, by direct inspection.
So the essential content of the next result is when the dimension of G
exceeds the dimension of H.  


Let H be a closed subgroup of a compact Lie group G, 
A a based H-space and B a based G-space.
Then the following triangle commutes up to H-equivariant based homotopy:

    @C=18mm B (G⋉_H A) [d]^_ shear[dr]^-B l_A   
    
     G⋉_H(i^*B A) [r]_-l_i^* B A    B A S^L



We write down an explicit homotopy: we define the map

    K  :  ( B (G⋉_H A) )× [0,1] → B A S^L

by

    K(b [g,a], t)  =  
      s(t l)^-1 b h al/1-|l|    if g=s(l)· h for (l,h)∈ D(L)× H,
      ∗    if g is not of this form.

Then for all (l,h)∈ D(L)× H with g=s(l)· h we have

    K(b [g, a],0) =  
       b h a (l/(1-|l|))  =  (B l_A)(b [g, a])

(because s(0)=1), and

    K(b [g, a],1) =  
      h g^-1 b h a (l/(1-|l|)  =  l_i^*B A[g, g^-1b  a])

(because s(l)^-1=h g^-1). So K is the desired H-equivariant homotopy.


The restriction functor from orthogonal G-spectra to orthogonal H-spectra 
has a left and a right adjoint, and both are given by
applying the space level adjoints G⋉_H - respectively ^H(G,-) levelwise.
We will mostly be concerned with the left adjoint, and so we spell out
the construction in more detail.

[Induced spectrum]
We let H be a closed subgroup of a compact Lie group G 
and Y an orthogonal H-spectrum. 
We denote by G⋉_H Ysubjectinduced spectrumsymbolG⋉_H Y - induced spectrum 
the induced G-spectrum whose value at an inner product space V is
G⋉_H Y(V).
When G-acts on V by linear isometries, 
then this value has the diagonal G-action, 
through the action on the external G and by functoriality in V.
With this diagonal G-action, (G⋉_H Y)(V) 
is equivariantly homeomorphic to G⋉_H Y(i^* V) 
(where H acts diagonally on Y(i^* V)), via

    G⋉_H Y(i^* V)  (G⋉_H Y)(V) ,   
    [g,y]  ⟼ [g, Y(l_g)(y)] .

Under this isomorphism the structure map of the spectrum G⋉_H Y
becomes the combination of the shearing isomorphism
and the structure map of the H-spectrum Y, 
i.e., for every G-representation W the following square commutes:

    @C13mm  S^V  ( G⋉_H Y(i^* W)) 
    [r]^-(<ref>)_-[d]_shear^   
    S^V (G⋉_H Y)(W) [dd]^σ_V,W^G⋉_H Y
    
    G⋉_H ( S^i^* V Y(i^* W)) [d]_G⋉_H (σ^Y_V,W)   
    
    G⋉_H ( Y(i^* V⊕ i^* W )) [r]_-(<ref>)^-    
    (G⋉_H Y)(V⊕ W)




If the group H has finite index in G, then 
the tangent representation L=T_e H(G/H) is trivial and
l_A:G⋉_H A → A is the projection onto the wedge summand indexed by e H;
the maps l_Y(V):G⋉_H Y(V)→ Y(V) then
form a morphism of orthogonal H-spectra l_Y:G⋉_H Y→ Y.
This morphism induces a map of H-equivariant homotopy groups.
In general, however, the diagram

    @C=18mm 
    S^V (G⋉_H Y(W))[r]^-S^V l_Y(W)@<-8ex>@/_2pc/[dd]_(.7)σ_V,W^G⋉_H Y[d]^shear_    
    S^V Y(W) S^L[dd]^σ_V,W^Y S^L
    
    G⋉_H (S^V Y(W))[ur]_l_S^V Y(W)[d]^G⋉_H σ_V,W^Y   
    
    G⋉_H Y(V⊕ W)[r]_-l_Y(V⊕ W)    Y(V⊕ W) S^L

does not commute on the nose (because the upper triangle does not commute);
so if the dimension of G is bigger than the dimension of H we do not
obtain a morphism of orthogonal H-spectra in the strict sense.
Still, Proposition <ref> shows 
that the above diagram does commute up to based H-equivariant homotopy,
and this is good enough to yield a well-defined homomorphism 

    (l_Y)_*  :  π^H_k(G⋉_H Y)  → π_k^H(Y S^L)  .

As we just explained, this is an abuse of notation, since (l_Y)_*
is in general not induced by a morphism of orthogonal H-spectra.

We can now consider the composite  

    _H^G  :  π^G_k(G⋉_H Y)   ^G_H π^H_k(G⋉_H Y)   (l_Y)_*  π_k^H(Y S^L)  ,

which we call the Wirthmüller map.subjectWirthmüller map
We will show in Theorem <ref> below that the Wirthmüller map is
an isomorphism.




[Transfers]
We let H be a closed subgroup of a compact Lie group G,
and we let Y be an orthogonal H-spectrum.
The external transfersubjecttransfer!external 

    symbolG⋉_H - - external transfer
     G⋉_H -  :  π_0^H(Y S^L)  → π_0^G(G⋉_H Y)

is a map in the direction opposite to the Wirthmüller map.
The construction involves another equivariant Thom-Pontryagin construction.subjectThom-Pontryagin construction
We choose a G-representation V and a vector v_0∈ V 
whose stabilizer group is H;
this is possible, for example by <cit.>,
<cit.>
or <cit.>.
This data determines a G-equivariant smooth embedding

    i :  G/H  → V  ,    g H  ⟼ g v_0

whose image is the orbit G v_0. We let

    W  =  V - (d i)_e H( L )  =  V - T_v_0( G· v_0)

denote the orthogonal complement of the image of the tangent space at e H;
this is an H-subrepresentation of V, and

    L⊕ W   V ,   (x,w) ⟼ (d i)_e H(x)  +  w

is an isomorphism of H-representations.

By multiplying the original vector v_0 with a sufficiently large scalar,
if necessary, we can assume that the embedding i is `wide', 
in the sense that the exponential map

    j  :  G×_H D(W)  → V  ,    
    [g , w]  ⟼ g· ( v_0  + w )

is an embedding, where D(W) is the unit disc of the normal H-representation,
compare <cit.>.
So j defines an equivariant tubular neighborhood of
the orbit G v_0 inside V.
The associated collapse map 

    c  :   S^V  → G⋉_H S^W

then becomes the G-map defined by 

    c(v)  = [g, w/1-|w|]    if v= g·(v_0 +w) for some (g,w)∈ G× D(W), and
      ∞   else.

With the collapse map in place, we can now define 
the external transfer (<ref>).
We let U be an H-representation and f:S^U → Y(U) S^L an H-equivariant 
based map that represents a class in π_0^H(Y S^L).
By enlarging U, if necessary, we can assume that it
is underlying a G-representation, 
see for example <cit.>
or <cit.>.
We stabilize f by W from the right to obtain the H-map

    f♢ W  :  
     S^U S^W  f S^W    Y(U) S^L S^W
        _(<ref>) 
     Y(U) S^V σ^_U,V  Y(U⊕ V)  .

The composite G-map

    S^U⊕ V  S^U c   
    S^U (G⋉_H S^W)     []shear  G⋉_H(S^U  S^W) 
    G⋉_H( f♢ W)    G⋉_H ( Y(U⊕ V)) _(<ref>)  (G⋉_H Y)(U⊕ V)

then represents the external transfer

    G⋉_Hf     in   π_0^G(G⋉_H Y) .




The next proposition provides the main geometric input for
the Wirthmüller isomorphism.


Let H be a closed subgroup of a compact Lie group G.


  * The composite

    S^V  c  G⋉_H S^W  l_H^G_H S^W  S^L S^W

is H-equivariantly homotopic to the map induced by
the inverse of the isometry (<ref>).

  * Let A be a based H-space and f,f':B→ G⋉_H A
two continuous based G-maps. If the composites
l_A∘ f, l_A∘ f':B→ A S^L are H-equivariantly homotopic,
then the maps f S^V,f' S^V:B S^V→ (G⋉_H A) S^V 
are G-equivariantly homotopic.



(i)
The composite (l_H^G_H S^W)∘ c
is the collapse map based on the composite closed embedding

    ζ :   D(L)× D(W)  (l,w)↦ [s(l), w]  G×_H D(W) [g,w]↦ g·(v_0+w) V  .

Then ζ(0,0)=v_0 and we let 

    D =  (d ζ)_(0,0) :  L× W → V

denote the differential of ζ at (0,0). We observe that
D(0,w) =  w because ζ(0,w)=v_0+w; 
on the other hand, ζ(l,0)=s(l)· v_0,
so the restriction of D to L is the differential at 0 of the composite

    D(L)  s  G  proj G/H   i   V  .

Since the differential of the composite proj∘ s
is the identity, we obtain 

    D(l,0)  =  (d i)_e H(l)  .

Since the differential is additive we conclude that
D(l,w)=(d i)_e H(l)+w, i.e., D equals the isomorphism (<ref>).

We consider the H-equivariant homotopy

    K  :  D(L)× D(W)× [0,1]    →   V 
    
    K(l,w,t)     =ζ(t· (l,w))- v_0/t +  t· v_0     for t>0, and
        D(l,w)     for t=0.

That this assignment is continuous (in fact smooth) when t approaches 0
is the defining property of the differential.  
Moreover, for every t∈ [0,1] the map K(-,-,t):D(L)× D(W)→ V
is a smooth equivariant embedding, 
so it gives rise to a collapse map c_t:S^V→ S^L S^W defined by

    c_t(v)  = ( l/1-|l|,w/1-|w|)    if v= K(l,w,t) for some (l,w)∈ D(L)× D(W), and
        ∞   else.

The passage from the embedding to the collapse map is continuous,
so the 1-parameter family c_t provides an H-equivariant
based homotopy from the collapse map c_0
to the collapse map c_1
associated to the embedding ζ=K(-,-,1):D(L)× D(W)→ V,
i.e., to the map (l_H^G_H S^W)∘ c.
Since D=(d ζ)_(0,0) is the isometry (<ref>),
another scaling homotopy compares the collapse map c_0 to the one-point
compactification of the inverse of (<ref>).

(ii) We define a based continuous G-map

    r  :  ^H(G,A S^L) S^V → G⋉_H( A S^L S^W)

as the composite

    ^H(G,A S^L) S^V      c  ^H(G,A S^L) (G⋉_H S^W)
        shear  
     G⋉_H( ^H(G,A S^L) S^W)
       G⋉_H(ϵ S^W) 
     G⋉_H( A S^L S^W)  .

Here ϵ:^H(G,A S^L)→ A S^L 
is the adjunction counit, i.e., evaluation at 1∈ G.
We expand this definition. We let ψ:G→ A S^L be an H-equivariant
based map and v∈ S^V. If v is not in the image of the tubular
neighborhood j:G×_H D(W)→ V, then r(ψ v) is the basepoint.
Otherwise, v=j[g,w]=g·(v_0+w) for some (g,w)∈ G× D(W), and then

    r(ψ v)     =  ((G⋉_H (ϵ S^W))∘shear)( ψ [g,w/(1-|w|)]) 
       =  [g, ϵ (g^-1·ψ)  w/(1-|w|)]  
    =  [g, ψ(g^-1)w/1-|w|]  .

We denote by l_A^♯: A→^H(G,A S^L)
the H-map defined by l_A^♯(a)(g)=l_A[g,a],
and we let φ:V→ L⊕ W be the inverse of the isometry (<ref>).
Now we argue that the following square commutes up to H-equivariant based
homotopy:

    @C=10mm A S^V [r]^-[1,-][dd]_l_A^♯ S^V    
     G⋉_H(A S^V)[d]_^G⋉_H S^φ
       G⋉_H (A S^L S^W) [d]_^G⋉_H(S^-_L S^W)
    ^H(G,A S^L) S^V[r]_-r    G⋉_H (A S^L S^W)

To see this we define an H-homotopy

    K :   (A S^V) × [0,1] → G⋉_H(A S^L S^W)

as follows. We exploit that the map

    ζ :   D(L)× D(W)  → V  ,  ζ(l,w) =  s(l)·(v_0 + w )

is a smooth embedding.
This map already featured in the proof of part (i),
because the collapse map based on ζ is the composite (l_H^G_H S^W)∘ c.
If a vector is of the form v=ζ(l,w)=s(l)· (v_0+w)
for some (l,w)∈ D(L)× D(W) (necessarily unique), then we set

    K( a v,t)  = 
     K( a (s(l)·(v_0 +w)), t)  = [ s(t· l),  a-l/1-|l|w/1-|w|]  .

For |l|=1 or |w|=1 this formula yields the basepoint,
so we can extend this definition by sending all elements a v
to the basepoint whenever v is not in the image of the embedding ζ.
We claim that for t=0 we obtain K(-,1)=r∘(l_A^♯ S^V).
Indeed, if v=s(l)·(v_0+w) for (l,w)∈ D(L)× D(W)
(necessarily unique), then 

    ( r∘(l_A^♯ S^V))(a v)   
    _(<ref>)   = [s(l), l_A^♯(a)(s(l)^-1)w/1-|w|] 
       =  [s(l),l_A[s(-l),a]w/1-|w|]
    
       =  [s(l), a-l/1-|l|w/1-|w|]
     =  K(a v,1) .

On the other hand, the map K(-,0) agrees with the composite

    A S^V     A ((l_H^G_H S^W)∘ c)  
    A S^L S^W 
        [1,-]  G⋉_H( A S^L S^W)  
     G⋉_H(S^-_L S^W) G⋉_H( A S^L S^W)   ;

part (i) provides an H-homotopy between (l_H^G_H S^W)∘ c and
the homeomorphism S^φ; so this proves the claim.

We let Λ:G⋉_H(A S^V)→^H(G,A S^L) S^V
be the G-equivariant extension of the H-map l_A^♯ S^V.
Since the square (<ref>) commutes up to H-equivariant homotopy,
the composite

    G⋉_H(A S^V)  Λ  ^H(G,A S^L) S^V  r  
    G⋉_H(A S^L S^W)

is G-equivariantly homotopic to the G-homeomorphism 
G⋉_H ((S^-_L S^W)∘ S^φ), by adjointness.
So the composite of r Λ with the shearing homeomorphism
(G⋉_H A) S^V  G⋉_H(A S^V)
is also G-homotopic to a homeomorphism.
This composite equals
the composite

    ( G⋉_H A) S^V Ψ_A S^V ^H(G,A S^L) S^V  r  
    G⋉_H(A S^L S^W)   ,

where Ψ_A: G⋉_H A→^H(G,A S^L)
is the adjoint of the H-map l_A:G⋉_H A→ A S^L.

Now we consider based G-maps f,f':B→ G⋉_H A such that
l_A∘ f and l_A∘ f':B→ A S^L are H-equivariantly homotopic.
Then the two composites

    B  f, f'   G⋉_H A  Ψ_A ^H(G, A S^L)

are G-equivariantly homotopic, by adjointness.
The composite (Ψ_A∘ f) S^V=(Ψ_A S^V)∘(f S^V)
is then G-equivariantly homotopic 
to (Ψ_A∘ f') S^V=(Ψ_A S^V)∘(f' S^V).
The map Ψ_A S^V has a G-equivariant retraction,
up to G-homotopy, by the previous paragraph.
So already the maps f S^V and f' S^V are G-homotopic.




Now we can establish the Wirthmüller isomorphism,
which first appeared in <cit.>.
Wirthmüller attributes parts of the ideas to tom Dieck,
and his statement that G-spectra define a `complete G-homology theory'
amounts to Theorem <ref>.
Our proof is essentially Wirthmüller's original argument,
adapted to the context of equivariant orthogonal spectra.
We recall that

    ε_L :  π_0^H(Y S^L)  → π_0^H(Y S^L)

denotes the effect of the involution of Y S^L
induced by the linear isometry -_L:L→ L given by multiplication by -1.

subjectWirthmüller isomorphism
Let H be a closed subgroup of a compact Lie group G
and Y an orthogonal H-spectrum.
Let L=T_e H(G/H) denote the tangent H-representation.
Then the maps

    _H^G  :  π^G_0(G⋉_H Y)  → π^H_0(Y S^L)

and

    (G⋉_H -)∘ε_L  :  π^H_0(Y S^L)  → π^G_0(G⋉_H Y)

are independent of the choices made in their definitions, 
and they are mutually inverse isomorphisms.


We prove the various claims in a specific order.
In a first step we show that the Wirthmüller map
is left inverse to the map (G⋉_H -)∘ε_L,
independently of all the choices made in the definitions.
We let U be a G-representation and f:S^U → Y(U) S^L an H-equivariant 
based map that represents a class in π_0^H(Y S^L).
We also choose a wide G-equivariant embedding
i: G/H → V as in Construction <ref>.
This provides a decomposition L⊕ W V 
of H-representations as in (<ref>)
and a G-equivariant Thom-Pontryagin collapse map c:S^V→ G⋉_H S^W.
We let φ:V L⊕ W denote the inverse 
of the linear isomorphism (<ref>).
We contemplate the diagram of based H-maps:

    @C=20mm
    S^U S^V @/^1pc/[dr]^(.6)S^U  S^φ@<-6ex>@/_3pc/[ddd]
    [d]_S^U c
    
    S^U (G⋉_H S^W)  [r]_-S^U( l_H^G_H S^W)[d]_shear^    S^U S^L S^W [d]^S^Uτ_L,W
    
     G⋉_H ( S^U S^W) [d]^G⋉_H(f♢ W)[r]_-l_S^U S^W   
     S^U S^W S^L [d]^(f♢ W) S^L
    
      G⋉_H Y(U⊕ V )[r]_-l_Y(U⊕ V)    Y(U⊕ V) S^L

The left vertical composite represents the external transfer G⋉_Hf,
so the composite around the lower left corner represents
(l_Y)_*(^G_H(G⋉_Hf)).
The upper triangle commutes up to H-equivariant homotopy by
Proposition <ref> (i).
The middle square commutes up to H-homotopy by
Proposition <ref> and the fact that l_A:G⋉_H A→ A S^L
is defined as the composite of (l_H^G_H A) and the twist isomorphism
S^L A A S^L.
The lower square commutes by naturality of the maps (<ref>).
Upon expanding the definition (<ref>) 
of f♢ W, the diagram shows that 
the class (l_Y)_*(^G_H(G⋉_Hf)) is also represented by the composite

    S^U⊕ V     f S^φ Y(U) S^L S^L S^W  Y(U) S^Lτ_L,W Y(U) S^L S^W S^L 
        Y(U) S^φ^-1 S^L Y(U) S^V S^L σ^_U,V S^L Y(U⊕ V)  S^L  .

The isometry 

    (φ^-1⊕ L)∘(L⊕τ_L,W)∘(L⊕φ)  :  
    L⊕ V → V⊕ L

is not the twist isometry τ_L,V;
however (φ^-1⊕ L)∘(L⊕τ_L,W)∘(L⊕φ)
is equivariantly homotopic to the composite

    L⊕ V τ_L,V V⊕ L  V⊕(-_L) V⊕ L .

Hence the class (l_Y)_*(^G_H(G⋉_Hf)) is also represented by the composite

    S^U⊕ V  f S^V Y(U) S^L S^V  Y(U)τ_L,V Y(U) S^V S^L σ^_U,V S^- Y(U⊕ V)  S^L  .

So altogether this shows the desired relation

    _H^G(G⋉_Hf)  =  
    (l_Y)_*(^G_H(G⋉_Hf)) =  ε_Lf .

In particular, the Wirthmüller map is surjective.

Now we show that the Wirthmüller map is injective.
We let f,f':S^U→ G⋉_H Y(U) be two G-maps that represent
classes with the same image under the Wirthmüller map
_H^G :π^G_0(G⋉_H Y) →π^H_0(Y S^L).
By increasing the G-representation U, if necessary, we can assume
that the composites

    S^U  f, f'   G⋉_H Y(i^* U)l_Y(i^* U) Y(i^* U) S^L

are H-equivariantly homotopic.
Then by Proposition <ref> (ii) the two maps

    S^U⊕ V  f S^V, f' S^V   ( G⋉_H Y(i^* U)) S^V 
      ( G⋉_H Y)(U) S^V

are G-equivariantly homotopic.
The maps remain G-homotopic if we furthermore postcompose
with the opposite structure 
map σ^_U,V:(G⋉_H Y)(U) S^V→(G⋉_H Y)(U⊕ V).
This shows that the stabilizations from the right of f and f' by V
become G-homotopic. 
Since such a stabilization represents the same class 
in π_0^G(G⋉_H Y) as the original map, this shows that f=f',
i.e., the Wirthmüller map is injective.

Now we know that the Wirthmüller map and the map (G⋉_H -)∘ε_L
are inverse to each other, no matter which choices of slice s:D(L)→ G, 
representation V and G-embedding G/H→ V we made.
Since the choices for the Wirthmüller map and the choices for
the external transfer are independent of each other, the two resulting maps
are independent of all choices.

The last thing to show is that the Wirthmüller map and the external
transfer are additive maps. If H has finite index in G, the Wirthmüller
map is the composite of a restriction homomorphism and the effect of
a morphism of orthogonal H-spectra, both of which are additive.
In general, however, we need an additional argument, namely naturality.
Indeed, for a fixed choice of slice s:D(L)→ G, the Wirthmüller
map π^G_0(G⋉_H Y) →π^H_0(Y S^L) is natural in the orthogonal
H-spectrum Y.
Since source and target are reduced additive functors
from orthogonal H-spectra to abelian groups, any natural transformation
is automatically additive, by Proposition <ref>.
Since the Wirthmüller map is additive, so is its inverse, and
hence also the external transfer.



Theorem <ref> states the Wirthmüller isomorphism
only for 0-dimensional homotopy groups.
We now extend it to homotopy groups in all integer dimensions.
This extension is a rather formal consequence of the fact that 
the Wirthmüller maps
commute with the loopsubjectloop isomorphism 
and suspension isomorphismssubjectsuspension isomorphism

    α :  π^G_k(Ω X) → π^G_k+1 (X)     and  
     - S^1  :  π^G_k (X) → π^G_k+1(X S^1)

defined in (<ref>) respectively (<ref>).


Let H be a closed subgroup of a compact Lie group G and Y
an orthogonal H-spectrum. 


  * The following diagrams commute for
all integers k:

    @C=15mm@R=8mmπ_k^G(G⋉_H (Ω Y))  [d]_ assembly^[r]^-_H^G   π_k^H( (Ω Y) S^L)[d]_^ assembly
    π_k^G(Ω(G⋉_H Y))[d]^_α   π_k^H(Ω(Y S^L)) [d]^α_
    π_k+1^G(G⋉_H Y) [r]_-_H^G   π_k+1^H(Y S^L)


    @C=15mm@R=8mmπ_k^G(G⋉_H Y)  [d]_- S^1^[r]^-_H^G   π_k^H(Y S^L) [d]_^- S^1
    π_k+1^G((G⋉_H Y) S^1)[d]_b_*^   π_k+1^H(Y S^L S^1) [d]^(Yτ_S^L,S^1)_*_-
    π_k+1^G(G⋉_H (Y S^1)) [r]_-_H^G   π_k+1^H(Y S^1 S^L)

Here τ_S^L,S^1:S^L S^1→ S^1 S^L is the twist homeomorphism,
and the G-equivariant isomorphism

    b  :  (G⋉_H Y) S^1  G⋉_H (Y S^1)

is given by b([g,y] t)=[g, y t].

  * The Wirthmüller map

    _H^G  :  π_k^G(G⋉_H Y)  → π_k^H(Y S^L)

is an isomorphism for all k∈.



(i) The loop and suspension isomorphisms commute with restriction from G to H, 
by direct inspection. So the proof comes down
to checking that the following diagrams commute:

    @C=15mm@R=8mmπ_k^H(G⋉_H (Ω Y))  [d]_assembly^[r]^-(l_Ω Y)_*   π_k^H( (Ω Y) S^L)[d]_^assembly
    π_k^H(Ω(G⋉_H Y))[d]^_α   π_k^H(Ω(Y S^L)) [d]^α_
    π_k+1^H(G⋉_H Y) [r]_-(l_Y)_*   π_k+1^H(Y S^L) 
    
    π_k^H(G⋉_H Y)  [d]_- S^1^[r]^-(l_Y)_*   π_k^H(Y S^L) [d]_^- S^1
    π_k+1^H((G⋉_H Y) S^1)[d]_b_*^   π_k+1^H(Y S^L S^1) [d]^(Yτ_S^L,S^1)_*_-
    π_k+1^H(G⋉_H (Y S^1)) [r]_-(l_Y S^1)_*   π_k+1^H(Y S^1 S^L)

This in turn follows from the fact – again verified by direct inspection – 
that for every based H-space A the following two squares commute:

    G⋉_H(Ω A) [r]^-l_Ω A[d]_assembly    
    (Ω A) S^L[d]^assembly   
    (G⋉_H A) S^1 [r]^-l_A S^1[d]_b     
    A S^L S^1[d]^Aτ_S^L.S^1
    Ω( G⋉_H A) [r]_-Ω l_A   Ω(A S^L)
       
    G⋉_H (A S^1) [r]_-l_A S^1   
    A S^1 S^L


(ii) We argue by induction over |k|, the absolute value of the integer k.
The induction starts with k=0, where Theorem <ref> 
provides the desired conclusion.
If k is positive, the compatibility of the Wirthmüller map with the loop isomorphism,
established in part (i), provides the inductive step.
If k is negative, 
the compatibility of the Wirthmüller map with the suspension isomorphism,
also established in part (i), provides the inductive step.



[External transfer in integer degrees]
So far we only defined the external transfer for 0-dimensional homotopy groups.
Theorem <ref> shows that in dimension 0
the external transfer is inverse to the map
ε_L∘_H^G:π^G_0(G⋉_H Y) →π^H_0(Y S^L).
We want the same property in all dimensions, and since the
Wirthmüller map is an isomorphism by Proposition <ref>
we simply define the external transfer isomorphism

    G⋉_H -  :  π^H_k(Y S^L)  → π^G_k(G⋉_H Y)

as the composite

    π^H_k(Y S^L) 
      ε_L  π^H_k(Y S^L)   (_H^G)^-1 π^G_k(G⋉_H Y)  .

The compatibility of the  Wirthmüller isomorphism with the 
loopsubjectloop isomorphism and suspension isomorphismssubjectsuspension isomorphism
then directly implies the analogous compatibility for the
external transfer G⋉_H -, by reading the diagrams of
Proposition <ref> (i) backwards.
For easier reference, we explicitly record this compatibility
in Proposition <ref> below.




In the next proposition we use the Wirthmüller isomorphism
to show that smashing with a cofibrant G-space is homotopical.
In <cit.>, Mandell and May give
a different proof of this fact which does not use 
the Wirthmüller isomorphism.




Let G be a compact Lie group and A a cofibrant based G-space. 


  * Let f:X→ Y be a morphism of orthogonal G-spectra with the
following property: for every closed subgroup H of G that 
fixes some non-basepoint of A,
the map π_*^H(f):π_*^H(X)→π_*^H(Y) is an isomorphism.
Then the morphism f A:X A→ Y A is a
_*-isomorphism of orthogonal G-spectra.

  * The functor - A preserves _*-isomorphisms of orthogonal G-spectra.



(i) Smashing with A commutes with mapping cones, so by the long exact
homotopy group sequence of Proposition <ref>
it suffices to show the following special case. Let X be an orthogonal G-spectrum
with the following property: for every closed subgroup H of G that occurs
as the stabilizer group of a non-basepoint of A,
the groups π_*^H(X) vanish.
Then for all closed subgroups K of G 
the equivariant homotopy groups π_*^K(X A) vanish.

In a first step we show this when A is a finite-dimensional G-CW-complex.
We argue by contradiction. If the claim were false,
we could find a compact Lie group G of minimal dimension for which it fails.
We let A be a G-CW-complex whose dimension n is minimal among all counterexamples.
Then A can be obtained from an (n-1)-dimensional subcomplex B
by attaching equivariant cells G/H_i× D^n, 
for i in some indexing set I, where H_i is a closed subgroup of G.
Then X (A/B) is isomorphic to a wedge, over the set I,
of orthogonal G-spectra X (G/H_i)_+ S^n.
Since equivariant homotopy groups take wedges to sums, the 
suspension isomorphism and the Wirthmüller
isomorphism allow us to rewrite the equivariant homotopy groups 
of X A/B as

    π_*^G(X A/B)     ⊕_i∈ I π_*^G(X (G/H_i)_+ S^n)
       ⊕_i∈ I π_*-n^G(X (G/H_i)_+)⊕_i∈ I π_*-n^H_i(X S^L_i) ,

where L_i is the tangent representation of H_i in G.
Since H_i is the stabilizer of a non-basepoint of A,
the groups π_*^K(X) vanish for all closed subgroups K of H_i, by hypothesis.
If H_i has finite index in G, then L_i=0 and the 
respective summand thus vanishes.
The representation sphere S^L_i admits a finite H_i-CW-structure,
so if H_i has strictly smaller dimension than G, then 
the respective summand vanishes by the minimality of G.
So altogether we conclude that the groups 
π_*^G(X A/B) vanish. The groups π_*^G(X B)
vanish by the minimality of A. The inclusion of B into A is
an h-cofibration of based G-spaces, so the induced morphism
X B→ X A is an h-cofibration of orthogonal G-spectra.
Hence the groups π_*^G(X A) vanish by the long exact sequence of
Corollary <ref> (i).
Since A was supposed to be a counterexample to the proposition,
we have reached the desired contradiction.
Altogether this proves the claim when A admits the structure of a finite-dimensional
G-CW-complex.

If A admits the structure of a G-CW-complex, possibly infinite
dimensional, we choose a skeleton filtration by G-subspaces A_n.
Then the G-homotopy groups of X A_n vanish for all n≥ 0,
and all the morphisms X A_n→ X A_n+1 are h-cofibrations
of orthogonal G-spectra. Since equivariant
homotopy groups commute with such sequential colimits
(see Proposition <ref> (i)), 
the groups π_*^G(X A) vanish as well.
An arbitrary cofibrant based G-space is G-homotopy equivalent
to a based G-CW-complex, so 
the groups π_*^G(X A) vanish for all cofibrant A.

Now we let K be an arbitrary closed subgroup of G. 
The underlying K-space of A is again cofibrant
by Proposition <ref> (i),
so we can apply the previous reasoning to K instead of G and
conclude that the groups π_*^K(X A) vanish.





Let G and K be compact Lie groups and A a cofibrant based (G× K)-space
such that the G-action is free away from the basepoint
and the K-action is free away from the basepoint.
Then the functor

    A_K -  :  K → G

takes _*-isomorphisms of orthogonal K-spectra to
_*-isomorphisms of orthogonal G-spectra.


The functor A_K- preserves mapping cones, so by the long exact
homotopy group sequence of Proposition <ref>
it suffices to show the following special case. Let C be an orthogonal K-spectrum
all of whose equivariant homotopy groups vanish, for all closed subgroups of K.
Then all equivariant homotopy groups of the orthogonal G-spectrum A_K C
vanish, for all closed subgroups of G.

In a first step we show that the G-equivariant stable homotopy groups
of A_K C vanish.
A cofibrant based (G× K)-space is equivariantly homotopy
equivalent to a based (G× K)-CW-complex, so it is no loss of generality
to assume an equivariant CW-structure with skeleton filtration

    ∗ =  A_-1 ⊂ A_0  ⊂ … ⊂ A_n ⊂ … .

We show first, by induction on n, that the orthogonal G-spectrum
(A_n)_K C has trivial G-equivariant homotopy groups.
The induction starts with n=-1, where there is nothing to show. 
For n≥ 0 the quotient A_n/A_n-1 is (G× K)-equivariantly isomorphic
to a wedge of summands of the form ((G× K)/Γ_i)_+ S^n,
for certain closed subgroups Γ_i of G× K.
Since the K-action on A is free (away from the basepoint), 
each isotropy group Γ_i that occurs 
is the graph of a continuous homomorphism α_i:H_i→ K
defined on a closed subgroup H_i of G.
Since the G-action on A is free (away from the basepoint),
all the homomorphisms α_i that occur are injective.

Since equivariant homotopy groups take wedges to sums, the 
suspension isomorphism and the Wirthmüller
isomorphism allow us to rewrite the equivariant homotopy groups 
of (A_n/A_n-1)_K C as

    π_*^G((A_n/A_n-1)_K C)     ⊕ π_*^G(((G× K)/Γ_i)_+_K C S^n)
        ⊕ π_*-n^G( G⋉_H_iα_i^*(C))⊕ π_*-n^H_i(α_i^*(C) S^L_i) ,

where L_i is the tangent representation of H_i in G.
By hypothesis on C and because α_i is injective, the orthogonal H-spectrum
α_i^*(C) is H_i-stably contractible. So α_i^*(C) S^L_i 
is H_i-stably contractible 
by Proposition <ref> (ii). 
Altogether this shows that the orthogonal G-spectrum (A_n/A_n-1)_K C 
has vanishing G-equivariant homotopy groups.

The inclusion A_n-1→ A_n is an h-cofibration of based (G× K)-spaces,
so the induced morphism ( A_n-1)_K C→ (A_n)_K C is an h-cofibration
of orthogonal G-spectra, giving rise to a long exact sequence of equivariant
homotopy groups (Corollary <ref>).
By the previous paragraph and the inductive hypothesis, the
orthogonal G-spectrum (A_n)_K C 
has vanishing G-equivariant homotopy groups.
This completes the inductive step.

Since A is the sequential colimit, along h-cofibrations
of based (G× K)-spaces, of the skeleta A_n,
the orthogonal G-spectrum A_K C is the sequential colimit, 
along h-cofibrations of orthogonal G-spectra, 
of the sequence with terms (A_n)_K C. 
Equivariant homotopy groups commute with such sequential colimits 
(Proposition <ref> (i)), 
so also A_K C has vanishing G-equivariant homotopy groups.

Now we let H be any closed subgroup of G. 
The underlying (H× K)-space of A is again cofibrant
by Proposition <ref> (i),
so we can apply the previous reasoning to H instead of G and
conclude that the groups π_*^H(A_K C) vanish.




Let H be a closed subgroup of a compact Lie group G. Then the induction functor  

    G⋉_H -  :  H → G

takes _*-isomorphisms of orthogonal H-spectra to
_*-isomorphisms of orthogonal G-spectra.


We let G× H act on G by left and right translation, i.e., via

    (g,h)·γ =  g γ h^-1 .

With this action, G is (G× H)-equivariantly isomorphic to
the homogeneous space (G× H)/Δ
for Δ={(h,h) :  h∈ H}.
In particular, G is (G× H)-cofibrant, and both partial actions
are free. So Proposition <ref> 
applies to the functor (G_+)_H- = G⋉_H- and yields the desired conclusion.



Now we discuss the transfer maps of equivariant homotopy groups.

[Transfers]subjecttransfer!on equivariant homotopy groups
We let H be a closed subgroup of a compact Lie group G.
As before we write L=T_e H(G/H) for the tangent space of G/H at the coset e H,
which inherits an H-action from the H-action on G/H.
We let X be an orthogonal G-spectrum. We form the composite

    π_k^H(X S^L)  [] G⋉_H -  π_k^G(G⋉_H X) 
      act_*  π_k^G(X)

of the external transfer (<ref>) and the effect of
the action map (i.e., the adjunction counit) G⋉_H X→ X. We call this composite
the dimension shifting transfersubjecttransfer!dimension shifting 
and denote itsymbol_H^G - dimension shifting transfer

    _H^G :  π_k^H (X S^L)  → π_k^G (X)  .

The (degree zero) transfer is then defined as the compositesubjecttransfer!degree zero,

    _H^G  :  π_k^H(X)  (X i)_*   π_k^H(X S^L)   _H^G  π_k^G(X) ,
   
where i:S^0→ S^L is the `inclusion of the origin', the based map sending 0 to 0.
Both kinds of transfer are natural for morphisms of 
orthogonal G-spectra.symbol_H^G - degree zero transfer
For finite index inclusions, L=0 and there is no difference between
the dimension shifting transfer and the degree zero transfer.
The  external transfer is additive;
since the dimension shifting and degree zero transfers are obtained from
there by applying morphisms of equivariant spectra, they are also additive.

The key properties of these transfer maps are:


  * transfers are transitive (Proposition <ref>);

  * transfers commute with inflation maps (Proposition <ref>);

  * the restriction of a degree zero transfer to a closed subgroup satisfies
a double coset formula (Theorem <ref>).





[Infinite Weyl group transfers]  
If H has infinite index in its normalizer, then the 
degree zero transfer _H^G is trivial. Indeed, the inclusion of the normalizer
N_G H of H into G induces a smooth embedding

    W_G H = (N_G H) / H  → G/H

and thus a monomorphism of tangent H-representations

    T_e H(W_G H)  → T_e H( G/H ) = L  .

If n∈ N_G H is an element of the normalizer and h∈ H, then

    h· n H  =  n· (n^-1h n) H  =  n H ,

so W_G H is H-fixed inside G/H. Consequently, the tangent space
T_e H (W_G H) is contained in the H-fixed space L^H.
If H has infinite index in its normalizer, then the Weyl group W_G H 
and the tangent space T_e H(W_G H) have positive dimension.
In particular, L has non-zero H-fixed points.
The point 0 in S^L can thus be moved through H-fixed points 
to the basepoint at infinity.
The first map in the composite (<ref>) is thus the zero map,
hence so is the transfer _H^G.




 As the previous example indicates,
the passage from the dimension shifting transfer _H^G to the
degree zero transfer _H^G loses information.
An extreme case is when the subgroup H is normal in G. 
Then the action of the group H on G/H is trivial;
hence also the H-action on the tangent space L is trivial.
Upon choosing an isomorphism L^d,
the Wirthmüller isomorphism identifies
π_0^G(G⋉_H Y) with π_0^H(Y S^d),
where d=(G/H)=(G)-(H).
The dimension shifting transfer then becomes a natural map

    π_0^H ( Y )   π_d^H(Y S^L)  _H^G π_d^G (Y)  .

This transformation is generically non-trivial.




Now we recall some important properties of the transfer maps.
We start with the compatibility with the loop and suspension isomorphisms.


Let H be a closed subgroup of a compact Lie group G,


  * For every orthogonal H-spectrum Y and all k∈, 
the following diagrams commute:

    @C=15mm@R=7mmπ_k^H( (Ω Y) S^L) [r]^-G⋉_H -_-[d]^_ assembly   π_k^G(G⋉_H (Ω Y))  [d]^ assembly_
    π_k^H(Ω(Y S^L)) [d]_α^   π_k^G(Ω(G⋉_H Y))[d]_^α
    π_k+1^H(Y S^L) [r]_-G⋉_H-^-   π_k+1^G(G⋉_H Y)


    @C=15mm@R=7mmπ_k^H(Y S^L) [r]^-G⋉_H -_-[d]^_- S^1   π_k^G(G⋉_H Y)  [d]^- S^1_
    π_k+1^H(Y S^L S^1) [d]_(Yτ_S^L,S^1)_*^-   π_k+1^G((G⋉_H Y) S^1)[d]_^b_*
    π_k+1^H(Y S^1 S^L) [r]_-G⋉_H-^-   π_k+1^G(G⋉_H (Y S^1))

Here τ_S^L,S^1:S^L S^1→ S^1 S^L is the twist isomorphism,
and the G-equivariant isomorphism

    b  :  (G⋉_H Y) S^1  G⋉_H (Y S^1)

is given by b([g,y] t)=[g, y t].

  * For every orthogonal G-spectrum X and all k∈, 
the following diagrams commute:

    @C=7mm@R=7mmπ_k^H( (Ω X) S^L) [r]^-_H^G[d]^_ assembly   π_k^G(Ω X) [dd]_^α   π_k^H(X S^L) [r]^-_H^G[d]^_- S^1   π_k^G(X)  [dd]^- S^1_
    π_k^H(Ω(X S^L))  [d]^_α      π_k+1^H(X S^L S^1) [d]_(Xτ_S^L,S^1)_*^-   
    π_k+1^H(X S^L) [r]_-_H^G   π_k+1^G(X)  
       π_k+1^H(X S^1 S^L) [r]_-_H^G   π_k+1^G(X S^1)


    @C=10mmπ_k^H( Ω X) [r]^-_H^G[d]^_α   π_k^G(Ω X) [d]_^α   π_k^H(X) [r]^-_H^G[d]^_- S^1   π_k^G(X)  [d]^- S^1_
    π_k+1^H(X) [r]_-_H^G   π_k+1^G(X)  
       π_k+1^H(X S^1) [r]_-_H^G   π_k+1^G(X S^1)




The first two diagrams commute because we can read the diagrams of
Proposition <ref> (i) backwards.
The commutativity of the other diagrams then follows by naturality.


Now we establish the transitivity property for a 
nested triple of compact Lie groups K≤ H≤ G.
We continue to denote by L=T_e H(G/H) the tangent H-representation in G,
and we write L̅=T_e K(H/K) for the tangent K-representation in H.
We choose a slice 

    s :  D( L )  → G

as in the construction of the map l_H^G:G→ S^L H_+ in 
Construction <ref>; so s is a wide smooth embedding
of the unit disc of L satisfying

    s(0) = 1      and       s(h· l) =  h· s(l)· h^-1

for all (h,l)∈ H× D(L), and 
the differential at 0∈ D(L) of the composite
proj_H∘ s:D(L)→  G/H is the identity of L.
The differential at 0∈ D(L) of the composite

    D(L)  G   G/K

is then a K-equivariant linear monomorphism

    d(proj_K∘ s)_0  :  L → T_e K(G/K)  =  L(K,G)

that splits the differential at e K of the projection q:G/K→ G/H.
So the combined map

    ( d(proj_K∘ s)_0,  (d q )_e H)  : L⊕L̅    =  T_e H(G/H)⊕ T_e K(H/K) 
       → T_e K(G/K)  =  L(K,G)

is an isomorphism of K-representations.
Upon one-point compactification this isomorphism induces a homeomorphism
of K-spaces

    S^L S^L̅  S^L(K,G) .

Any two slices are H-equivariantly isotopic 
(compare <cit.>),
so the K-equivariant homotopy class of the latter isomorphism is independent
of the choice of slice. 




Let G be a compact Lie group, K≤ H≤ G nested closed subgroups
and X an orthogonal G-spectrum.
Then the composite

    π_k^K(X S^L(K,G))  _(<ref>) π_k^K(X S^L S^L̅)   _K^H  π_k^H(X S^L)   _H^G  π_k^G(X)

agrees with the transfer _K^G.
Moreover, the degree zero transfers satisfy

    _H^G∘_K^H  =  _K^G  :  π_k^K (X)  → π_k^G (X)  .



We start by establishing transitivity of the Wirthmüller maps.
We choose a slice for the inclusion of K into H, 
i.e., a wide smooth embedding s̅ : D( L̅ ) →  H satisfying

    s̅(0) = 1      and    s̅(k·l̅) =  k·s̅(l̅)· k^-1

for all (k,l̅)∈ K× D(L̅), and 
such that the differential at 0∈ D(L̅) lifts the identity of L̅.
We combine the two slices into a slice for K inside G:
the K-equivariant map

    D( L ⊕L̅) → G ,  
    (l,l̅) ⟼  s(l)·s̅(l̅)

sends (0,0) to 1, and its differential at (0,0) is exactly the
identification (<ref>).
So we can – and will – define the map l_K^G:G→ S^L(K,G) K_+
from the slice

    s'  :  D(L(K,G))  [(<ref>)^-1]D( L ⊕L̅) (l,l̅) ↦ s(l)·s̅(l̅) G .

The maps l_H^G:G→ S^L H_+ respectively
l_K^H:H→ S^L̅ K_+ are the Thom-Pontryagin collapses 
based on the H^2-equivariant smooth embedding

    s̃ :   D(L)× H  → G  ,    (l,h) ⟼ s(l)· h

respectively 
the K^2-equivariant smooth embedding

    ŝ :   D(L̅)× K  → H  ,    
    (l̅,k) ⟼ s̅(l̅)· k .

The composite (S^L l_K^H)∘ l_H^G is thus K^2-equivariantly homotopic
to the Thom-Pontryagin collapse 
based on the K^2-equivariant smooth embedding

    D(L)× D(L̅)× K  → G  , 
       (l,l̅,k) ⟼ s(l)·s̅(l̅)· k  .

The following diagram of K^2-equivariant smooth embeddings
then commutes by construction:

    @C=12mm@R=7mm
    D(L⊕L̅)×  K[d]_incl[r]_-^-(<ref>)    D(L(K,G))× K [dd]^(l,k)↦ s'(l)· k
    
    D(L)× D(L̅)×  K
    [d]_(l,l̅, k)↦ (l,s̅(l̅)· k)   
    
    D(L)×  H  [r]_-(l,h)↦ s(l)· h    G

So the associated diagram of K^2-equivariant collapse maps also commutes:

    @C=12mm@R=7mm
    G_+ [dd]_l_K^G[r]^-l_H^G   
    S^L H_+[d]^S^L l_K^H
        
    S^L S^L̅ K_+[d]^Ψ
    
     S^L(K,G) K_+ [r]_-^-(<ref>)^-1     S^L⊕L̅ K_+

Here Ψ:S^L S^L̅→ S^L⊕L̅ is the collapse map
for the inclusion D(L⊕L̅)→ D(L)× D(L̅).
A rescaling homotopy connects Ψ to the canonical homeomorphism
S^L S^L̅ S^L⊕L̅, so the following square
commutes up to K^2-equivariant based homotopy:

    @C=12mm
    G_+ [d]_l_K^G[r]^-l_H^G   
    S^L H_+[d]^S^L l_K^H
    
     S^L(K,G) K_+ [r]_-^-(<ref>)^-1     S^L⊕L̅ K_+

We can thus conclude that the map l_K^G / K 
is K-equivariantly homotopic to the composite

    G / K_+      G⋉_H(H /K _+)  l_H^G_H (H / K_+)(H /K _+) S^L  
        H⋉_K S^L  l_K^H_K S^L  
    S^L  S^L̅ _(<ref>) S^L(K,G) .

Naturality and transitivity of restriction maps then show that the following
diagram commutes:

    @C=7mm@R=7mmπ_k^G(X G / K_+) [rr]^-^G_K[d]_      π_k^K(X G / K_+) [ddd]^l_K^G / K
    π_k^G(X G⋉_H (H / K_+)) [d]^-^G_H@<-8ex>@/_2pc/[dd]_(.3)_H^G      
    π_k^H(X G⋉_H(H / K_+)) [d]^l_H^G_H(H / K_+)      
    π_k^H(X (H/K_+) S^L) [d]^      π_k^K(X S^L(K,G)) [d]^
    π_k^H(X H⋉_K S^L)) [r]^-^H_K@/_2pc/[rr]_(.8)_K^H   π_k^K(X H⋉_K S^L) [r]^-l_K^H_K S^L   π_k^K(X S^L S^L̅)

By Theorem <ref>
the Wirthmüller map is inverse to the composite

    π^K_k(X S^L(K,G))  ε_L(K,G) π^K_k(X S^L(K,G))  G⋉_K - π^G_k(G⋉_K X)   .

The map ε_L(K,G) is induced by the antipodal map of S^L(K,G).
Under the homeomorphism between S^L(K,G) and S^L S^L̅,
this becomes the smash product of the antipodal maps of L and L̅.
So reading the diagram backwards gives a commutative diagram of external transfers

    @C=7mm@R=7mmπ_k^K(X S^L S^L̅) [r]^-H⋉_K-[dd]_   π_k^H(X H⋉_K S^L)  [r]^-   π_k^H(X H/K_+ S^L) [d]^G⋉_H-
          π_k^G(X G⋉_H H/K_+) [d]^
    π_k^K(X S^L(K,G)) [rr]_G⋉_K-      π_k^G(X G/K_+)

Postcomposing with the effect of the projection G/K→∗
and exploiting naturality gives the claim about the dimension shifting transfer.
The second claim follows by precomposing with the inclusion of the origin of L(K,G).




 
We let K≤ H≤ G be nested closed subgroups and X an orthogonal G-spectrum.
We let p:G/K→ G/H denote the projection.
For later reference we show that under the external transfer
isomorphisms the effect of morphism X p_+:X G/K_+→ X G/H_+
corresponds to the transfer from K to H. For simplicity
we restrict to the case where (K)=(H), i.e., when
K has finite index in H; the general case only differs 
by more complicated notation. If K has finite index in H,
then the differential of the projection G/K→ G/H 
is an isomorphism from the K-representation T_e K(G/K)
to the underlying K-representation of L=T_e H(G/H).
We identify these two representations via this isomorphism.
We claim that then the following square commutes:

    @C=15mmπ_*^K(X S^L)[r]^-_K^H[d]_G⋉_K -^   π_*^H(X S^L)[d]^G⋉_H -_
    π_*^G(X G/K_+)[r]_- (X p_+)_*   π_*^G(X G/H_+)

To see this we compose the commutative 
diagram (<ref>)
in the proof of Proposition <ref>
that encodes the transitivity of external transfers
with the map π_*^G(X p_+) and arrive at another commutative diagram:

    @C=7mm@R=7mmπ_k^K(X S^L) [d]^H⋉_K-@=[rr] @<-4ex>@/_3pc/[ddd]_(.3)_K^H      π_k^K(X S^L) [dd]^G⋉_K-
    π_k^H(X H⋉_K S^L)  [d]^   
    π_k^H(X H/K_+ S^L) [r]^-G⋉_H-[d]    π_k^G(X G⋉_H H/K_+) [r]^-   π_k^G(X G/K_+) [d]^(X p_+)_*
    π_k^H(X  S^L) [rr]_-G⋉_H-      π_k^G(X G/H_+)

The lower left vertical map is induced by the projection H/K→∗
and the bottom part of the diagram commutes by naturality of the
external transfer.



Now we prove the compatibility of transfers with inflations, i.e., restriction along
continuous epimorphisms α:K→ G. For every closed 
subgroup H of G, the map

    α̅ :  K/J  → G/H  ,    k J ⟼ α(k) H

is a diffeomorphism, where J=α^-1(H).
The differential at the coset e J is an isomorphism

    (dα̅)_e J :  L̅ = T_e J(K/J)  → 
    (α|_J)^*(T_e H(G/H)) =  (α|_J)^*( L )

of J-representations.
In the statement and proof of the following proposition 
a couple of unnamed isomorphisms occur. One of them is the natural isomorphism
of K-spaces

    K⋉_J (α|_J)^*(A)   α^*(G⋉_H A)  ,   
    [k, a] ⟼ [α(k), a] .




Let K and G be compact Lie groups and α:K→ G 
a continuous epimorphism. 
Let H be a closed subgroup of G,
set J=α^-1(H), and let α|_J:J→ H 
denote the restriction of α.  


  * For every orthogonal H-spectrum Y
the following diagram commutes:

    @C=10mmπ_k^H(Y S^L) [d]_G⋉_H -[r]^-(α|_J)^*   π_k^J( (α|_J)^*(Y S^L))  [r]^-   π_k^J( (α|_J)^*(Y) S^L̅) [d]^K⋉_J- 
    π_k^G(G⋉_H Y) [r]_-α^*   π_k^K(α^*(G⋉_H Y)) [r]_-   π_k^K(K⋉_J(α|_J)^*( Y) )


  * For every orthogonal G-spectrum X
the following diagram commutes:

    @C=10mmπ_k^H(X S^L) [d]__H^G[r]^-(α|_J)^*   π_k^J( (α|_J)^*(X S^L))  [r]^-   π_k^J( (α|_J)^*(X) S^L̅) [d]^_J^K
    π_k^G(X) [rr]_-α^*      π_k^K(α^* (X) )

Moreover, the degree zero transfers satisfy the relation

    α^*∘_H^G  =  _J^K ∘ (α|_J)^*

as maps π_k^H (X)→π_k^K(α^*(X)).

  * For every orthogonal G-spectrum X, every g∈ G
and all closed subgroups K≤ H of G the following diagram commutes:

    π_k^K^g(X) [d]__K^g^H^g[r]^-g_⋆   π_k^K(X)  [d]^_K^H
    π_k^H^g(X) [r]_-g_⋆   π_k^H(X)




(i) The restriction maps commute with the loop and suspension isomorphisms,
and so do the external transfer maps (by Proposition <ref>).
So it suffices to prove the claim in dimension k=0.
We choose a wide G-equivariant embedding i:G/H→ V into some
G-representation; from this input data we form the collapse map

    c_G/H :  S^V  → G⋉_H S^W

and the external transfer G⋉_H-. 
Then the composite

    α^*(i)∘α̅ :   
    K/J  → α^*(G/H) → α^*(V)

is a wide K-equivariant embedding, and we can (and will) base the 
external transfer K⋉_J- on this embedding.
Since i:G/H→ V and α^*(i)∘α̅ have the same image,
they define the same decomposition of V into tangent and normal subspaces,
i.e.,

    W̅ =  α^*(V) - d(α^*(i)∘α̅)_e J(L̅)
     =  (α|_J)^*(W) .

Moreover, the composite

    S^α^*(V) =  α^*(S^V) α^*(c_G/H) α^*(G⋉_H S^W)
      K⋉_J (α|_J)^*(S^W)  =  K⋉_J S^W̅

is precisely the collapse map based on the wide embedding (<ref>).
From here the commutativity of the square is straightforward from the
definitions.

(ii) The dimension shifting transfer _H^G is defined as the composite
of the external transfer G⋉_H - and the effect of the action map
G⋉_H X → X, and similarly for _J^K. 
The action map for the orthogonal K-spectrum K⋉_J α^*(X)
coincides with the composite

    K⋉_J α^*(X)   α^*(G⋉_ H X) α^*(act) α^*(X) .

So the following diagram commutes by naturality of the restriction map: 

    @C=15mmπ_k^G(G⋉_H X) [d]_π_k^G(act)[r]^-α^*   π_k^K( α^*(G⋉_H X)) [dr]_π_k^K(α^*(act))[r]^-   π_k^K(K⋉_J α^*(X)) [d]^π_k^K(act)
    π_k^G(X) [rr]_-α^*      π_k^K(α^* (X) )

Part (ii) then follows by stacking this commutative diagram to the
one of part (i).
The second claim follows by precomposing with the inclusion of the origin of L.

Part (iii) follows from part (ii) for the epimorphism c_g:H→ H^g and
the closed subgroup K of H, and naturality of the transfer:

    g_⋆∘_K^g^H^g    =  (l_g^X)_*∘ (c_g)^*∘_K^g^H^g
    
    _(ii)    =  (l_g^X)_*∘_K^H∘ (c_g)^*=  _K^H∘  (l_g^X)_*∘ (c_g)^* = _K^H∘ g_⋆



Now we know how transfers compose and interact with inflations.
The remaining compatibility issue is to rewrite the composite 
of a transfer map followed by a restriction map.
The answer is given by the double coset formula
that we will prove in Theorem <ref> below.



subjectgeometric fixed points|(



§ GEOMETRIC FIXED POINTS



In this section we study the 
geometric fixed point homotopy groups Φ^G_*(X) 
of an orthogonal G-spectrum X.
We establish the isotropy separation sequence (<ref>) 
that is often useful for inductive arguments, 
and we prove that equivariant equivalences can
also be detected by geometric fixed points, 
see Proposition <ref>.
In Proposition <ref> we show that 
geometric fixed points annihilate transfers from proper subgroups.
Theorem <ref>  provides
a functorial description of the 0-th equivariant stable homotopy groups 
of a G-space Y in terms of the path components of the 
H-fixed points spaces Y^H for closed subgroups H of G with finite Weyl group.



We define the geometric fixed point homotopy groups of an orthogonal G-spectrum X.
As before we let s(_G) denote the set 
of finite-dimensional G-subrepresentations 
of the complete G-universe _G, considered as a poset under inclusion. 
We obtain a functor from s(_G) to sets by

    V  ⟼ [S^V^G, X(V)^G]  ,

the set of (non-equivariant) homotopy classes of based maps
from the fixed point sphere S^V^G to the fixed point space X(V)^G.
An inclusion V⊆ W in s(_G) is sent to the map

    [S^V^G, X(V)^G]  → [S^W^G,X(W)^G]

that takes the homotopy class of f:S^V^G→ X(V)^G
to the homotopy class of the composite

    S^W^G  S^(V^⊥)^G S^V^G    f  S^(V^⊥)^G X(V)^G  
       =   (S^V^⊥ X(V))^G  (σ_V^⊥,V)^G X(V^⊥⊕ V)^G  = X(W)^G  .

 

 
Let G be a compact Lie group and X an orthogonal G-spectrum.
The  0-th geometric fixed point homotopy groupsubjectgeometric fixed points subjectfixed points!geometric  
is defined as symbolΦ^G_k -  geometric fixed point homotopy group  

    Φ_0^G(X)  =  _V∈ s(_G)  [S^V^G, X(V)^G]   .

If k is an arbitrary integer, we define the k-th
geometric fixed point homotopy group Φ_k^G(X)
by replacing S^V by S^V⊕^k (for k>0)
or  replacing X(V) by X(V⊕^-k) (for k<0),
analogous to the definition of π_k^G(X) in (<ref>).
 

The construction comes with a 
geometric fixed point mapsubjectgeometric fixed point homomorphism

    Φ^G :  π_0^G (X)      →    Φ_0^G (X) 
    
    [f:S^V→ X(V)]    ⟼ [f^G:S^V^G→ X(V)^G]

from the G-equivariant homotopy group to the 
geometric fixed point homotopy group.
For a trivial group, equivariant and geometric fixed point groups
coincide and the geometric fixed point map Φ^e:π_0^e(X)→Φ^e_0(X)
is the identity.

[Geometric fixed points of suspension spectra]
subjectgeometric fixed points!of suspension spectra 
If A is any based G-space,
then the geometric fixed points Φ_*^G(Σ^∞ A) 
of the suspension spectrum are given by

    Φ_k^G(Σ^∞ A) 
     =  _V∈ s(_G)  [S^V^G⊕^k, S^V^G A^G]   .

As V ranges over s(_G) the dimension of the fixed points
grows to infinity. So the composite

    π_k^e (Σ^∞ A^G)  p_G^* π_k^G (Σ^∞ A^G) incl_*π_k^G (Σ^∞ A)  Φ^G  Φ_k^G (Σ^∞ A)

is an isomorphism, where p_G^* is inflation along the 
unique group homomorphism p_G:G→ e.
 We will sometimes refer to this isomorphism by saying that
`geometric fixed points commute with suspension spectra'.





We let X be a G-orthogonal spectrum and α:K→ G a continuous epimorphism.
We define inflation mapssubjectinflation map!of geometric fixed point homotopy groups  

    α^*  :  Φ^G_0 (X)  → Φ^K_0 (α^* X)

on geometric fixed point homotopy groups.
We choose a K-equivariant linear isometric embedding ψ:α^*(_G)→_K
of the restriction along α of the complete G-universe into
the complete K-universe. We let f:S^V^G→ X(V)^G be a based map representing
an element of Φ_0^G(X), for some V∈ s(_G).
Since α is surjective, V^G=(α^* V)^K and
X(V)^G=(α^*(X(V)))^K=((α^* X)(α^* V))^K. We use ψ to identify
α^* V with ψ(V) as K-representations, and 
hence also (α^* V)^K with ψ(V)^K. 
This turns f into a based map

    S^ψ(V)^K   S^(α^* V)^K =  S^V^G  f  
    X(V)^G  =  ((α^* X)(α^*V))^K   ((α^* X)(ψ(V)))^K  .

Any two equivariant embeddings of  α^*(_G) into _K
are homotopic through K-equivariant linear isometric embeddings, so
the restriction map is independent of the choice of ψ.
This latter map represents the element α^*[f] in Φ_0^K(α^* X).
The element α^*[f] depends only on the class of f in Φ_0^G(X),
so the inflation map α^* is a well-defined homomorphism.


The surjectivity of α is essential to obtain an inflation map α^*
on geometric fixed point homotopy groups, 
and geometric fixed points do not have natural restriction maps to subgroups.
These inflation maps between the geometric fixed point homotopy groups
are clearly natural in the orthogonal G-spectrum.
The next proposition lists the other naturality properties.


 
Let G be a compact Lie group and X an orthogonal G-spectrum.
  
  
  * For every pair of composable continuous epimorphisms α:K→ G
    and β:L→ K we have
    
    β^*∘α^*  =  (αβ)^*  :  Φ_0^G (X) → Φ_0^L( (αβ)^* X) .

  
  * For every element g∈ G the composite
    
    Φ_0^G(X)   (c_g)^* Φ_0^G (c_g^* X)   (l_g^X)_*  Φ_0^G (X)

    is the identity.
  
  * For every continuous epimorphism α:K→ G
    of compact Lie groups the following square commutes:
    
    π_0^G(X) [r]^-Φ^G[d]_α^*   Φ_0^G(X) [d]^α^*
    π_0^K(α^* X) [r]_-Φ^K   Φ_0^K (α^* X)

  


(i) We choose a K-equivariant linear isometric embedding ψ:α^*(_G)→_K
and an L-equivariant linear isometric embedding φ:β^*(_K)→_L.
If we then use the L-equivariant linear isometric embedding

    φ∘β^*(ψ)  :  (α∘β)^*(_G) → _L

for the calculation of (α∘β)^*, the desired equality even holds
on the level of representatives.

(ii) We let V be a finite-dimensional G-subrepresentation of _G
and f:S^V^G→ X(V)^G a based map representing an element of Φ_0^G(X).
We use the G-equivariant linear isometry l_g:c_g^*(_G)→_G
given by left multiplication by g. 
Then c_g^*(V) and l_g(V) have the same underlying sets 
and the restriction of l_g to (c_g^*(V))^G is the identity onto (l_g(V))^G.

The class c_g^*[f] is represented by the composite

    S^(l_g(V))^G =  S^(c_g^* V)^G = S^V^G  f  
    X(V)^G    = ((c_g^* X)(c_g^*V))^G 
        ((c_g^*X)(l_g^V))^G ((c_g^* X)(l_g(V)))^G  .

Consequently, (l_g^X)_*(c_g^*[f]) is represented by the composite

    S^(l_g(V))^G =   S^V^G     f  
    X(V)^G  =  ((c_g^* X)(c_g^*V))^G 
        ((c_g^*X)(l_g^V))^G ((c_g^* X)(l_g(V)))^G 
      ((l_g^X)(l_g(V)))^G (X(l_g(V)))^G   .

The G-action on X(V) is diagonally,
from the external G-action on X and the internal G-action on V.   
Hence the map l_g^X(V):c_g^*(X(V))→ X(V)
is the composite of (c_g^* X)(l_g^V):(c_g^*X)(c_g^* V)→ (c_g^* X)(V)
and (l_g^X)(V):(c_g^* X)(V)→ X(V).
Since the restriction of l_g^X(V) to the G-fixed points is the
identity, the composite of ( (c_g^* X)(l_g^V))^G
and ((l_g^X)(V))^G is the identity.
This shows that (l_g^X)_*(c_g^*[f]) is again represented by f,
and hence (l_g^X)_*∘ (c_g)^*=.

(iii) 
We consider a based continuous G-map f:S^V→ X(V); 
then α^*[f] is represented by the K-map

    α^*(f) :  S^α^*(V) =  α^*(S^V)  → α^*(X(V)) =  (α^* X)(α^*(V)) ,

and so Φ^K(α^*[f]) is represented by

    (α^*(f))^K :  (S^α^*(V))^K =  (α^*(S^V))^K  → (α^*(X(V)))^K =  ((α^* X)(α^*(V)))^K .

Since α is surjective, this is the same map as

    f^G :  (S^V)^G → X(V)^G .

To calculate α^*(Φ^G([f])) 
we choose a K-equivariant linear isometric embedding 
ψ:α^*(_G)→_K and conjugate f^G by the isometry

    (ψ|_α^*(V))^K :  (α^*(V))^K   ψ(V)^K .

But conjugation by an isometry does not change the stable homotopy class,
by Proposition <ref> (ii)
for the trivial group. So 

    Φ^K(α^*[f]) = [α(f)^K] =  [f^G] =  α^*(Φ^G[f]) .





[Weyl group action on geometric fixed points]

We let H be a closed subgroup of a compact Lie group G, and X an
orthogonal G-spectrum. Every g∈ G gives rise to a conjugation homomorphism
c_g:H→ H^g by c_g(h)=g^-1h g. Moreover,
left translation by g is a homomorphism of orthogonal H-spectra
l_g^X:c_g^*(X)→ X. So combining inflation along c_g
with the effect of l_g^X gives a homomorphismsubjectconjugation homomorphism!on geometric fixed point homotopy groups  

    g_⋆ :  Φ_0^H^g(X)  (c_g)^*  Φ_0^H(c_g^*(X))  (l^X_g)_* Φ_0^H(X) .

In the special case when g normalizes H, this is a self-map
of the geometric fixed point group Φ_0^H(X).
If moreover g belongs to H, then g_⋆ is the identity
by Proposition <ref> (ii).
So the maps g_⋆ define an action of the Weyl group
W_G H=N_G H/H on the geometric fixed point homotopy group Φ_0^H(X).
By the same arguments as for equivariant homotopy groups
in Remark <ref>, the identity path component of the 
Weyl group acts trivially, so the action factors over an action
of the discrete group π_0(W_G H)=W_G H/(W_G H)^∘.
Since the geometric fixed point map Φ:π_0^H(X)→Φ_0^H(X)
is compatible with inflation and natural in X, this map
is π_0(W_G H)-equivariant.


Now we recall the interpretation of geometric fixed point homotopy groups 
as the equivariant homotopy groups 
of the smash product of X with a certain universal G-space.
We denote by _G the family of proper closed subgroups of G,
and by E_G a universal space for the family _G.subjectuniversal space!for the family of proper subgroups
So E_G is a cofibrant G-space without G-fixed points and such that the
fixed point space (E_G)^H is contractible for every closed proper subgroup H
of G.
These properties determine E_G uniquely up to G-homotopy equivalence,
see Proposition <ref>.

We denote by Ẽ_G the reduced mapping cone of the based G-map 
(E_G)_+→ S^0 that sends E_G to the non-basepoint of S^0. 
So Ẽ_G is the unreduced suspension of the universal space E_G.
The G-fixed points of E_G are empty and 
fixed points commute with mapping cones, so the map
S^0→ (Ẽ_G)^G is an isomorphism. 
For all proper subgroups H of G the map (E_G)^H_+→ (S^0)^H=S^0 
is a weak equivalence, so (Ẽ_G)^H is contractible.



We let _G^⊥=_G-(_G)^G be the orthogonal complement of
the G-fixed points in the complete G-universe _G. 
We claim that the unit sphere S(_G^⊥)
of this complement is a universal space E_G.
Indeed, the unit sphere S(_G^⊥) is G-equivariantly homeomorphic
to the space (,_G^⊥), so it is cofibrant as a G-space
by Proposition <ref> (ii). 
Since S(_G^⊥) has no G-fixed points, any stabilizer group
is a proper subgroup of G, i.e., in the family _G.
On the other hand, for every proper subgroup H of G 
there is a G-representation V with V^G=0 but V^H 0,
see for example <cit.>.
Since ^⊥_G contains infinitely many isomorphic copies of V,
the H-fixed points

    (S(_G^⊥))^H  =   S((_G^⊥)^H)

form an infinite-dimensional sphere, and hence are contractible.
So S(_G^⊥) is a universal G-space for the family of proper subgroups.

Since Ẽ_G is an unreduced suspension of E_G,
it is equivariantly homeomorphic to 

    S(⊕_G^⊥)  ,

the unit sphere in ⊕_G^⊥.
So S(⊕_G^⊥) is a model for Ẽ_G.



The inclusion i:S^0→Ẽ_G induces an isomorphism of
G-fixed points S^0(Ẽ_G)^G. 
So for every based G-space A the map A i :A→ AẼ_G
induces an isomorphism of G-fixed points. Hence also for every orthogonal G-spectrum
the induced map of geometric fixed point homotopy groups

    Φ^G_k(X i) :  Φ_k^G(X)   Φ_k^G(XẼ_G)

is an isomorphism. If we compose the inverse with the geometric
fixed point homomorphism (<ref>), we arrive at a homomorphism
Φ:π_k^G(XẼ_G)→Φ^G_k(X).

 
For every orthogonal G-spectrum X and every integer k,
the geometric fixed point map

    Φ :  π_k^G (XẼ_G)  → Φ_k^G (X)

is an isomorphism.

 
We claim that for every finite based G-CW-complex A
and every based G-space Y the map

    (-)^G  :  _*^G(A,YẼ_G)  → _*(A^G,Y^G)

that takes a G-map f:A→ YẼ_G to the induced map
on G-fixed points

    f^G : A^G →  (YẼ_G)^G=Y^G (Ẽ_G)^G Y^G

is a weak equivalence and Serre fibration.

Indeed, since A is a G-CW-complex, 
the inclusion of fixed points
A^G→ A is a G-cofibration and induces a Serre fibration 
of equivariant mapping spaces

    _*^G(A,YẼ_G)  → _*^G(A^G,YẼ_G)  .

Since every G-map from A^G lands in the G-fixed points of
YẼ_G and because (YẼ_G)^G=Y^G, 
the target space is the non-equivariant mapping space _*(A^G,Y^G). 
The G-space A is built from its fixed points 
by attaching G-cells G/H× D^n whose isotropy H is a proper subgroup.
Since the H-fixed points of YẼ_G are contractible 
for all proper subgroups H of G, the fibration is also a weak equivalence.

Now we consider a finite-dimensional G-representation V.
When applied to A=S^V and Y=X(V), the previous claim implies that the fixed point map

    [S^V,X(V)Ẽ_G]^G  → [S^V^G, X(V)^G]

is bijective for every G-representation V.
Passing to colimits over the poset s(_G) proves
the result for k=0. The argument in the other dimensions is similar, and
we leave it to the reader.


 
A consequence of the previous proposition is the following
isotropy separation sequence.subjectisotropy separation sequence 
The mapping cone sequence of based G-space

    (E_G)_+ → S^0  → Ẽ_G

becomes a mapping cone sequence of G-spectra 

    X (E_G)_+  → X  → XẼ_G

after smashing with any given orthogonal G-spectrum X. So
taking equivariant homotopy groups gives a long exact sequence

    ⋯ →π_k^G (X (E_G)_+)     →π_k^G (X)   Φ 
       Φ_k^G (X) → π_k-1^G (X (E_G)_+)  → ⋯

where we exploited the identification 
of Proposition <ref>. 


 
Let G be a compact Lie group.
For a morphism f:X→ Y of orthogonal G-spectra the following are equivalent:


  * The morphism f is a _*-isomorphism.

  * For every closed subgroup H of G and every integer k the map 
Φ_k^H(f) of geometric H-fixed point homotopy groups is an isomorphism.


 
(i)⟹(ii) If f is
is an equivalence of orthogonal G-spectra, then so is fẼ_G
by Proposition <ref> (ii).
Proposition <ref>
then implies that Φ_k^H(f):Φ_k^H(X) →Φ_k^H(Y)
is an isomorphism for all k. 

(ii)⟹(i) We argue by induction on the size of the group G
(i.e., of the dimension of G and the order of π_0 G).
If G is the trivial group, then the geometric fixed point map 
Φ:π_k^e(X)→Φ_k^e(X) does not do anything, and is an isomorphism.
Since Φ_*^e(f) is an isomorphism, so is π_*^e(f).

If G is a non-trivial group we know by induction hypothesis that f is an equivalence
of orthogonal H-spectra for every proper closed subgroup H of G. 
Since E_G is a cofibrant G-space without G-fixed points, 
Proposition <ref> (i)
lets us conclude that f (E_G)_+ is an
equivalence of orthogonal G-spectra.
Since Φ_*^G(f):Φ_*^G(X)→Φ_*^G(Y) 
is also an isomorphism, the isotropy separation sequence
and the five lemma let us conclude that 
π_*^G(f):π_*^G(X)→π_*^G(Y) is an isomorphism.





The next proposition shows that `geometric fixed points vanish on transfers'.
In fact, it is often a helpful slogan to think of geometric fixed points
as `dividing out transfers from proper subgroups' – despite the fact
that the kernel of the geometric fixed point map
Φ:π_0^G (X) →Φ_0^G (X) is in general larger than the subgroup
generated by proper transfers. For finite groups G, the slogan is 
in fact true up to torsion, i.e., 
the geometric fixed point map Φ:π_0^G (X) →Φ_0^G (X) is 
rationally surjective and its kernel is rationally generated by transfers
from proper subgroups, compare 
Proposition <ref> below.
subjectgeometric fixed points!and transfers
A more general version of part (iii) below will
appear in Proposition <ref> (ii).


 
Let K be a closed subgroup of a compact Lie group G
and X an orthogonal G-spectrum.


  * Let H be a closed subgroup of G such that K is not subconjugate to H.
Then the composite

    π_0^G (X)  ^G_K π_0^K (X)   Φ^K  Φ_0^K (X)

annihilates the images of the dimension shifting transfer 
_H^G:π_0^H (X S^L) →π_0^G (X)
and of the degree zero transfer _H^G:π_0^H (X) →π_0^G (X). 

  * The geometric fixed point map Φ^G:π_0^G (X) →Φ_0^G (X) 
annihilates the images of the dimension shifting transfer 
and of the degree zero transfer from all proper closed subgroups of G.

  * If the Weyl group W_G K is finite, then the relation
  
    Φ^K∘^G_K∘_K^G  =  ∑_ g K∈ W_G K  Φ^K∘ g_⋆

holds as natural transformations from π_0^K(X) to Φ_0^K(X).



(i) Let V be any G-representation. The G-space
(G⋉_H X)(V) is isomorphic to G⋉_H X(i^* V).
If K is not subconjugate to H, then both G-spaces 
have only one K-fixed point, the base point. 
So the geometric fixed point homotopy group Φ_0^K(G⋉_H X) vanishes.
The dimension shifting transfer is defined as the composite

    π_0^H (X S^L)  G⋉_H- π_0^G (G⋉_H X)  act_* π_0^G (X) .

The geometric fixed point map is natural for G-maps,
so the 
composite Φ^K∘^G_K∘act_*:π_0^G (G⋉_H X) →Φ_0^K (X)
factors through the trivial group Φ_0^K(G⋉_H X).
Thus the dimension shifting transfer vanishes.
The degree 0 transfer factors through the dimension shifting transfer,
so it vanishes as well.
Part (ii) is the special case of (i) for K=G.

(iii)
Since the functor π_0^K is
represented by the suspension spectrum of G/K 
(in the sense of Proposition <ref>),
is suffices to check the relation for the orthogonal G-spectrum Σ^∞_+ G/K
and the tautological class e_K defined in (<ref>).

For every g∈ N_G K and every class x∈π_0^K(X) we have 

    g_⋆( Φ^K(^G_K(_K^G(x))))  =  Φ^K( ^G_K( g_⋆(_K^G(x))))  =  Φ^K(  ^G_K(_K^G(x)))

because g_⋆ is the identity on π_0^G.
So all classes of the form 
Φ^K(  ^G_K(_K^G(x))) are invariant under the action
of the Weyl group W_G K specified in 
Remark <ref>.
In the universal case this class lives in the group Φ_0^K(Σ^∞_+ G/K)
which is W_G K-equivariantly isomorphic to
π_0^e(Σ^∞_+ (G/K)^K)=π_0^e(Σ^∞_+ W_G K)
and hence a free module of rank 1 over the integral group ring of
the Weyl group W_G K, generated by the class Φ^K(e_K).
So the class Φ^K(^G_K(_K^G(e_K)))
is an integer multiple of the norm element, i.e.,

    Φ^K(^G_K(_K^G(e_K)))  =  λ·∑_g K∈ W_G K g_⋆ (Φ^K(e_K ))

for some λ∈.

It remains to show that λ=1.
We let 1∈π_0^K() be the class represented by the identity of S^0.
Inspection of Construction <ref>
reveals that the transfer _K^G(1) in π_0^G()
is represented by the G-map

    S^V   c   G⋉_K S^W  a   S^V

where c is the collapse map based on any wide embedding
of i:G/K→ V into a G-representation,
W is the orthogonal complement of the image of T_e K(G/K)
under the differential of i, and a[g,w]=g w.
The class Φ^K_0(^G_K(_K^G(1))) is then represented by the restriction
to K-fixed points of the above composite, i.e., by the map

    S^V^K  c^K   (G⋉_K S^W)^K  a^K   S^V^K .

Every K-fixed point of G⋉_K S^W is of the form
[g,w] with g∈ N_G K and w∈ S^W^K, i.e., the map

    (W_G K)_+ S^W^K → (G⋉_K S^W)^K ,    
    g K w  ⟼ [g, w]

is a homeomorphism.

Since the Weyl group W_G K is finite we have (T_e K(G/K))^K=0.
Indeed, the translation action of K on the homogeneous space G/K is smooth,
so by the differentiable slice theorem, 
the K-fixed point e K has an open K-invariant neighborhood
inside G/K that is K-equivariantly diffeomorphic 
to the tangent space T_e K (G/K), compare <cit.>
or <cit.>.
Since W_G K=(G/K)^K is finite, e K is an isolated K-fixed point in G/K,
and hence 0 is an isolated K-fixed point in T_e K (G/K),
i.e., (T_e K (G/K))^K=0.

Since (T_e K(G/K))^K=0 we have W^K=V^K.
Under these identifications, the map c^K becomes a pinch map

    S^V^K → (W_G K)_+ S^V^K  ⋁_g K∈ W_G K  S^V^K ,

i.e., the projection to each wedge summand has degree 1.
On the other hand, the map a^K becomes the fold map

    (W_G K)_+ S^V^K → S^V^K .

So the degree of the composite a^K∘ c^K is the order of the Weyl group W_G K.
We have thus shown that

    Φ^K(^G_K(^G_K(1))) =  |W_G K|·Φ^K(1)

in the group Φ^K_0().
On the other hand, the class 1 is invariant under the action of the Weyl group,
and hence

    Φ^K(^G_K(^G_K(1))) =  λ·∑_ g K∈ W_G K   
    g_⋆ (Φ^K(1))  =  λ· |W_G K|·Φ^K(1) .

Since the abelian group Φ_0^K() is freely generated by Φ^K(1), we can compare
coefficients in the last two expressions and deduce that λ=1.






The 0-th equivariant homotopy groups of equivariant spectra 
have two extra pieces of structure, compared to equivariant spaces: 
an abelian group structure and transfers.
Theorem <ref> 
and Proposition <ref> make precise,
first for suspension spectra of G-spaces and then for suspension spectra of
orthogonal spaces, that at the level of 0-th equivariant homotopy sets,
the suspension spectrum `freely builds in' the extra structure
that is available stably.

We introduce specific stabilization maps that relate unstable homotopy sets 
to stable homotopy groups.
We let H be a compact Lie group and Y an H-space.
We define a mapsymbolσ^G - stabilization map π_0^G (Y)→π_0^G(Σ^∞_+ Y)

    σ^H :  π_0(Y^H)  → π_0^H(Σ^∞_+ Y)

by sending the path component of an H-fixed point y∈ Y^H
to the equivariant stable homotopy class 
σ^H[y] represented by the H-map

    S^0 - y S^0 Y_+  =  (Σ^∞_+ Y)(0)  .

By direct inspection, the map σ^H can be factored as the composition

    π_0(Y^H)   σ^e  π_0^e(Σ^∞_+ Y^H)  
      p_H^*  π_0^H(Σ^∞_+ Y^H)  
     incl_* π_0^H(Σ^∞_+ Y)   ,

where p_H:H→ e is the unique group homomorphism.

We recall that for every space Z the non-equivariant stable homotopy group
π_0^e(Σ^∞_+ Z) is free abelian generated by the 
classes σ^e(y) for all y∈π_0(Z), i.e,

    π_0^e(Σ^∞_+ Z)  {π_0(Z)} .

Indeed, for all n≥ 2 the group π_n(S^n Z_+,∗) is free abelian,
with basis the classes of the maps - y:S^n→ S^n Z_+
as y runs over the path components of Z, 
see for example <cit.>.
Passing to the colimit over n proves the claim.

If H is a closed subgroup of a compact Lie group G, 
and Y is the underlying H-space of a G-space,
then the normalizer N_G H leaves Y^H invariant,
and the action of N_G H factors over an
action of the Weyl group W_G H=N_G H/H on Y^H.
This, in turn, induces an action of the component group π_0(W_G H)
on the set π_0(Y^H).
For all g∈ G, the following square commutes, again by direct inspection:

    π_0(Y^H^g)[r]^-σ^H^g[d]_π_0(l_g)   π_0^H^g(Σ^∞_+ Y)[d]^g_⋆
    π_0(Y^H)[r]_-σ^H   π_0^H(Σ^∞_+ Y)

Here l_g:Y^H^g→ Y^H is left multiplication by g.
In particular, the map σ^H is equivariant for the
action of the group π_0(W_G H).

After stabilizing along the map
σ^H:π_0(Y^H)→π_0^H(Σ^∞_+ Y), we can then transfer from H to G.
For an element g∈ N_G H and a class x∈π_0(Y^H) we have

    _H^G(σ^H(π_0(l_g)(x)))     =  _H^G(g_⋆(σ^H(x))) 
       =  g_⋆(_H^G(σ^H(x)))  =  _H^G(σ^H(x))  ,

because transfer commutes with conjugation,
and inner automorphisms act as the identity.
So the composite _H^G∘σ^H coequalizes 
the π_0(W_G H)-action on π_0(Y^H).


Our proof of the following theorem is based 
on an inductive argument with the isotropy separation sequence.
A different proof, based on the tom Dieck splitting, 
can be found in <cit.>.


 
Let G be a compact Lie group and Y a G-space.subjectsuspension spectrum!of a G-space 


  * The equivariant homotopy group π_0^G(Σ_+^∞ Y) 
is a free abelian group with a basis given by the elements

    _H^G(σ^H(x)) ,

where H runs through all conjugacy classes of closed subgroups of G
with finite Weyl group and x runs through a set of representatives 
of the W_G H-orbits of the set π_0(Y^H).

  * Let z∈π_0^G(Σ^∞_+ Y) be a class
such that for every closed subgroup K of G with finite Weyl group
the geometric fixed point class

    Φ^K( ^G_K(z)) ∈ Φ_0^K(Σ^∞_+ Y)

is trivial. Then z=0.



(i)
In (<ref>) we recalled property (i) when G is a trivial group.
For the trivial group the geometric fixed point map
Φ^e:π_0^e(Σ^∞_+ Y)→Φ_0^e(Σ^∞_+ Y) 
is the identity, so part (ii) is tautologically true.

Now we let G be any compact Lie group.
We let H be a closed subgroup of G with finite Weyl group.
By (<ref>) we know that the group
π_0^e(Σ^∞_+ Y^H) is free abelian 
on the set of path components of Y^H; moreover, the Weyl group W_G H
permutes the basis elements, i.e., 
π_0^e(Σ^∞_+ Y^H) is an integral permutation representation of
the group W_G H. For any representation M of a finite group W
the norm map 

    N  :  M  → M  ,    x ⟼ ∑_w∈ W w· x

factors over the group of coinvariants

    M_W =  M /  x- w x  |  x∈ M, w∈ W .

For the integral permutation representation M=[S] of a W-set S,
a special feature is that the induced map N̅:M_W → M is injective.

So part (i) is equivalent to the claim that the map

    T  :  ⊕_(H)  ( π_0^e(Σ^∞_+ Y^H))_W_G H → π_0^G(Σ_+^∞ Y)

is an isomorphism, where the sum is indexed by representatives of the
conjugacy classes of closed subgroups with finite Weyl group,
and the restriction of T to the H-summand is induced by the composite

    π_0^e(Σ^∞_+ Y^H)  p_H^*  π_0^H(Σ^∞_+ Y^H) incl_* π_0^H(Σ^∞_+ Y)  _H^G  π_0^G(Σ^∞_+ Y) .

We consider the total geometric fixed point homomorphism

    Φ^total :  π_0^G(Σ_+^∞ Y)  → ∏_K:  W_G K finiteΦ_0^K(Σ^∞_+ Y)
     ,    y ⟼ (Φ^K(^G_K(y)))_K .

Property (ii) is equivalent to the claim that Φ^total is injective.

We show now that the composite Φ^total∘ T is injective.
We argue by contradiction and suppose that

    z  =  (z_H)_H ∈ ⊕_(H)  ( π_0^e(Σ^∞_+ Y^H))_W_G H

is a non-zero element in the source of T
such that Φ^total( T(z)) vanishes.
As an element in a direct sum, z has only finitely many non-zero components z_H.
We let K be of maximal dimension and with maximal
number of path components among all indexing subgroups 
such that z_K 0.
Then

    T(z)  =  _K^G(incl_*(p_K^*(y))) + ∑_i=1^m _H_i^G(y_i)

with y an element of π_0^e(Σ^∞_+ Y^K)
with non-zero image in the W_G K-coinvariants,
and with certain closed subgroups H_i of G
that are not conjugate to K and `no larger' in the sense that
either (H_i) <(K), or (H_i) =(K) and |π_0(H_i)|≤ |π_0(K)|.
This means in particular that K is not subconjugate to any of the
groups H_1,…,H_m. Thus

    Φ^K(^G_K(_H_i^G(y_i)))  =  0

for all i=1,…,m, by Proposition <ref> (i).
Hence

    0  =  Φ^K(T(z))    =  Φ^K(^G_K(_K^G(incl_*(p_K^*(y))))) 
        =  ∑_g K∈ W_G KΦ^K(g_⋆(incl_*(p_K^*(y)))) 
        =  ∑_g K∈ W_G KΦ^K(incl_*(g_⋆(p_K^*(y)))) 
       =  Φ^K(incl_*(p_K^*(∑_g K∈ W_G K  (l_g)_*(y))))  .

The third equation is Proposition <ref> (iii).
Since the composite

    π_0^e(Σ^∞_+ Y^K)  p_K^*  π_0^K(Σ^∞_+ Y^K) incl_* π_0^K(Σ^∞_+ Y)  Φ^K  Φ_0^K(Σ^∞_+ Y)

is an isomorphism, we conclude that the 
norm of the element y∈π_0^e(Σ^∞_+ Y^K) is zero. 
But since π_0^e(Σ^∞_+ Y^K) is an integral permutation
representation of the Weyl group, this only happens if
y maps to 0 in the coinvariants, which contradicts our assumption.

Now we show that the classes _H^G(σ^H(x)) in the statement of (i)
generate the group π_0^G(Σ^∞_+ Y)
(i.e., the homomorphism T is surjective).
We argue by induction on the size of G, 
i.e., by a double induction over the dimension and number of path components of G. 
The induction starts when G is the trivial group, which we dealt with above.
Now we let G be a non-trivial compact Lie group.
We start with the special case Y=G/K_+
for a proper closed subgroup K of G. 
The composite

    G⋉_K G⋉_K (e K)_* 
    G⋉_K (Σ^∞_+ G/K)   act  Σ^∞_+ G/K

is an isomorphism of orthogonal G-spectra.
Hence the composite

    π_0^K(Σ^∞ S^V) [] G⋉_K -    π_0^G(G⋉)  π_0^G(G⋉_K (e K)_*)
       π_0^G( G⋉_K (Σ^∞_+ G/K))  π_0^G(act) π_0^G(Σ^∞_+ G/K)

is an isomorphism of abelian groups, 
where V=T_e K(G/K) is the tangent representation and
the first map is the external transfer 
(an isomorphism by Theorem <ref>). 

The inclusion S^0→ S^V is an equivariant h-cofibration
and its quotient S^V/S^0 is G-homeomorphic to the unreduced suspension
of the unit sphere S(V)
(with respect to any K-invariant scalar product on V). So the group

    π_0^G(Σ^∞ (S^V/S^0))   π_0^G(Σ^∞ (S(V)_+ S^1))
      π_-1^G(Σ^∞_+ S(V))

vanishes by the suspension isomorphism and 
Proposition <ref> (ii).
The long exact sequence of 
Corollary <ref> (i)
then shows that the map

    incl_*  : π_0^K(Σ^∞_+ {0}) → π_0^K(Σ^∞ S^V)

is surjective.
Since K is a proper closed subgroup of G,
it either has smaller dimension or fewer path components, so we
know by the inductive hypothesis that the group
π_0^K(Σ^∞_+ {0}) is generated by the elements _L^K(σ^L[0])
where L runs through all conjugacy classes of closed subgroups of K
with finite Weyl group.

Putting this all together lets us conclude 
that the group π_0^G(Σ^∞_+ G/K) is generated by the classes 

    ( π_0^G(act∘ (G⋉_K(e K)_*))∘ 
    (G⋉_K-)∘_L^K∘σ^L)[0]

for all K-conjugacy classes of closed subgroups L≤ K 
that have finite Weyl group in K.
However, this long expression in fact defines a familiar class, as we shall now see.
Indeed, the following diagram commutes by the various naturality properties:

    @C=11mm@R=8mmπ_0({0})@<-4ex>@/_1pc/[ddd]_(.3)σ^L[rr]^-π_0((e K)_*)[d]^p_L^*∘σ^e      π_0( (G/K)^L) [d]_-p_L^*∘σ^e@<0ex>@/^2pc/[ddr]^(.7)σ^L
    π_0^L(Σ^∞ S^0)[dd]^incl_*[rr]^-π_0^L((e K)_*)      π_0^L( Σ^∞_+ (G/K)^L) [dr]_incl_*
             π_0^L(Σ^∞_+ G/K)[d]^_L^K[dl]_incl_*
    π_0^L(Σ^∞_+ S^V)[d]__L^K[rr]^-π_0^L((e K)_* S^V)      π_0^L(Σ^∞_+ G/K S^V)[d]__L^K   π_0^K(Σ^∞_+ G/K)[dl]_incl_*[dd]^_K^G
    π_0^K(Σ^∞_+ S^V)[d]^_G⋉_K -[rr]^π_0^K((e K)_* S^V)      π_0^K(Σ^∞_+ G/K S^V)[d]_G⋉_K -
    π_0^G(G⋉_K  )[rr]_-π_0^G(G⋉_K (e K)_*))      π_0^G(G⋉_K Σ^∞_+ G/K)  [r]_-π_0^G(act)   π_0^G(Σ^∞_+ G/K)

So 

    ( π_0^G(act∘ (G⋉_K(e K)_*))∘ 
    (G⋉_K-)∘_L^K∘σ^L)[0]   
    
    =  _K^G(_L^K(σ^L( (e K)_*[0] )))    = _L^G( σ^L[e K] )  .

So the group π_0^G(Σ^∞_+ G/K) is generated by the classes 
_L^G( σ^L[e K] ) for all K-conjugacy classes of closed subgroups L≤ K 
that have finite Weyl group in K.
If the Weyl group of L in the ambient group G  happens to be infinite, 
then _L^G=0 and the generator is redundant. 
Otherwise e K is an L-fixed point of G/K, so the generator is one
of the classes mentioned in the statement of (i).
This shows the generating property for the G-space G/K.


Next we observe that whenever the claim is true for a family
of G-spaces, then it is also true for their disjoint union;
this follows from the fact that both fixed points and π_0
commute with disjoint unions, and
that equivariant homotopy groups take wedges to direct sums.
In particular, the claim holds when Y is a disjoint union
of homogeneous space G/H for varying proper closed subgroups H of G.

Now we prove the claim when Y is any G-space
without G-fixed points.
For every proper closed subgroup H of G we choose representatives 
for the path components of the H-fixed point space Y^H. 
These choices determine a continuous G-map

    f :  Z  =  ∐_H ∐_[x]∈π_0(Y^H)  G/H  → Y

by sending g H in the summand indexed by x∈ Y^H to g x.
Because Y has no G-fixed points, the induced map π_0(f^H):π_0(Z^H)→π_0(Y^H)
is then surjective for every closed subgroup H of G, by construction.
In the commutative square

    ⊕_(H)  ( π_0^e(Σ^∞_+ Z^H))_W_G H[r]^-T
    [d]_⊕ (π_0^e(Σ^∞_+ f^H))_W_G H   π_0^G(Σ_+^∞ Z) [d]^π_0^G(Σ^∞_+ f)
    ⊕_(H)  ( π_0^e(Σ^∞_+ Y^H))_W_G H[r]_-T    π_0^G(Σ_+^∞ Y)

the right vertical map is then surjective by
Proposition <ref> (i) for m=0,
and the upper horizontal map is surjective by the above.
So the lower horizontal map is surjective as well.

Now we let Y be any G-space, possibly with G-fixed points. 
The composite

    π_*^e(Σ^∞_+ Y^G)   p_G^*  π_*^G(Σ^∞_+ Y^G)  incl_* π_*^G(Σ^∞_+ Y)   Φ^G   Φ_*^G(Σ^∞_+ Y)

is an isomorphism, i.e., the map incl_*∘ p_G^* splits
the geometric fixed point homomorphism.
The long exact isotropy separation sequence (<ref>) 
thus decomposes into short exact sequences and the map

    (q_*, incl_*∘ p_G^*) : π_*^G(Σ^∞_+ (Y× E_G) ) ⊕π_*^e(Σ^∞_+ Y^G) 
     → π_*^G(Σ^∞_+ Y)

is an isomorphism, where q:Y× E_G→ Y is the projection. 
The G-space Y× E_G has no G-fixed points.
So by the previous part, the group 
π_0^G(Σ^∞_+ (Y× E_G)) is generated by
the classes _H^G(σ^H(x) ) for H with finite Weyl group as above 
and x∈π_0( (Y× E_G)^H).
If H is a proper subgroup, then (E_G)^H is contractible and
so the projection q:Y× E_G→ Y induces a bijection on π_0((-)^H).
So the generators coming from Y× E_G
map to the desired basis elements for Y that are
indexed by proper subgroups of G.
On the other hand, the group π_*^e(Σ^∞_+ Y^G) is free
with basis the classes σ^e(x) for all components x∈π_0(Y^G),
by (<ref>).
Because

    incl_*(p_G^*(σ^e(x))) =  σ^G(x)  ,

the basis of the second summand precisely maps
to the desired basis elements for Y that are
indexed by the group G itself. This completes the argument 
that the map T is surjective.

Now we can complete the proofs of part (i) and (ii).
Since the composite Φ^total∘ T is injective, 
the homomorphism T is injective; since T is also surjective, 
it is bijective, which shows (i) for the group G.
Since T is bijective and Φ^total∘ T is injective, 
the homomorphism Φ^total is injective, which shows (ii) for the group G.




[Equivariant 0-stems and Burnside rings]
We recall that for a finite group G, the Burnside ring A(G)
is the Grothendieck group of the abelian monoid, under disjoint union,
of isomorphism classes of finite G-sets.subjectBurnside ringsubjectsphere spectrum
Every finite G-set is the disjoint union of transitive G-sets,
so A(G) is a free abelian group generated by the classes
of the G-sets G/H, as H runs over all conjugacy classes
of subgroups of G.

The sphere spectrum  is also the unreduced suspension spectrum
of the one-point G-space, Σ^∞_+{0}.
So Theorem <ref> (i) 
says that the map

    ψ_G  :  A(G) → π_0^G() ,    [G/H] ⟼_H^G(1)

is an isomorphism, a result that is originally due to Segal <cit.>.
The Burnside rings for different groups are related by restriction 
homomorphisms α^*:A(G)→ A(K)
along homomorphisms α:K→ G, induced by restriction of
the action along α.
The Burnside rings also enjoy transfer maps

    _H^G :  A(H) → A(G)

induced by sending an H-set S to the induced G-set G×_H S.

The compatibility of transfers with inflations 
(Proposition <ref> (iii)) 
implies that for every surjective homomorphism α the relation

    α^*(ψ_G[G/H])    =  α^*(_H^G(1)) =  _L^K((α|_L)^*(1))
       =  _L^K(1) =  ψ_K([K/L])  =  ψ_K(α^*[G/H])

holds. In other words, the isomorphisms ψ_G are compatible
with inflation. 
The double coset formula (see Example <ref> below),
and the double coset formula in the Burnside rings imply
that the isomorphisms ψ_G are also compatible with transfers. 

If G is a compact Lie group of positive dimension, there is no direct
interpretation of π_0^G() in terms of finite G-sets;
in that situation, some authors define the Burnside ring of G
as the group π_0^G().
In <cit.>, 
tom Dieck gives an interpretation of this Burnside ring 
in terms of certain equivalence classes of compact G-ENRs.
subjecteuclidean neighborhood retract!equivariant 
Theorem <ref> (i) identifies π_0^G()
as a free abelian group with basis the classes _H^G(1)
for all conjugacy classes of closed subgroups H with finite Weyl group.


Theorem <ref> (ii) 
provides a convenient detection criterion for elements in the equivariant
0-stems; as we explain now, this can be rephrased as a degree function.
For every compact Lie group K the 
geometric fixed point group Φ^K_0()
is the 0-th non-equivariant stable stem, and hence free abelian of rank 1,
generated by the class Φ^K(1). 
Moreover, if z∈π_0^K() is represented by the K-map
f:S^V→ S^V for some K-representation V, then

    Φ^K(z) =  [f^K :  S^V^K→ S^V^K] = (f^K)·Φ^K(1) .

So in terms of the basis Φ^K(1), the geometric fixed point
homomorphism Φ^K:π_0^K()→Φ_0^K()
is extracting the degree of the restriction of any representative
to K-fixed points.
We let C(G) denote the set of integer valued, conjugation-invariant functions
from the set of closed subgroup of G with finite Weyl group.
We define a homomorphism

    :  π_0^G() → C(G)

by

    [f:S^V→ S^V](K) =  (f^K :  S^V^K→ S^V^K) .

Theorem <ref> (ii) 
then says that this degree homomorphism is injective.
Whenever the group G is non-trivial, the degree homomorphism
is not surjective. 
In <cit.>,
tom Dieck exhibits an explicit set of congruences that 
together with a certain continuity condition characterize the 
image of the degree homomorphism (<ref>).
When G is finite, these congruences – combined with the isomorphism ψ_G
to the Burnside ring A(G) – specialize to the
congruences between the cardinalities of the various fixed point sets
of a finite G-set, specified for example 
in <cit.>.

We will revisit the equivariant 0-stems from a global perspective in
Example <ref> and with a view towards products
and multiplicative power operations in Example <ref>.




Theorem <ref>
gives a functorial description of the 0-th equivariant stable homotopy
group of an unbased G-space. We will now deduce a similar result
for reduced suspension spectra of based G-spaces.
This version, however, needs a non-degeneracy hypothesis,
i.e., we must restrict to well-pointed G-spaces.




Let G be a compact Lie group and Y a well-pointed G-space.subjectwell-pointed 
Then the group π_0^G(Σ^∞ Y) 
is a free abelian group with a basis given by the elements

    _H^G(σ^H(x)) ,

where H runs through all conjugacy classes of closed subgroups of G
with finite Weyl group and x runs through a set of representatives 
of the W_G H-orbits of the non-basepoint components of the set π_0(Y^H).


If the inclusion of the basepoint i:{y_0}→ Y is an unbased h-cofibration,
then the based map i_+:{y_0}_+→ Y_+ is a based h-cofibration.
The induced morphism of suspension spectra 
Σ^∞_+ i:Σ^∞_+{y_0}→Σ^∞_+ Y 
is then an h-cofibration of orthogonal G-spectra,
so it gives rise to a long exact sequence 
of homotopy groups as in Corollary <ref> (i).
The cokernel of i_+:{y_0}_+→ Y_+ is G-homeomorphic to Y
(with the original basepoint). Also, the map i has a section,
so the long exact sequence degenerates into a short exact sequence:

    0 → π_0^G (Σ^∞_+ {y_0})  (Σ^∞_+ i)_* π_0^G (Σ^∞_+ Y) proj_* π_0^G (Σ^∞ Y)  → 0

Theorem <ref> (i),
applied to the unbased G-spaces {y_0} and Y,
provides bases of the left and middle group; by naturality
the map  (Σ^∞_+ i)_* hits precisely
the subgroup generated by the basis elements coming from basepoint components
of the space Y^H.
So the cokernel of (Σ^∞_+ i)_*,
and hence the group π_0^G (Σ^∞ Y), has a basis of the desired form.


subjectgeometric fixed points|)





§ THE DOUBLE COSET FORMULA


  
The main  aim of this section is to establish
the double coset formula for the composite of a transfer 
followed by a restriction to a closed subgroup,
see Theorem <ref> below.
We also discuss various examples and special cases in 
Examples <ref> through <ref>.
For finite groups (or more generally for transfers along finite index inclusions),
the statement and proof of the double coset formula are significantly simpler,
see Example <ref>,
as this special case does not need any equivariant differential topology.
We end the section with a discussion of Mackey functors for
finite groups, and prove that rationally and for finite groups, 
geometric fixed point homotopy groups can be obtained from equivariant homotopy groups 
by dividing out transfers
from proper subgroups (Proposition <ref>).



For use in the proof of the double coset formula,
and as an interesting result on its own, we calculate
the geometric fixed points of the restriction of any transfer.
In fact, when K=e is the trivial subgroup of G,
then Φ_0^e(X)=π_0^e(X), the map Φ^e is the identity
and the following proposition reduces to a special case of the double coset formula.
For the statement we use that the action of a closed subgroup K
on a homogeneous space G/H by left translation is smooth,
and hence the fixed points (G/H)^K form a disjoint union of
closed smooth submanifolds, possibly of varying dimensions.

We let K be a compact Lie group and B a closed smooth K-manifold.
The Mostow-Palais embedding theorem <cit.>
provides a smooth K-equiva­riant embedding i: B→ V, 
for some K-representation V. 
We can assume without loss of generality that V is a subrepresentation
of the chosen complete K-universe _K.
We use the inner product on V to define the normal bundle ν 
of the embedding at b∈ B by

    ν_b  =  V - (d i)(T_b B) ,

the orthogonal complement of the image of the tangent space T_b B in V.
By multiplying with a suitably large scalar, if necessary, 
we can assume that the embedding
is widesubjectwide embeddingsubjectembedding!wide|seewide embedding
in the sense that the exponential map

    D(ν)  → V  ,    (b,v) ⟼ i(b)   +   v

is injective on the unit disc bundle of the normal bundle, 
and hence a closed K-equivariant embedding.
The image of this map is a tubular neighborhood of radius 1 around i(B),
and it determines a K-equivariant Thom-Pontryagin collapse map

    c_B :  S^V  →  S^V B_+

as follows: every point outside of the tubular neighborhood is sent
to the basepoint, and a point i(b)+ v, for (b,v)∈ D(ν),
is sent to 

    c_B(i(b) +  v)  =  ( v/1-|v|)   b  .

The homotopy class of the map c_B is then an element in
the equivariant homotopy group π_0^K(Σ^∞_+ B).
The next result determines the image of the class [c_B]
under the geometric fixed point map.

In (<ref>) 
we defined the map σ^K:π_0(B^K)→π_0^K(Σ^∞_+ B) that
produces equivariant stable homotopy classes from fixed point information.


Let K be a compact Lie group.


  * For every closed smooth K-manifold B the relation
  
    Φ^K[c_B] =  ∑_ M∈π_0( B^K)  χ(M)·Φ^K(σ^K[M])

  holds in the group Φ_0^K(Σ^∞_+ B),
  where the sum runs over the connected components of the fixed point manifold B^K.

  * Let G be a compact Lie group containing K and let H be another
  closed subgroup of G. Then
  
    Φ^K∘^G_K∘_H^G  =  ∑_ M∈π_0( (G/H)^K)  χ(M)·Φ^K∘ g_⋆∘^H_K^g ,

  where the sum runs over the connected components of the fixed point
  space (G/H)^K and g∈ G is such that g H∈ M.



(i)
Since the composite

    π_0^e(Σ^∞_+ B^K )  p_K^*  π_0^K(Σ^∞_+ B^K)  incl_* π_0^K(Σ^∞_+ B) 
      Φ^K  Φ_0^K(Σ^∞_+ B)

is an isomorphism and the source is free abelian on the path components
of the space B^K,
the group Φ_0^K(Σ^∞_+ B) is free abelian with basis
given by the classes

    Φ^K(σ^K[M])  =  Φ^K(incl_*(p_K^*(σ^e[M])))

for M∈π_0(B^K).
We let i: B→ V be a wide smooth K-equivariant embedding into a K-representation. 
Then the class Φ^K[c_B] is represented by the non-equivariant map

    S^V^K  (c_B)^K   S^V^K B^K_+  .

Since K acts smoothly on B, the fixed points B^K 
are a disjoint union of finitely many closed smooth submanifolds.
For each component M of B^K we let p_M:B^K_+→ M_+ denote
the projection, i.e., p_M is the identity on M and sends all
other path components of B^K to the basepoint.
Then the composite

    S^V^K  (c_B)^K   S^V^K B^K_+  S^V^K p_M  S^V^K M_+

coincides with 

    S^V^K  c_M   S^V^K M_+  ,

the collapse map (<ref>) 
based on the non-equivariant wide smooth embedding (i^K)|_M:M→ V^K.
Since M is path connected, the group of based homotopy classes
of maps [S^V^K,S^V^K M_+] is isomorphic to ,
by (<ref>),
and an element is determined by the degree of the composite
with the projection S^V^K M_+→ S^V^K.
It is a classical fact that the degree of the composite

    S^V^K  c_M   S^V^K M_+  proj S^V^K

is the Euler characteristic of the manifold M, 
see for example <cit.>.
So the summand indexed by M precisely contributes the term
χ(M)·Φ^K(σ^K[M]), which proves the desired relation.

(ii)
Both sides of the equation are natural transformations 
on the category of orthogonal G-spectra from the functor π_0^H
to the functor Φ^K_0. Since the functor π_0^H is
represented by the suspension spectrum of G/H 
(in the sense of Proposition <ref>),
is suffices to check the relation for the orthogonal G-spectrum Σ^∞_+ G/H
and the tautological class e_H defined in (<ref>).
Inspection of the definition in Construction <ref>
reveals that the transfer 
of the tautological class _H^G(e_H) in π_0^G(Σ^∞_+ G/H)
is represented
by the G-map

    S^V   c   G⋉_H S^W  a   S^V G/H_+

where c is the collapse map based on any wide embedding
of i:G/H→ V into a G-representation,
and a[g,w]=g w g H.
So the class ^G_K(_H^G(e_H)) is represented by the underlying
K-map of the above composite, which is precisely the map c_G/H 
for the underlying K-manifold of G/H. Part (i) thus provides the relation
  
    Φ^K(^G_K(_H^G(e_H))) =  Φ^K[c_G/H] =  ∑_ M∈π_0( (G/H)^K)  χ(M)·Φ^K(σ^K[M])

  where the sum runs over the connected components of (G/H)^K. 
  On the other hand, if g∈ G is such that g H∈ M⊂ (G/H)^K, then
  K^g≤ H and
  
    σ^K [M]     =  g_⋆ (σ^K^ge H)     =  g_⋆( ^H_K^g(e_H)) ;

  this proves the desired relation for the universal class e_H.





Now we can proceed towards the double coset formula
for a transfer followed by a restriction.
The double coset formula was first proved by Feshbach 
for Borel cohomology theories <cit.> 
and later generalized to equivariant cohomology theories by 
Lewis and May <cit.>. 
I am not aware of a published account of the double coset formula
in the context of orthogonal G-spectra.
Our method of proof for the double coset formula is different
from the approach of Feshbach and Lewis-May: 
we verify the double coset formula after passage to geometric fixed points 
for all closed subgroups. 
The detection criterion of Theorem <ref> (ii) 
then lets us deduce the  double coset formula for the universal example,
the tautological class e_H in π_0^H(Σ^∞_+ G/H).


Before we can state the double coset formula we have to recall 
some additional concepts, such as orbit type submanifolds and 
the internal Euler characteristic.


Let L be a closed subgroup of a compact Lie group K and B a K-space.
The orbit type space associated to the conjugacy class of L is
the K-invariant subspace

    B_(L) = {  x ∈  B  |  the stabilizer group of x is conjugate to L} .



The points in the space B_(L) are said to have `orbit type' the conjugacy
class (L).
We mostly care about orbit type subspaces for  equivariant smooth manifolds.
Two sources that collect general information about orbit type manifolds
are Chapters IV and VI of Bredon's book <cit.>
and Section I.5 of tom Dieck's book <cit.>.
If B is a smooth K-manifold, then the subspace B_(L)
is a smooth submanifold of B, and locally closed in B, 
compare <cit.>
or <cit.>.subjectorbit type manifold
Thus B_(L) is called the orbit type manifold.
If the smooth K-manifold is compact, 
then it only has finitely many orbit types –
this was first shown by Yang <cit.>; 
other references 
are <cit.>, 
<cit.>, 
or <cit.>.
The orbit type manifolds B_(L) need not be closed inside B;
but if one orbit type manifold B_(L) lies in the closure of another one
B_(L'), then L' is subconjugate to L in K, 
compare <cit.>.
For every conjugacy class (L), the quotient map B_(L)→ K B_(L)
is a locally trivial smooth fiber bundle with fiber K/L and structure group
the normalizer N_K L, compare <cit.>.
Thus every path component of every orbit space K B_(L) of
an orbit type manifold is again a smooth manifold, 
in such a way that the quotient maps are smooth submersions.
In particular, if the action happens to have only one orbit type
(i.e., all stabilizer groups are conjugate), then the quotient space K B 
is a manifold and inherits a smooth structure from B.

The terms in the double coset formula will be indexed by path components
of the orbit type orbit manifolds K B_(L).
These manifolds need not be connected, and different components may
have varying dimensions, but each K B_(L)
has only finitely many path components.
Indeed, by <cit.>
the orbit space K B admits a triangulation
such that the orbit type is constant on every open simplex.
So if a path component M of K B_(L) intersects a particular open simplex,
then that simplex is entirely contained in M, i.e.,
M is a union of open simplices in such a triangulation.
Since K B is compact, any triangulation has only finitely many simplices.


[Internal Euler characteristic]
We continue to consider a compact Lie group K acting smoothly on
a closed smooth manifold B. We let L be a closed subgroup of K
and M⊂ K B_(L) a connected component of
the orbit type manifold for the conjugacy class (L).
We let M̅ denote the closure of M inside K B
and δ M=M̅-M the complement of M inside its closure. 
Since K B is compact, so is M̅. Since K B_(L)
is locally closed in K B, the set M is open inside its closure M̅.
So the complement δ M is closed in M̅, hence compact.
One should beware that while M is a topological manifold (without boundary,
and typically not compact),
M̅ need not be a topological manifold with boundary.

By <cit.>
the orbit space K B admits a triangulation (necessarily finite)
such that the orbit type is constant on every open simplex.
So if the orbit type component M intersects a particular open simplex,
then that simplex is entirely contained in M, i.e.,
M is a union of open simplices in such a triangulation.
This means that M̅ is obtained from M by adding
some simplices of smaller dimension. 
Hence M is a simplicial subcomplex of any such triangulation,
and δ M is a subcomplex of M.
In particular, M̅ and δ M admit the structure of finite simplicial complexes
of dimension less than or equal to the dimension of B.
Thus the integral singular homology groups 
of M̅ and δ M are finitely generated in every degree, and they
vanish above the dimension of B.
So the Euler characteristics of M̅ and δ M
are well-defined integers. 
The internal Euler characteristicsubjectEuler characteristic!internal
of M is then defined as 

    χ^♯(M) = χ(M̅) - χ(δ M) .

The internal Euler characteristic χ^♯(M) also
has an intrinsic interpretation that does not refer to the ambient space K B.
Indeed since the integral homology groups of M̅ and δ M
are finitely generated and vanish for almost all degrees, 
the same is true for the relative singular homology groups H_*(M̅,δ M;),
and the internal Euler characteristic satisfies

    χ^♯(M) = ∑_n≥ 0 (-1)^n·rank(H_n(M̅,δ M;))
     = ∑_n≥ 0 (-1)^n·rank(H^n(M̅,δ M;)) .

The second equality uses that by the universal coefficient theorem,
we can use co-homology instead of homology to calculate the Euler characteristic.
Since δ M is a simplicial subcomplex of M̅,
it is a neighborhood deformation retract inside M̅.
So the relative cohomology group H^n(M̅,δ M;)
is isomorphic to the reduced cohomology group of the quotient space
M̅/δ M, by excision. This quotient space is homeomorphic
to M̂, the one-point compactification of M=M̅-δ M.
Since M is a topological manifold, it is locally compact, and so 
the cohomology groups of M̂ are isomorphic 
to the compactly supported cohomology groups of M. 
Altogether, this provides isomorphisms

    H^n(M̅,δ M;)  H^n_c(M;) .

So we conclude that the internal Euler characteristic χ^♯(M) 
can also be defined as the compactly supported Euler characteristic of M, 
i.e., 

    χ^♯(M) =  ∑_n≥ 0 (-1)^n·rank(H^n_c(M;)) .





Now we let K and H be two closed subgroups of a compact Lie group G.
Then the homogeneous space G/H is a smooth manifold
and the K-action on G/H by left translations is smooth.
The double coset spacesubjectdouble coset space 
K\ G/HsymbolK\ G/H - double coset space 
is the quotient space of G/H by this K-action, 
i.e., the quotient of G by the left K- and right H-action by translation.
In contrast to a homogeneous space G/H, the double coset space is in general
not a smooth manifold. However, the orbit type decomposition expresses
the double coset space as the union of certain subspaces that are
manifolds of varying dimensions.
Indeed, G/H decomposes into orbit type manifolds with respect to K:

    K G/H  =  ⋃_(L)  K (G/H)_(L) ,

where the set-theoretic union is indexed by conjugacy classes of closed
subgroups of L; all except finitely many of the subspaces K (G/H)_(L) are empty.
The double coset formula expresses the composite ^G_K∘_H^G
as a sum of terms, indexed by all connected components M 
of orbit type orbit manifolds K (G/H)_(L).
The coefficient of the contribution of M is the 
internal Euler characteristic χ^♯(M). 



Our next result is a technical proposition that contains the essential
input to the double coset formula from equivariant differential topology.


Let K be a compact Lie group, B a closed smooth K-manifold and
L≤ J≤ K nested closed subgroups. 
Let N be a connected component of the fixed point space B^L,
and M a connected component of the orbit space K B_(J).
Let b∈ B be any point with stabilizer group J and with K b∈ M.
Let W[M,N] be the preimage of N∩ B_(J) under the map

    (K/J)^L  → (B_(J))^L  ,   k J  ⟼ k b .

Then

    χ^♯(N∩ B_(J))  =  ∑_M∈π_0(  K B_(J))χ^♯(M)·χ(W[M,N]) .



The projection

    B_(J)  p   K B_(J)

is a smooth fiber bundle with fiber K/J and structure group N_K J, compare
<cit.> or <cit.>.
Since L is contained in J the composite

    (B_(J))^L  incl B_(J)  p   K B_(J)

is surjective and the restriction of p to
(B_(J))^L is another smooth fiber bundle

    (B_(J))^L  →  K B_(J) ,

now with fiber (K/J)^L, and the map

    Ψ_M :  (K/J)^L  → ( B_(J))^L  ,     k J  ⟼ k b

is the inclusion of the fiber over K b ∈ M.

For every connected component M of the base K B_(J)
the inverse image p^-1(M) is open and closed in (B_(J))^L.
For every connected component N of B^L
the intersection N∩ B_(J) is open and closed in (B_(J))^L.
So the subset p^-1(M)∩ N is open and closed in (B_(J))^L.
As M varies, the subsets p^-1(M)∩ N 
cover all of N∩ B_(J).
Moreover, the restriction of the projection to a map

    p  :  p^-1(M)∩ N  → M

is a smooth fiber bundle with connected base and with fiber W[M,N],
by definition of the latter.
The internal Euler characteristic is multiplicative on smooth fiber bundles
with closed fiber, so

    χ^♯(p^-1(M)∩ N) = χ^♯(M)·χ(W[M,N]) .

The internal Euler characteristic is additive on disjoint unions, so

    χ^♯ ( N∩  B_(J)  )    =  ∑_M∈π_0 ( K B_(J))χ^♯ ( N∩ p^-1(M) )    =   
    ∑_M∈π_0 ( K B_(J))χ^♯ (M)·χ(W[M,N]) .





For later reference we point out a direct consequence of 
Proposition <ref>
for L=e, the trivial subgroup of K.
We let J be any closed subgroup of K, and M a connected component
of K B_(J).
The space K/J is the disjoint union of its subspaces W[M,N], 
as N varies over the connected components of B.
So summing the formula of Proposition <ref>
over all components of B yields

    χ^♯(B_(J))  =  ∑_M∈π_0(  K B_(J))χ^♯(M)·χ( K/J ) ,

by additivity of Euler characteristics for disjoint unions.
Because χ(B) is the sum of the internal Euler characteristics
χ^♯(B_(J)) over all conjugacy classes of closed subgroups of K,
summing up over conjugacy classes gives the formula

    χ(B)  =  ∑_(J)≤ K ∑_M∈π_0(  K B_(J))χ^♯(M)·χ( K/J ) .





We let K be a compact Lie group and B a closed smooth K-manifold.
The Mostow-Palais embedding theorem <cit.>
provides a wide smooth K-equivariant embedding i: B→ V, 
for some K-representation V. 
The associated collapse map 

    c_B :  S^V  →  S^V B_+

was defined (<ref>).
The homotopy class of the map c_B is then an element in
the equivariant homotopy group π_0^K(Σ^∞_+ B).
Theorem <ref> (i)
exhibits a basis of the  group π_0^K(Σ^∞_+ B),
and the next theorem expands the class of c_B in that basis. 
As we shall explain in the proof of Theorem <ref> below, 
the double coset formula for ^G_K∘_H^G
is essentially the special case B=G/H of the following theorem.
A compact smooth K-manifold only has finitely many orbit types,
so the sum occurring in the following theorem is finite.



Let K be a compact Lie group and B a closed smooth K-manifold.
Then the relation

    [c_B]  =  ∑_(J)≤ K∑_M∈π_0(K B_(J))  χ^♯(M)·_J^K (σ^J b_M )

holds in the group π_0^K(Σ^∞_+ B).
Here the sum runs over all connected components M 
of all orbit type orbit manifolds K B_(J), 
and the element b_M∈ B that occurs is such that K b_M∈ M
and with stabilizer group J, and b_M is the path component of b_M
in the space B^J.


Because the desired relation lies in the 0-th equivariant homotopy group 
of a suspension spectrum, 
Theorem <ref> (ii) 
applies and shows that we only need to verify the formula
after taking geometric fixed points to any closed subgroup L of K.
We let J be another closed subgroup of K containing L.
We calculate the contribution of the conjugacy class (J)
to the effect of Φ^L on the right hand side of the formula.
For every connected component M of K B_(J) we choose an 
element b_M∈ B with stabilizer group J and K b_M∈ M.
Then

    ∑_M ∈π_0( K B_(J))   χ^♯(M)·Φ^L (^K_L( _J^K(σ^Jb_M)))
     
    =     ∑_M ∈π_0( K B_(J)) ∑_W∈π_0( (K/J)^L) χ^♯(M)·χ(W)·Φ^L ( k_⋆(^J_L^k(σ^Jb_M))) 
    
    =     ∑_M ∈π_0( K B_(J)) ∑_W∈π_0( (K/J)^L) χ^♯(M)·χ(W)·Φ^L (σ^L(k_⋆(^J_L^kb_M))) 
    
    =     ∑_M ∈π_0( K B_(J))∑_N∈π_0(B^L)χ^♯(M)·χ(W[M,N])·Φ^L (σ^L[N]) 
    
    =     ∑_N∈π_0(B^L)χ^♯ ( N∩  B_(J))  ·Φ^L (σ^L[N])  .

The first equation is Proposition <ref> (ii),
applied to the closed subgroups L and J of K.
The element k∈ K is chosen so that k J∈ W⊂ (K/J)^L;
in particular this forces the relation L≤^k J≤^kstab(b_M),
and thus k b_M∈ B^L.
The third equation uses that for every M, 

    (K/J)^L  =  ∐_N∈π_0(B^L)  W[M,N]  ,

that the Euler characteristic is additive on disjoint unions,
and that for all k J∈ W[M,N] the element k b_M 
lies in the path component N of B^L, by the very definition of W[M,N].
The fourth equation is Proposition <ref>.

Now we sum up the contributions from the different conjugacy classes of
subgroups of K. 
The smooth K-manifold B is stratified by the 
relatively closed smooth submanifolds B_(J),
indexed over the poset of conjugacy classes of subgroup of K, 
where only finitely many orbit types occur.
Intersecting the strata with a connected component N 
of the fixed point submanifold B^L
gives a stratification of N by relatively closed
submanifolds, still indexed over the conjugacy classes of subgroup of K. 
The internal Euler characteristic is 
additive for such stratifications, i.e.,

    χ( N )  =  ∑_(J)≤ K χ^♯( N∩  B_(J) )  .

An application of Proposition <ref> (i) gives

    Φ^L(^K_L [c_B] )     = ∑_N∈π_0(B^L)χ(N)·Φ^L(σ^L[N]) 
       =  ∑_(J)≤ K∑_N∈π_0(B^L)χ^♯( N∩ B_(J))·Φ^L(σ^L[N]) 
       =  ∑_(J)≤ K∑_M ∈π_0( K B_(J))χ^♯(M)·Φ^L(^K_L(_J^K(σ^Jb_M))) 
       =  Φ^L(^K_L( ∑_(J)≤ K∑_M ∈π_0( K B_(J)) χ^♯(M)·_J^K(σ^Jb_M) )) .

This proves the desired formula after composition
with the geometric fixed point map to any closed subgroup of K.



Now we have all ingredients for the statement and proof of the double
coset formula in place, which gives an expression for
the operation ^G_K∘_H^G on the equivariant homotopy
groups of an orthogonal G-spectrum X, as a sum indexed by the path components M
of the orbit type orbit manifolds K (G/H)_(L).
The operation associated to M 
involves the transfer _K∩^g H^K, where g∈ G 
is an element such that K g H∈ M;
since K∩^g H is the stabilizer of the coset g H, 
the group K∩^g H lies in the conjugacy class (L).
The group K∩^g H may have infinite index in its normalizer in K, 
in which case _K∩^g H^K=0, and then the corresponding
summand in the double coset formula vanishes.



Let H and K be closed subgroupssubjectdouble coset formula
of a compact Lie group G. Then for every orthogonal G-spectrum X the relation

    ^G_K∘_H^G  =  ∑_Mχ^♯(M)·_K∩^g H^K  ∘ g_⋆∘^H_K^g∩ H

holds as homomorphisms π_0^H(X)→π_0^K(X).
Here the sum runs over all connected components M 
of all orbit type orbit manifolds K  (G/H)_(L), 
and g∈ G is an element such that K g H∈ M.


Both sides of the double coset formula are natural transformations 
on the category of orthogonal G-spectra from the functor π_0^H
to the functor π^K_0. Since the functor π_0^H is
represented by the suspension spectrum of G/H 
(in the sense of Proposition <ref>),
is suffices to check the relation for the orthogonal G-spectrum Σ^∞_+ G/H
and the tautological class e_H defined in (<ref>).

Inspection of the definition in Construction <ref>
reveals that the transfer 
of the tautological class _H^G(e_H) in π_0^G(Σ^∞_+ G/H)
is represented
by the G-map

    S^V   c   G⋉_H S^W  a   S^V G/H_+

where c is the collapse map based on any wide embedding
i:G/H→ V into a G-representation, and a[g,w]=g w g H.
So the class ^G_K(_H^G(e_H)) is represented by the underlying
K-map of the above composite, which is precisely the map c_G/H 
for the underlying K-manifold of G/H.

Theorem <ref> thus yields the formula

    ^G_K(_H^G(e_H)) =  [c_G/H]  =  ∑_(J)≤ K∑_M∈π_0(K (G/H)_(J)) χ^♯(M)·_J^K (σ^J g_M H )

in the group π_0^K(Σ^∞_+ G/H).
Here g_M∈ G is such that K g_M H∈ M
and K∩^g_M H=J. On the other hand,

    σ^J g_M H    =  (g_M)_⋆ (σ^J^g_Me H)
       =  (g_M)_⋆( ^H_J^g_M(σ^He H))=  (g_M)_⋆( ^H_K^g_M∩ H(e_H)) ,

so this proves the double coset formula for the universal class e_H.





 
A special case of the double coset formula is when K=e,
i.e., when we restrict a transfer all the way to the trivial subgroup of G.
In this case the orbit type component manifolds are the path components
of the coset space G/H, and the sum in the double coset
formula is indexed by these.
Since all path components of G/H are homeomorphic, 
they have the same Euler characteristic χ(G/H)/|π_0(G/H)|, 
so the double coset formula specializes to

    ^G_e∘_H^G     =  ∑_ M∈π_0 (G/H)  χ(M)·  g_⋆∘^H_e 
       =  χ(G/H)/|π_0(G/H)| ·∑_ [g H]∈π_0 (G/H)    g_⋆∘^H_e  .

In the special case where the conjugation action of π_0(G) on π_0^e(X)
happens to be trivial (for example for all global stable homotopy types),
all the maps g_⋆ are the identity and the formula simplifies to

    ^G_e∘_H^G  =  χ( G/H)·^H_e  .




[Double coset formula for finite index transfers]
If H has finite index in G,subjectdouble coset formula!finite index 
then the double coset formula simplifies.
In this situation the homogeneous space G/H, and
hence also the double coset space K G/H, is discrete,
all orbit type manifold components are points, and all internal Euler characteristics
that occur in the double coset formula are 1.
Since the intersection K∩^g H also has finite index in K, 
only finite index transfers show up in the double coset formula, 
which specializes to the relation

    ^G_K∘_H^G  =  ∑_[g]∈ K\ G/H_K∩^g H^K ∘ g_⋆∘^H_K^g∩  H .






We calculate the double coset formula for the maximal torus 

    H = {[ λ 0; 0 μ ] :  λ,μ∈ U(1) }

of G=U(2).
We take K=N_U(2) H, the normalizer of the maximal torus; 
this is Example VI.2 in <cit.>.subjectunitary group!U(2)
Then K is isomorphic to Σ_2⋉ H, the semidirect product 
with Σ_2 permuting the two diagonal entries of matrices in H;
the group K is generated by H and the involution 
([ 0 1; 1 0 ]).

We calculate the double coset space K  U(2) / H
by identifying the space H U(2) / H 
and the residual action of the symmetric group Σ_2 on this space.
A homeomorphism from H U(2) / H to the unit interval [0,1] is induced by

    h  :  U(2)   → [0,1]  ,   
     A  ⟼ |a_11|^2  ,

the square of the length of the upper left entry a_11 of A.
Lengths are non-negative and every column of a unitary matrix is a unit vector,
so the map really lands in the interval [0,1].
The number h(A) only depends on the
double coset of the matrix A, so the map h factors over
the double coset space H U(2) / H.
For every x∈ [0,1] 
the vector (√(x),√(1-x))∈^2 
has length 1, so it can be complemented to an orthonormal basis,
and so it occurs as the first column of a unitary matrix;
the map h is thus surjective.
On the other hand, if h(A)=h(B) for two unitary matrices A and B,
then left multiplication by an element in H makes the first row
of B equal to the first row of A. Right multiplication by
an element of 1× U(1) then makes the matrices equal.
So A and B represent the same element in the double coset space.
The induced map

    h̅ :  H  U(2) / H  → [0,1]

is thus bijective, hence a homeomorphism.
The action of Σ_2 on the orbit space
H  U(2) / H permutes the two rows of a matrix; 
under the homeomorphism h̅, this action thus 
corresponds to the involution of [0,1] sending x to 1-x.
Altogether this specifies a homeomorphism 
from the double coset space to the interval [0,1/2]
that sends a double coset K· A· H
to the minimum of |a_11|^2 and |a_21|^2.
The inverse homeomorphism is 

    g  :  [0, 1/2]  K  U(2) / H ,   
    g(t) =   K·[ √(1-t)   √(t);  -√(t) √(1-t) ]· H .

The orbit type decomposition is as

    {0}∪ (0,1/2)∪{1/2} .

So as representatives of the orbit types we can choose

    g(0)    = [ 1 0; 0 1 ] ,  
     g(1/5)= 1/√(5)[  2  1; -1  2 ]   and  
     g(1/2)= 1/√(2)[  1  1; -1  1 ] .

For the intersections (which are representatives of the conjugacy classes
of those subgroups of K with non-empty orbit type manifolds) we get

     K       ∩  ^g(0) H       =  H  , 
  
     K       ∩^g(1/5) H       =  Δ , and
  
     K       ∩^g(1/2) H       =  Σ_2×Δ .

Here Δ={ 
([ λ 0; 0 λ ])  :  λ∈ U(1) }
is the diagonal copy of U(1).
The group Δ is normal in K, so 
it has infinite index in its normalizer, and the corresponding 
transfer does not contribute to the double coset formula.
The internal Euler characteristic of a point is 1, so
the double coset formula has two non-trivial summands, and it specializes to 

    ^U(2)_K ∘ _H^U(2) =  _H^K +  _Σ_2×Δ^K ∘γ_⋆∘^H_(Σ_2×Δ)^γ ,

where γ=g(1/2)=1/√(2)([ -1  1; -1  1 ]).




 
For later use in the `explicit Brauer induction'
(see Remark <ref> below), 
we work out another double coset formula for
the unitary group G=U(n).
We write U(k,n-k) for the subgroup of those elements of U(n) 
that leave the subspaces ^k⊕ 0
and 0⊕^n-k invariant. We also use the analogous notation for more than two
factors. As subgroups we take

    H = U(1,n-1)      and     
    K  =  U(k,n-k) ,

where 1≤ k≤ n-1.subjectunitary group
Again the relevant double coset space `is' an interval.
Indeed, the continuous map

    U(n)   → [0,1]   ,   
     A  ⟼ |a_1 1|^2 + … +  |a_k 1|^2  ,

the partial length of the entries in the first column of A,
is invariant under right multiplication by elements of U(1,n-1),
and under left multiplication by elements of U(k,n-k).
Every column of a unitary matrix is a unit vector,
so the map really lands in the interval [0,1].
The map factors over a continuous surjective map

    h  :  U(k,n-k)  U(n) / U(1,n-1)   → [0,1]  .

On the other hand, if h[A]=h[B] for two unitary matrices A and B,
then left multiplication by an element in U(k,n-k) makes the first
columns of A and B equal.
Right multiplication by an element of U(1,n-1) then makes the matrices equal.
So A and B represent the same element in the double coset space.
This shows that the map h is bijective, hence a homeomorphism.
The inverse homeomorphism is 

    [0, 1]  U(k,n-k)  U(n) / U(1,n-1)  ,  
    t  ↦ U(k,n-k)· g(t) · U(1,n-1)

with

    g(t) =  [    √(t)       0       ⋯       0  √(1-t);       0       1       ⋯       0       0;       ⋮       ⋮       ⋱       ⋮       ⋮;       0       0       ⋯       1       0; -√(1-t)       0       ⋯       0    √(t) ] .

The orbit type decomposition is as {0}∪ (0,1)∪{1} .
As representatives of the orbit types we can choose 

    g(0)= [  0  0  ⋯  0  1;  0  1  ⋯  0  0;  ⋮  ⋮  ⋱  ⋮  ⋮;  0  0  ⋯  1  0; -1  0  ⋯  0  0 ] ,   
    g(1/2)= 1/√(2)[  1  0  ⋯  0  1;  0  1  ⋯  0  0;  ⋮  ⋮  ⋱  ⋮  ⋮;  0  0  ⋯  1  0; -1  0  ⋯  0  1 ]

and the identity g(1).
For the intersections (which are representatives of the conjugacy classes
of those subgroups of U(k,n-k) with non-empty orbit type manifolds) we get

    U(k,n-k)        ∩  ^g(0) U(1,n-1) 
      =  U(k, n-k-1, 1)
  
    U(k,n-k)        ∩  ^g(1/2) U(1,n-1)       =  Δ
  
    U(k,n-k)        ∩  ^g(1) U(1,n-1) 
      =  U(1, k-1, n-k) .

Here Δ is the subgroup of U(n) consisting of matrices of the form

    (λ    0     0     0
     
        0     A     0     0
     
        0     0     B     0
     
        0     0     0    λ)

for (λ,A,B)∈ U(1)× U(k-1)× U(n-k-1).
The subgroup Δ is normalized by U(1,  k-1,  n-k-1,  1),
so it has infinite index in its normalizer, and the corresponding 
transfer does not contribute to the double coset formula.
The double coset formula thus
has two non-trivial summands, and comes out as

    ^U(n)_U(k,n-k) ∘ _U(1,n-1)^U(n)    = 
    _U(1,k-1,n-k)^U(k,n-k)   ∘^U(1,n-1)_U(1,k-1,n-k)  + _U(k,n-k-1,1)^U(k,n-k)∘ g(0)_⋆∘^U(1,n-1)_U(1,k,n-k-1) .









subjectMackey functor|(
For the rest of this section we restrict our attention to finite groups.
For easier reference we record that the equivariant homotopy groups
of an orthogonal G-spectrum form a Mackey functor.
We first recall one of several equivalent definitions of this concept.

subjectMackey functor
Let G be a finite group. A G-Mackey functorsubjectG-Mackey functor@G-Mackey functor|seeMackey functor 
M consists of the following data:


  * an abelian group M(H) for every subgroup H of G,

  * conjugation homomorphisms g_⋆:M(H^g)→ M(H) for all H≤ G 
and g∈ G,subjectconjugation homomorphism!in a Mackey functor  

  * restriction homomorphisms ^H_K:M(H)→ M(K) for all K≤ H≤ G, andsubjectrestriction homomorphism!in a Mackey functor  

  * transfer homomorphisms ^H_K:M(K)→ M(H) for all K≤ H≤ G.subjecttransfer!in a Mackey functor  

This data has to satisfy the following conditions:


  * (Unit conditions)

    _H^H  =  _H^H =  _M(H)

for all subgroups H, and h_⋆ =_M(H) for all h∈ H.

  * (Transitivity conditions)

    ^K_L∘^H_K  =  ^H_L 
       and   ^H_K∘_L^K  =  ^H_L

for all L≤ K≤ H≤ G.

  * (Interaction conditions)

    g_⋆∘ g'_⋆ =  (g g')_⋆ ,   ^H_K∘ g_⋆ =  g_⋆∘^H^g_K^g   and   ^H_K∘ g_⋆ =  g_⋆∘^H^g_K^g

for all g,g'∈ G and K≤ H≤ G.

  * (Double coset formula)
for every pair of subgroups K, L of H the relation

    ^H_L∘_K^H   =  ∑_[h]∈ L\ H/K ^L_L∩ ^h K∘ h_⋆∘^K_L^h∩ K

holds as maps M(K)→ M(L);
here [h] runs over a set of representatives for the
L-K-double cosets.

A morphism of G-Mackey functorssubjectmorphism!of G-Mackey functors
 f:M→ N is a collection
of group homomorphisms f(H):M(H)→ N(H), for all subgroups H of G,
such that

    ^H_K∘ f(H)  =  f(K)∘^H_K   and  ^H_K∘ f(K)  = f(H)∘^H_K

for all K≤ H≤ G, and

    g_⋆∘ f(H^g)  =  f(H)∘ g_⋆

for all g∈ G.



We denote the category of G-Mackey functors by G.symbol  G - category of G-Mackey functors 
To my knowledge, the concept of a Mackey functor goes back to Dress <cit.>
and Green <cit.>.
Green in fact defined what is nowadays called a Green functorsubjectGreen functor
(which he called G-functor in <cit.>),
which amounts to G-Mackey functor whose values underlie commutative rings,
where restriction and conjugation maps are ring homomorphisms,
and where the transfer maps satisfy Frobenius reciprocity.
Definition <ref> is the `down to earth' definition
of Mackey functor, and the one that is most useful for concrete calculations.
There are two alternative (and equivalent) definitions that are often used:


  * As a pair (M_*,M^*) of additive functors on the category of finite G-sets,
where M_* is covariant and M^* is contravariant;
the two functors must agree on objects
and for every pullback diagram of finite G-sets
    
    A [r]^-f [d]_g     B [d]^h
    
          C [r]_-k     D

the relation

    M_*(f)∘ M^*(g)  =  M^*(h)∘ M_*(k) :  M(C) → M(B)

holds. This is (a special case of) the definition introduced by Dress <cit.>.

  * As an additive functor on the category of spans of finite G-sets;
the equivalence with Dress' definition is due to Lindner <cit.>.



Our main source of examples of G-Mackey functors comes from 
orthogonal G-spectra: as we verified 
in Sections <ref>
and <ref>
and Theorem <ref>,
the restriction, conjugation and transfer maps 
make the homotopy groups π_k^H(X) for varying subgroups H
into a G-Mackey functor.



Proposition <ref> above
shows that the geometric fixed point map (<ref>)
factors over the quotient of π_0^G(X) by the subgroup
generated by proper transfers. This should serve as motivation for the
following algebraic interlude about Mackey functors for finite groups,
where we study the process of `dividing out transfers' systematically.



We let G be a finite group and M a G-Mackey functor.
For a subgroup H of G we let  t_H M be the subgroup of M(H) 
generated by transfers from proper subgroups of H, and we define

    τ_H M  =  M(H) / t_H M .

For g∈ G, the conjugation isomorphism
g_⋆: M(H^g)→ M(H) descends to a homomorphism 

    g_⋆ :  τ_H^g M → τ_H  M  .

Moreover, the transitivity 
relation g_⋆∘ g'_⋆=(g g')_⋆
still holds. In particular, the action of the Weyl group W_G H on M(H)
descends to a W_G H-action on the quotient group τ_H M.

Now we recall how a G-Mackey functor M can rationally be
recovered from the W_G H-modules τ_H M for all subgroups H of G.
We let

    ψ̅^M_G  :  M(G) → ∏_H≤ G τ_H M

be the homomorphism whose H-component is the composite

    M(G) ^G_H M(H) proj τ_H M   ;

here the product is indexed over all subgroups H of G.
The group G acts on the product via the maps (<ref>),
permuting the factors within conjugacy classes.
Since inner automorphisms induce the identity we have
g_⋆∘^G_H^g=^G_H∘ g_⋆=^G_H, 
so the map ψ̅^M_G lands in the subgroup of G-invariant tuples.
We let

    ψ^M_G  :  M(G) → ( ∏_H≤ G τ_H M)^G

denote the same map as ψ̅^M_G, but now with image the subgroups
of G-invariants.
While the above description of the target of ψ^M_G is the most natural one,
it is somewhat redundant.
Indeed, for every H≤ G, the image of the restriction map 
^G_H:M(G)→ M(H) lands in the subgroup
M(H)^W_G H of invariants under the Weyl group W_G H.
Moreover, if we choose representatives of the conjugacy 
classes of subgroups, then projection from the full product
(over all subgroups of G) to the product indexed by the representatives restricts
to an isomorphism

    ( ∏_H≤ G τ_H M)^G     ∏_(H) (τ_H M)^W_G H .

For explicit calculations, the second description of the target of ψ^M_G
is often more convenient.


For a finite group G we set

    d_G =  ∏_(H)  |W_G H| ,

the product, over all conjugacy classes of subgroups of G,
of the orders of the respective Weyl groups.
Since the order of W_G H divides the order of G,
inverting |G| also inverts the number d_G.
The following result is well known;
the earliest reference that I am aware of is <cit.>.


subjectMackey functor
For every finite group G and every G-Mackey functor M,
the kernel and cokernel of the homomorphism ψ^M_G are annihilated by d_G.
In particular, ψ^M_G becomes an isomorphism after inverting the order of G.


We reproduce the proof given in <cit.>.
We choose representatives for the conjugacy classes of subgroups and number them
by non-decreasing order

    e = H_1  , H_2, …, H_n = G  .

Then


  * for all i<j and g∈ G the group H_j∩^g H_i
is a proper subgroup of H_j, and

  * every proper subgroup of H_i is conjugate to one of the groups
occurring before H_i.

We set

    K_j  =  (ψ^M_G) ∩ ∑_i=1^j _H_i^G(M(H_i))  ⊆ M(G) ;

this defines a nested sequence of subgroups

    0  =  K_0  ⊆K_1  ⊆K_2  ⊆ … 
    K_n-1  ⊆ K_n  =  (ψ^M_G) .

We show that

    |W_G H_j|· K_j  ⊆ K_j-1

for all j=1,…,n.
Altogether this means that

    d_G· K_n  =  ∏_i=1^n |W_G H_i|· K_n  ⊆ K_0  =  0  .

In the course of proving (<ref>)
we call an element of M(H) degenerate if it maps to
zero in τ_H M, i.e., if it is a sum of transfers from proper
subgroups of H. So the kernel of ψ^M_G is precisely the
subgroup of those elements of M(G) that restrict to degenerate
elements on all subgroups.


We write any given element x∈ K_j as

    x  =  _H_j^G(y)  +  x̅

for suitable y∈ M(H_j) and with x̅ a sum of transfers from the groups
H_1,…,H_j-1. 
For i=1,…, j-1 the double coset formula for ^G_H_j∘^G_H_i
lets us write ^G_H_j(x̅) as a sum of transfers from proper subgroups
of H_j; this uses the hypothesis on the enumeration of the subgroups.
So the class ^G_H_j(_H_i^G(x̅)) is degenerate.
Since x is in the kernel of ψ_G^M, the class ^G_H_j(x) is also degenerate.
So the class

    ^G_H_j(^G_H_j(y)) =  ^G_H_j(x)  - ^G_H_j(x̅)

is degenerate.
The double coset formula expresses 
^G_H_j(_H_j^G(y)) as the sum of the W_G H_j-conjugates
of y, plus transfers from proper subgroups of H_j. So the element

    ∑_g H_j∈ W_G H_j g_⋆(y) ∈ M(H_j)

is degenerate and hence the element

    |W_G H_j|·_H_j^G(y) =  _H_j^G( ∑_g H_j∈ W_G H_j  g_⋆(y) )  ∈ M(G)

is a sum of transfers from proper subgroups of H_j.
Every proper subgroup of H_j is conjugate to one of the group H_1,…,H_j-1,
so

    |W_G H_j|· x  =  
    |W_G H_j|·_H_j^G(y) +  |W_G H_j|·x̅

is both in the kernel of ψ^M_G and
a sum of transfer from the groups H_1,…,H_j-1.
This proves the claim that |W_G H_j|· x belongs to K_j-1.
Altogether this finishes the proof that the kernel of ψ^M_G
is annihilated by the number d_G.

Now we show that the cokernel of ψ^M_G is annihilated by d_G.
We let I_j denote the subgroup of ∏_i=1^n (τ_H_i M)^W_G H_i
consisting of those tuples

    x  =  (x_i)_1≤ i≤ n

such that x_j+1=x_j+2=…=x_n=0.
This defines a nested sequence

    0  =  I_0  ⊆I_1  ⊆I_2  ⊆ … 
    I_n-1  ⊆ I_n  =  ∏_i=1,…, n (τ_H_i M)^W_G H_i .

We show that

    |W_G H_j|· I_j  ⊆ Im(ψ^M_G)  + I_j-1

for all j=1,…,n.
Altogether this means that

    d_G· I_n  ⊆ Im(ψ^M_G)   ,

i.e., the cokernel of ψ^M_G is annihilated by d_G.

To prove (<ref>)
we consider a tuple x=(x_n) in I_j and choose a representative
y∈ M(H_j) for the `last' non-zero component, i.e., such that
y maps to x_j in τ_H_jM.
Since x_j is invariant under the action of the Weyl group W_G H_j, the element y
is at least W_G H_j-invariant modulo transfers from proper subgroups of H_j.
The double coset formula for ^G_H_j∘_H_j^G thus
gives that

    ^G_H_j(_H_j^G(y))  ≡ ∑_g H_j∈ W_G H_j g_⋆(y)  ≡ 
    |W_G H_j|· y  ,

with both congruences modulo transfers from proper subgroups of H_j.
So the composite

    M(G) ^G_H_j M(H_j) proj τ_H_jM

takes ^G_H_j(y) to |W_G H_j|· x_j.
For i>j the double coset formula
shows that ^G_H_i(_H_j^G(y)) is a sum of transfers
from proper subgroups of H_i. So the element
^G_H_i(_H_j^G(y)) maps to 0 in τ_H_i M for all i=j+1,…,n.
In other words, the tuple ψ^M_G(_H_j^G(y)) belongs to I_j.
Since ψ^M_G(_H_j^G(y)) and |W_G H_j|· x both belong to I_j
and agree at the component of H_j, we can thus conclude that 

    ψ^M_G(_H_j^G(y)) - |W_G H_j|· x  ∈ I_j-1 .

This proves (<ref>) and finishes the proof 
that the cokernel of ψ^M_G is annihilated by the number d_G.


The previous Proposition <ref>
shows that for every G-Mackey functor M
the group [1/|G|] (τ_G M) is a direct summand of the
group [1/|G|] M(G), and the splitting is natural for
morphisms of G-Mackey functors. So the functor

    G →   ,   
    M⟼ [1/|G|] (τ_G M)
 
is a natural direct summand of an exact functor. So we can conclude:


Let G be a finite group. The functor that assigns to a G-Mackey functor M
the abelian group [1/|G|] (τ_G M) is exact.  



Now we will show that after inverting the order of the finite group G,
the category of G-Mackey functors splits as a product of categories,
indexed by the conjugacy classes of subgroups of G; 
the factor corresponding to H≤ G is the category of W_G H-modules,
i.e., abelian groups with an action of Weyl group of G
(also known as modules over the group ring [W_G H]).
The argument is not particularly difficult, but somewhat lengthy
because the details involve a substantial amount of book keeping.
The following result is folklore, but I do not know a reference that 
states it in precisely this form. 
The discussion in Appendix A of <cit.> is closely related,
but the exposition of Greenlees and May differs from ours 
in that it emphasizes idempotents in the rationalized Burnside ring, 
as opposed to dividing out transfers.
One could deduce part (ii) of the following theorem
from the results in <cit.>
by using the action of the Burnside ring A(H) on the value
M(H) of any G-Mackey functor M, and showing that after inverting the group order,
the functor τ_H:G→ W_G H becomes isomorphic to
e_H· M(H), where e_H∈ A(H)[1/|G|] is the idempotent `supported at H'.

The special case Q= of the following theorem
in particular shows that rational G-Mackey functors form a semisimple abelian category, 
i.e., every object is both projective and injective. 


Let G be a finite group.


  * The functor

    τ =  (τ_H)_(H) :   G → ∏_(H)  W_G H

has a right adjoint ρ:∏_(H)  W_G H→  G.

  * Let Q be a subring of the ring of rational numbers
in which the order of G is invertible.
Then the adjoint functors (τ,ρ) 
restrict to inverse equivalences between the category
Q-local G-Mackey functors and the product of the categories
of Q-local W_G H-modules.



(i) We start by exhibiting a right adjoint ρ_H:W_G H→ G
to the functor τ_H.
We recall that the Weyl group W_G H acts
on the H-fixed points of every G-set S by

    W_G H× S^H  → S^H  ,    (g H, s)  ⟼ g s  .

We let N be a W_G H-module and K another subgroup of G. We set

    (ρ_H N)(K) =  ^W_G H((G/K)^H,N) ,

the abelian group of W_G H-equivariant functions from (G/K)^H to N,
under pointwise addition of functions.
The conjugation map
γ_⋆:(ρ_H N)(K^γ)→ (ρ_H N)(K),
for γ∈ G, is precomposition with the W_G H-map

    (G/K)^H  → (G/K^γ)^H  ,    
    g· K  ⟼ g γ· K^γ .

For L≤ K, the restriction map
^K_L:(ρ_H N)(K)→ (ρ_H N)(L) is precomposition with the W_G H-map

    (G/L)^H  → (G/K)^H  ,    g L ⟼ g K .

The transfer map
^K_L:(ρ_H N)(L)→ (ρ_H N)(K) is given by `summation over preimages', i.e.,

    _L^K(f)(g K)  =   ∑_γ L∈ (G/L)^H : γ K= g K f(γ L) ,

where f:(G/L)^H→ N is W_G H-equivariant and g K∈ (G/K)^H.
We omit the verification that _L^K(f) is again W_G H-equivariant.

With these definitions, the verification of properties (i), (ii) and (iii) 
of Definition <ref> is straightforward, and we also omit it.
Checking the double coset formula is more involved, so we 
spell out the argument. 
We consider a subgroup K of G and subgroups J and L of K. 
We let g∈ G be such that g J∈ (G/J)^H, i.e., H^g≤ J.
We let R be a set of representatives of the J-L-double cosets in K. 
For every k∈ R we let S_k be a set of representatives j
of those (J∩^k L)-cosets j (J∩^k L)∈ (G/(J∩^k L))^H such that 
j J=g J.
Then the sets S_k· k are pairwise disjoint for k∈ R, and their union

    ⋃_k∈ R  S_k· k

is a set of representatives of those H-fixed L-coset 
γ L∈ (G/L)^H such that γ K=g K.
Hence for every W_G H-map f:(G/L)^H→ N all g J∈ (G/J)^H we obtain the relation

    (^K_J(_L^K (f)))(g J)    =  ∑_γ L∈ (G/L)^H : γ K= g K f(γ L)
       =  ∑_k∈ R ∑_ j∈ S_k   f(j k L)
       =  ∑_k∈ R ∑_ j∈ S_k   k_⋆(^L_J^k∩ L(f))(j(J∩^k L))
       =  ∑_k∈ R_J∩^k L^J(k_⋆(^L_J^k∩ L(f)))(g J) .

For varying g J and f, this proves the double coset relation

    ^K_J∘_L^K = ∑_k∈ R _J∩^k L^J∘ k_⋆∘^L_J^k∩ L :  
    (ρ_H N)(L)→(ρ_H N)(J)  .

Now we have shown that ρ_H N is a G-Mackey functor, 
so this completes the definition of the functor ρ_H on objects;
on morphisms, ρ_H is simply given by postcomposition with
a W_G H-module homomorphism.

The action of W_G H on (G/H)^H is free and transitive, so the evaluation map

    (ρ_H N)(H) =  ^W_G H((G/H)^H,N)  → N  ,    
    f ⟼ f(e H)

is an isomorphism. Moreover, if H is not subconjugate to K, then
the set (G/K)^H is empty, and so (ρ_H N)(K)=0.
This holds in particular for all proper subgroups of H, so in
the G-Mackey functor ρ_H N, all transfers from proper subgroups 
to H are trivial. Hence the isomorphism (<ref>)
passes to a natural isomorphism of W_G H-modules

    ϵ_H^N  :  τ_H(ρ_H N)   N  , 
      ϵ_H^N[ f] =  f(e H) .

Here [y] denotes the class in τ_H M of an element y∈ M(H).
Given a G-Mackey functor M, we also define a homomorphism

    η_H^M  :  M  → ρ_H(τ_H M)

of G-Mackey functors as follows. For a subgroup K of G
we define

    η_H^M(K) :  M(K) → ^W_G H((G/K)^H,τ_H M)  = ρ_H(τ_H M)(K)

by

    η_H^M(K)(x)(g K) =   [ g_⋆( ^K_H^g(x) ) ]  .

We show that these additive maps indeed define a morphism of G-Mackey functors.
For L≤ K≤ G, x∈ M(K) and g L∈ (G/L)^H we have 

    ^K_L(η_H^M(K)(x))(g L)    =  η_H^M(K)(x)(g K) =   [ g_⋆(^K_H^g(x)) ] 
       =  [ g_⋆(^L_H^g(^K_L(x))) ]  =  η_H^M(L)( ^K_L(x))(g L) .

So ^K_L∘η_H^M(K)=η_H^M(L)∘^K_L, i.e.,
the η_H^M-maps are compatible with restrictions.
Similarly, for γ∈ G and x∈ M(K^γ) we have 

    γ_⋆(η_H^M(K^γ)(x))(g· K)    =  η_H^M(K^γ)(x)(gγ· K^γ) 
    =   [ (gγ)_⋆(^K^γ_H^gγ(x)) ] 
       =  [ g_⋆(^K_H^g(γ_⋆(x))) ] =  η_H^M(K)( γ_⋆(x))(g· K) .

So γ_⋆∘η_H^M(K^γ)=η_H^M(K)∘γ_⋆, i.e.,
the η_H^M-maps are compatible with conjugations.

For compatibility with transfers we consider g K∈ (G/K)^H 
(i.e., H^g≤ K), and use the double coset formula 

    g_⋆∘^K_H^g∘_L^K    =  ∑_(H^g)k L∈ H^g K/L 
    g_⋆∘_H^g∩^k L^H^g∘ k_⋆∘^L_H^g k∩ L
       ≡∑_(H^g) k L :  H^g k≤ L  ( g k)_⋆∘^L_H^g k
       =  ∑_γ L∈ (G/L)^H :  γ K = g Kγ_⋆∘^L_H^g k

as maps M(L)→τ_H^g( M). The second relation is modulo
transfers from proper subgroups of H^g, i.e., we have dropped all summands
where H^g is not contained in ^k L.
The third relation exploits that sending (H^g) k L to g k L
is a bijection between the set of those H^g-L-double cosets with H^g k≤ L
and the set of those γ L∈ (G/L)^H that satisfy γ K=g K.
So for all x∈ M(L) we obtain the relation

    ^K_L(η_H^M(L)(x))(g K)    =  ∑ η_H^M(L)(x)(γ L) 
    =  ∑  [ γ_⋆(^L_H^γ(x)) ] 
    
     _(<ref>)   =  [ g_⋆(^K_H^g(^K_L(x)))]  =  η_H^M(K)( ^K_L(x))(g K) .

The sums run over those H-fixed L-cosets γ L∈ (G/L)^H
such that γ K= g K.
So _L^K∘η_H^M(L)=η_H^M(K)∘_L^K, i.e.,
the η_H^M-maps are compatible with transfers.

It is now straightforward to check that the composite maps

    ρ_H N    η_H^ρ_H N ρ_H(τ_H(ρ_H N)) []ρ_H(ϵ_H^N) ρ_H N      and
    τ_H M    τ_H(η_H^M) τ_H(ρ_H(τ_H M)) []ϵ_H^τ_H M τ_H M

are the respective identity morphisms.
Indeed,

    ρ_H(ϵ_H^N)(η_H^ρ_H N(K)(f) )(g K)    =  ϵ_H^N(η_H^ρ_H N(K)(f)(g K)) =  ϵ_H^N[ g_⋆(^K_H^g(f))]
       =  g_⋆(^K_H^g(f))(e H) =  ^K_H^g(f)(g H) =  f(g K)

for all W_G H-equivariant f:(G/K)^H→ N and all g K∈(G/K)^H.
Moreover,

    ϵ_H^τ_H M(τ_H(η^M_H)[x])    =  ϵ_H^τ_H M[η^M_H(H)(x)] =  η^M_H(H)(x)(e H) =  [x]

for all x∈ M(H).
This shows that η_H and ϵ_H are the unit and counit 
of an adjunction between τ_H and ρ_H.

Now we can easily prove part (i). Since (τ_H,ρ_H)
is an adjoint pair for every subgroup H of G,
a right adjoint to the product functor τ is given by

    ρ :  ∏_(H)  W_G H→  G ,   ρ( (N_H)_(H)) =  ∏_(H) ρ_H(N_H) ,

the product of the G-Mackey functors ρ_H(N_H).
The universal property of the product of G-Mackey functors
and the previous adjunctions combine into a natural isomorphism

    G(M,ρ ( (N_H)_(H)) )        ∏_(H)  G(M,ρ_H (N_H))
         ∏_(H)   W_G H(τ_H M, N_H) .

This completes the construction of the adjunction between τ and ρ.

(ii)
We show first that for every G-Mackey functor M, the adjunction unit

    η^M  =  (η_H^M) :  M  → ρ(τ M)  =  ∏_(H) ρ_H(τ_H M)

becomes an isomorphism after inverting the order of G.
When evaluated at G, this is the content 
of Proposition <ref>.
At a general subgroup of G, we reduce the claim 
to Proposition <ref> as follows.
We let H and K be two subgroups of G.
We let R_H,K be a set of representatives of the K-conjugacy classes 
of subgroups L≤ K that are G-conjugate to H.
For each L∈ R_H,K we choose an element g_L∈ G such that H^g_L=L.
Then g_L K∈ (G/K)^H, and the map

    ∐_L∈ R_H,K (W_G H) / (W_^g_L K H) → (G/K)^H ,   
    [n H] ↦ n g_L K

is an isomorphism of W_G H-sets. So taking W_G H-equivariant maps into τ_H M
provides an isomorphism of abelian groups

    ρ_H(τ_H M)(K) =  ^W_G H( (G/K)^H, τ_H M)     → ∏_L∈ R_H,K  (τ_H M)^W_^g_L K H
    
    f        ⟼   ( f(g_L K) )_L .

Because H^g_L=L, the map (g_L)_⋆:τ_L M→τ_H M
defined in (<ref>) is an isomorphism of abelian groups,
equivariant for the isomorphism 
W_K L=W_K H^g_L W_^g_L KH
induced by conjugation by g_L.
So the map

    ζ_H :  ρ_H(τ_H M)(K) → ∏_L∈ R_H,K  (τ_L M)^W_K L ,   
    f  ⟼ ( (g_L)_⋆^-1 f(g_L K) )_L

is an isomorphism.


Now the following diagram commutes by direct inspection, where the right vertical map
is the product of the maps ζ_H over all conjugacy classes
of subgroups of G:

    @C=8mm 
    M(K)[rr]^-η_M(K)@/_1pc/[drr]_ψ^M_K      ρ(τ M)(K)@=[r]    ∏_(H) ρ_H( τ_H M)(K) [d]^∏ζ_H_
          ∏_(L)≤ K  (τ_L M)^W_K L@=[r]
       ∏_(H)≤ G∏_L∈ R_H,K (τ_L M)^W_K L

The map ψ^M_K becomes an isomorphism after inverting the
order of K, by Proposition <ref>,
applied to the underlying K-Mackey functor of M.
Since the right vertical map is an isomorphism and
the order of K divides the order of G, we conclude that the map η_M(K) 
becomes an isomorphism after inverting the order of G.
This completes the proof that η_M:M→ρ(τ M) becomes an isomorphism 
after inverting |G|. 

Now |G| is invertible in Q, so when restricted to Q-local objects, 
the adjunction unit η:→ρ∘τ is a natural isomorphism.
Hence the restriction of the functor τ to the full subcategory of 
Q-local G-Mackey functors is fully faithful.
We claim that additionally, every family of Q-local W_G H-modules
is the image of a Q-local G-Mackey functor.
Since the functor τ is additive, it commutes with finite products,
so it suffices to show that for every individual subgroup H of G
and every Q-local W_G H-module N, there is an
Q-local G-Mackey functor M such that τ M is isomorphic to N.
We let e:τ(ρ_H N)→τ(ρ_H N) be the idempotent
endomorphism whose H-component is the identity of τ_H(ρ_H N),
and such that e_K=0 for all K that are not conjugate to H.
Since ρ_H N is Q-local and τ is fully faithful on 
Q-local G-Mackey functors, there exists an idempotent endomorphism
ẽ:ρ_H N→ρ_H N such that e=τ(ẽ).
Then ρ_H N is isomorphic to the direct sum of the image of ẽ
and the image of -ẽ. Since the functor τ is additive,
τ(ẽ·ρ_H N) is isomorphic to 
e·τ(ρ_H N)=τ_H(ρ_H N).
As we showed in the proof of part (i), the W_G H-module τ_H(ρ_H N) 
is isomorphic to N, so this proves that N 
is isomorphic to τ(ẽ·ρ_H N).
Altogether this shows that the functor τ is essentially surjective on
Q-local objects. On this subcategory, τ is also fully faithful,
and hence an equivalence of categories.



Now we return to equivariant spectra.
For every orthogonal G-spectrum X, Proposition <ref> (ii)
shows that the geometric fixed point map (<ref>)
factors over the quotient of π_0^G(X) by the subgroup
generated by proper transfers. We denote by

    Φ̅ :  π_k^G(X) / t_G( _k(X))
     =  τ_G( _k(X) ) → Φ^G_k(X)

the induced map on the factor group, and call it the 
reduced geometric fixed point homomorphism.subjectgeometric fixed point homomorphism!reduced
Our next result shows that for finite groups G,
the `corrected' (i.e., reduced) geometric fixed point map
becomes an isomorphism after inverting the order of G.
The following proposition is well known,
and closely related statements appear in Appendix A of <cit.>;
however, I am not aware of a reference for the following statement in this form.

 
For  every finite group G, every orthogonal G-spectrum X
and every integer k the reduced geometric fixed point map

    Φ̅ :  τ_G( _k(X) ) → Φ^G_k(X)

becomes an isomorphism after inverting the order of G.subjectgeometric fixed points!rationallysubjectgeometric fixed points!and transfers 
 

We start by showing that for every orthogonal G-spectrum X and every
subgroup H of G the transfer map

    _H^G :  π_k^H(X G/H_+)  → π_k^G(X G/H_+)

for the spectrum X G/H_+ is surjective.
Indeed, the transfer is defined as the composite

    π_k^H(X G/H_+)  []G⋉_H - π_k^G(G⋉_H( X G/H_+)) 
     act_* π_k^G(X G/H_+)  .

The first map – the external transfer – is an isomorphism
by Theorem <ref>.
The second map is surjective because the action morphism has a G-equivariant section

    X G/H_+  → G⋉_H( X G/H_+) ,   
    x g H  ⟼ [g,  g^-1 x e H]  .


Now we let A be a G-CW-complex without G-fixed points.
We claim that for every orthogonal G-spectrum X,
the entire group π_k^G(X A_+) is generated by transfers from
proper subgroups after inverting |G|.
In a first step we show this when A is a finite-dimensional G-CW-complex.
We argue by induction over the dimension of A. The induction starts when A
is empty, in which case X A_+ is the trivial spectrum and there is nothing to show.
Now we consider n≥ 0 and assume the claim for all 
(n-1)-dimensional G-CW-complexes without G-fixed points. 
We suppose that A is obtained
from such an (n-1)-dimensional G-CW-complex B by attaching equivariant n-cells. 
After choosing characteristic maps for the n-cells we can identify
the quotient A/B as a wedge

    A/B   ⋁_i∈ I  S^n (G/H_i)_+

for some index set I and certain subgroups H_i of G.
Equivariant homotopy groups commute with wedges, so we can identify
the homotopy group Mackey functor of X (A/B) as

    _k(X A/B)  ⊕_i∈ I _k(X S^n (G/H_i)_+) .

Since A has no G-fixed points, the groups H_i are all proper subgroups of G,
so the group π_k^G(X S^n (G/H_i)_+) is generated by transfers
from the proper subgroup H_i, by the previous paragraph.
So altogether the group π_k^G(X A/B) is generated by transfers
from proper subgroups (even before any localization).

The inclusion B→ A is a G-equivariant h-cofibration, so it induces an
h-cofibration of orthogonal G-spectra X B_+→ X A_+
that results in a long exact sequence of homotopy group Mackey functors as in 
Corollary <ref> (i).
The functor sending a G-Mackey functor M to [1/|G|](τ_G M)
is exact (Corollary <ref>), 
the groups [1/|G|]τ_G(_*(X B_+))
vanish by induction, and the groups τ_G(_*(X A/B)) 
vanish by the previous paragraph.
So the groups [1/|G|]τ_G(_*(X A_+))
vanish by exactness, and this finishes the inductive step.

Now we suppose that A is an arbitrary G-CW-complex without G-fixed points,
possibly infinite dimensional. 
We choose a skeleton filtration by G-subspaces A^n.
Then the groups [1/|G|]τ_G(_*(X A^n_+))
vanish for all n≥ 0, by the previous paragraph.
All the morphisms X A^n_+→ X A^n+1_+ are h-cofibrations
of orthogonal G-spectra, hence levelwise closed embeddings. Since equivariant
homotopy groups and the functor [1/|G|]τ_G(-)
both commute with such sequential colimits, this shows that  
the groups τ_G(_k(X A_+)) vanish after inverting |G|.

Now we can prove the proposition.
The inclusion i:S^0→Ẽ_G gives rise to a commutative square:

    @C=15mmτ_G( _k(X))[r]^-τ_G(_k(X i))[d]_Φ̅   τ_G( _k(XẼ_G)) [d]^Φ̅
    Φ^G_k(X)[r]_-Φ^G_k(X i)   Φ^G_k(XẼ_G)

The lower horizontal map is an isomorphism because i 
identifies S^0 with the fixed points (Ẽ_G)^G.
Proposition <ref> 
shows that the geometric fixed point map Φ: π^G_k(XẼ_G)→Φ_k^G(XẼ_G) is an isomorphism before dividing out transfers.
So all transfers from proper subgroups are in fact trivial  in 
π^G_k(XẼ_G), and the right vertical map
in the square (<ref>) is an isomorphism 
(even before any localization).
By the previous paragraph, the group π_k^G(X (E_G)_+) 
is generated by transfers from proper subgroups after inverting |G|, so

    [1/|G|]τ_G( _k(X (E_G)_+) =  0  .

The functor sending a G-Mackey functor M to the group [1/|G|](τ_G M)
is exact (Corollary <ref>), 
so the isotropy separation sequence (<ref>) 
shows that the upper horizontal map in the square (<ref>)
becomes an isomorphism after inverting |G|.
So the left vertical map in the square (<ref>)
becomes an isomorphism after inverting |G|.




When X is an orthogonal G-spectrum, we can apply the earlier algebraic
Proposition <ref>
to the G-Mackey functor _k(X); this allows us – after inverting |G| – 
to reconstruct π_k^G(X) from the groups τ_H(_k(X)) 
with their Weyl group action.
Moreover, after inverting the group order we can use the reduced
geometric fixed point map (<ref>),
for the underlying H-spectrum of X, to identify
the group τ_H ( _k(X)) with the group Φ_k^H(X).
Under this identification, the map ψ̅^M_G 
becomes the product of the maps

    π_k^G(X)   ^G_H  π_k^H(X)  Φ^H Φ_k^H(X) .

So Propositions <ref> 
and <ref>  together prove:


For every finite group G, every orthogonal G-spectrum X
and every integer k the map

    (Φ^H∘^G_H)_H :  π^G_k(X)  → ∏_(H) ( Φ^H_k(X))^W_G H

becomes an isomorphism after inverting the order of G.
 

subjectMackey functor|)





§ PRODUCTS


In this section we recall the smash product of orthogonal spectra
and orthogonal G-spectra and study its formal and homotopical properties.
Like the box product of orthogonal spaces, the smash product 
of orthogonal spectra is a special case of Day's convolution product on categories
of enriched functors, compare Appendix <ref>.
The main homotopical result about the smash product is the `flatness theorem'
(Theorem <ref>), showing that smashing with 
G-flat orthogonal G-spectra (in the sense of Definition <ref>)
preserves _*-isomorphisms.

The smash product is intimately related to pairings of equivariant stable homotopy groups
that we recall in Construction <ref>;
the main properties of these pairings are summarized 
in Theorem <ref>. 
When specialized to equivariant ring spectra, these pairings turn 
the equivariant stable homotopy groups into a graded ring,
compare Corollary <ref>. 



The smash product of orthogonal spectra is characterized by a
universal property that we recall now.
The indexing category  for orthogonal spectra has a symmetric monoidal
product by direct sum as follows. We denote by  the
category enriched in based spaces whose objects are
pairs of inner product spaces and whose morphism spaces
are smash products of morphism spaces in . A based continuous functor

    ⊕ :   →

is defined on objects by orthogonal direct sum, and on morphism spaces by

    (V,W)(V',W')    → (V⊕ W,V'⊕ W')
    
    (w,φ) (w',φ')     ⟼   ((w,w'),φ⊕φ') .

A bimorphism b:(X,Y)→ Zsubjectbimorphism!of orthogonal spectra
from a pair of orthogonal spectra (X,Y)
to an orthogonal spectrum Z is a natural transformation 

    b  :  XY  → Z∘⊕

of continuous functors →_*;
here XY is the `external smash product'  defined by
(XY)(V,W)=X(V) Y(W). A bimorphism thus consists of
based continuous maps 

    b_V,W :  X(V)   Y(W)  → Z(V⊕ W)

for all inner product spaces V and W that form morphisms of orthogonal
spectra in each variable separately.
A smash product of two orthogonal spectra is now a universal example
of a bimorphism from (X,Y).


A smash productsubjectsmash productsymbol - smash product of orthogonal spectra
of two orthogonal spectra X and Y is a pair (X Y,i) 
consisting of an orthogonal spectrum X Y
and a universal bimorphism i:(X,Y)→ X Y,
i.e., a bimorphism 
such that for every orthogonal spectrum Z the map

    (X Y,Z)  → Bimor((X,Y),Z)  , f↦ f i = {f(V⊕ W)∘ i_V,W}_V,W

is bijective. 


Since the index category  is skeletally small and the base category _*
is cocomplete, every pair of orthogonal spectra has a smash product
by Proposition <ref>. 
Often only the object X Y will be referred to 
as the smash product, but one should keep in mind that it comes equipped
with a specific, universal bimorphism. We will often refer to the 
bijection (<ref>) as the
universal property of the smash product of orthogonal spectra.
subjectuniversal property!of the smash product

While the smash products for pairs of orthogonal spectra are choices, 
there is a preferred way to extend any chosen smash products 
to a functor in two variables
(Construction <ref>);
this functor has a preferred symmetric monoidal structure, i.e., 
distinguished natural associativity and symmetry isomorphisms

    α_X,Y,Z :  (X Y) Z  → X(Y Z)
       respectively   τ_X,Y :   X Y  → Y X

(see Construction <ref>).
Together with strict  unit isomorphisms X= X= X,
these satisfy the coherence conditions of a symmetric monoidal category,
compare Day's Theorem <ref>.
The smash product of orthogonal spectra is closed symmetric monoidal
in the sense that the smash product is adjoint to an
internal Hom spectrum.
We won't use the internal function spectrum, so we don't go into any details.

When a compact Lie group G acts on the orthogonal spectra X and Y,
then X Y becomes an orthogonal G-spectrum via the diagonal action.
So the smash product lifts to a symmetric monoidal closed structure

    :  G× G → G

on the category of orthogonal G-spectra.subjectsmash product!of orthogonal G-spectra


[Smash product with a free orthogonal spectrum]
We give a `formula' for the smash product with a free orthogonal spectrum,
i.e., the represented functor (W,-) for some inner product space W.
A general feature of Day type convolution products is that
the convolution product of represented functors is represented, 
see Remark <ref>.
In the case at hand this specializes to a preferred isomorphism

    (V,-)(W,-)  (V⊕ W,-)

specified, via the universal property (<ref>),
by the bimorphism with (U,U')-component

    ⊕ :  (V,U)(W,U') → (V⊕ W,U⊕ U') .

Now we let X be any orthogonal spectrum.
We give a formula for the value at V⊕ W of the smash product X(W,-). 
The composite 

    X(V) -_W X(V)(W,W)  i_V,W  
    (X(W,-))(V⊕ W)

is O(V)-equivariant, where O(V) acts on the target by restriction
along the homomorphism -⊕ W:O(V)→ O(V⊕ W).
So the map extends to an O(V⊕ W)-equivariant based map

    O(V⊕ W)⋉_O(V) X(V) → (X(W,-))(V⊕ W)  .

We claim that this map is a homeomorphism.
To see that we first check the special case where X=(U,-)
is itself a represented functor. The isomorphism (<ref>)
turns (<ref>) into the map

    O(V⊕ W)⋉_O(V)(U,V)    →  (U⊕ W,V⊕ W) 
    
    [A, (v,φ)]     ⟼ (A· (v,0),A∘(φ⊕ W))

which is a homeomorphism by inspection.
Source and target of the map (<ref>)
commute with smash products with based spaces
and colimits in the variable X. Since every orthogonal spectrum
is a coend of represented orthogonal spectra smashed with based spaces,
the map (<ref>) is an isomorphism in general.



Our next topic is a flatness result, proving that smashing with
certain orthogonal G-spectra preserves _*-isomorphisms.
To define the relevant class, the `G-flat orthogonal G-spectra', 
we introduce the skeleton filtration,
a functorial way to write an orthogonal spectrum as a 
sequential colimit of spectra which are made from the information
below a fixed level. The word `filtration' should be used with caution
because the maps from the skeleta to the orthogonal spectrum
need not be injective. Since an orthogonal G-spectrum is the same data as an orthogonal
spectrum with continuous G-action, and since skeleta are functorial,
the skeleta of an orthogonal G-spectrum automatically come as orthogonal G-spectra.

[Skeleton filtration of orthogonal spectra]
As in the unstable situation in Section <ref>,
the skeleton filtration is a special case of 
the general skeleton filtration on certain enriched functor categories
that we discuss in Appendix <ref>.
Indeed, if we specialize the base category to =_*,
the category of based spaces under smash product, and the index category
to =, then the functor category ^∗ becomes the
category  of orthogonal spectra. The dimension function needed in
the construction and analysis of skeleta is the vector space dimension.
For every orthogonal spectrum X and every m≥ 0,
the general theory provides 
an m-skeleton,subjectskeleton!of an orthogonal spectrum

    ^m X =  l_m(X^≤ m)  ,

the extension of the restriction of X to _≤ m,
and a natural morphism i_m:^m X→ X, the counit of
the adjunction (l_m,(-)^≤ m). The value

    i_m(V) : (^m X)(V) → X(V)

is an isomorphism for all inner product spaces V of dimension at most m.
The m-th latching spacesubjectlatching space!of an orthogonal spectrum
of X is the based O(m)-space

    L_m X  =  (^m-1 X)(^m)  ;

it comes with a natural based O(m)-equivariant map

    ν_m=i_m-1(^m) : L_m X → X(^m)  ,

the m-th latching map. 
We also agree to set ^-1 X=∗, the trivial orthogonal spectrum,
and L_0 X=∗, a one-point space. 

The different skeleta are related by natural morphisms
j_m:^m-1 X→^m X, for all m≥ 0, 
such that i_m∘ j_m=i_m-1.
The sequence of skeleta stabilizes to X in a 
very strong sense. For every inner product space V, the maps
j_m(V) and i_m(V) are isomorphisms as soon as m >(V).
In particular, X(V) is a colimit, with respect to the maps i_m(V), 
of the sequence of maps j_m(V). 
Since colimits in the category of orthogonal spectra are created objectwise,
we deduce that the orthogonal spectrum X is a colimit, 
with respect to the morphisms i_m, of the sequence of morphisms j_m. 




Now we can define the class of G-flat orthogonal G-spectra for which smash product
is homotopical. As we remarked earlier,
the skeleta of (the underlying orthogonal spectrum of)
an orthogonal G-spectrum inherit a continuous G-action by functoriality.
In other words, the skeleta and the various morphisms between them
lift to endofunctors and natural transformations on the category of
orthogonal G-spectra.
If X is an orthogonal G-spectrum, then the O(m)-space L_m X 
comes with a commuting action by G, again by functoriality of the latching space.
Moreover, the latching morphism ν_m:L_m X→ X(^m) is
(G× O(m))-equivariant.


  Let G be a compact Lie group.
  An orthogonal G-spectrum X is G-flatsubjectG-flat@G-flat!orthogonal G-spectrum if for every m≥ 0 the latching map ν_m:L_m X → X(^m)
  is a (G× O(m))-cofibration.


When G is the trivial group, we simply speak of flat 
orthogonal spectra.subjectflat!orthogonal spectrum
These flat orthogonal spectra play a special role as the cofibrant objects in the global
model structure on the category of orthogonal spectra, to be established
in Theorem <ref>  below.



The G-flat orthogonal spectra are the cofibrant objects in a certain model
structure on the category of orthogonal G-spectra, the  
-model structure of Stolz <cit.>.
This model structure has more cofibrant objects 
than the one of Mandell-May <cit.>,
whose `q-cofibrant' orthogonal G-spectra are precisely the ones 
for which the latching  map ν_m:L_m X → X(^m)
is a (G× O(m))-cofibration and in addition the group O(m)
acts freely on the complement of the image of ν_m.

A convenient feature of the class of flat equivariant spectra is that they
are closed under various change-of-group functors. 
For example, if α:K→ G is a continuous 
homomorphism and X an orthogonal G-spectrum, then the 
orthogonal K-spectrum α^* X satisfies

    L_m (α^* X)  =  (α× O(m))^*(L_m X)   and  
    (α^* X)(^m) =  (α× O(m))^*(X(^m))

as (K× O(m))-spaces. Since restriction along 
α× O(m) preserves cofibrations 
(see Proposition <ref> (i)),
the restriction functor α^*:G→ K
takes G-flat orthogonal G-spectra to
K-flat orthogonal K-spectra. Of particular relevance for global considerations
is the special case of the unique homomorphism G→ e to the
trivial group. In this case we obtain that a flat orthogonal spectrum 
is in particular G-flat when we give it the trivial G-action.

A similar argument applies to induction of equivariant spectra. Indeed, if H
is a closed subgroup of G and Y an orthogonal H-spectrum, then

    L_m (G⋉_H Y)      (G× O(m))⋉_H× O(m) (L_m Y)
       and
    
    (G⋉_H Y)(^m)     (G× O(m))⋉_H× O(m) Y(^m) .

Since induction from H× O(m) to G× O(m)
preserves cofibrations (see Proposition <ref> (ii)),
the induction functor G⋉_H -:H→ G
takes H-flat orthogonal H-spectra to
G-flat orthogonal G-spectra.


The main purpose of the latching object and latching morphism are to record
how one skeleton of an orthogonal G-spectrum is obtained from the previous one.
We denote by

    G_m   :  O(m)_*  →

the left adjoint to the evaluation functor X↦ X(^m).
Explicitly, this functor sends a based O(m)-space A to the orthogonal spectrum

    G_m A  =  (^m,-)_O(m) A .

Since G_m is a continuous functor, it takes (G× O(m))-spaces
to G-orthogonal spectra, by functoriality.
Proposition <ref> 
and the fact that pushouts of orthogonal G-spectra are created on
underlying orthogonal spectra yield:



For every orthogonal G-spectrum X and every m≥ 0 the commutative square

    @C=12mm G_m L_m X [r]^G_mν_m[d]     G_m X(^m) [d]
    ^m-1 X [r]_-j_m   ^m X

is a pushout of orthogonal G-spectra.
The two vertical morphisms are adjoint to the identity of L_m X 
respectively X(^m).





The next theorem shows that smashing with a G-flat orthogonal G-spectrum
preserves _*-isomorphisms.
This result is due to Stolz <cit.>,
because the G-flat orthogonal G-spectra are precisely the -cofibrant
objects in the sense of <cit.>;
a proof can also be found in <cit.>. 
The theorem is stronger than the earlier result 
of Mandell and May <cit.>
because the class of G-flat of orthogonal G-spectra
is strictly larger than the cofibrant G-spectra 
in the sense of <cit.>.

Since Stolz' thesis <cit.> is not published, the notation and 
level of generality in <cit.>
is different from ours, 
and the characterization of flat objects in terms of latching maps is not
explicitly mentioned in <cit.> nor <cit.>,
we spell out the argument.


subjectflatness theorem!for orthogonal G-spectra 
Let G be a compact Lie group and X a G-flat orthogonal G-spectrum.
Then the functor - X preserves _*-isomorphisms of orthogonal G-spectra.


We go through a sequence of six steps, 
proving successively more general cases of the theorem.
We call an orthogonal G-spectrum G-stably contractible
if all of its equivariant homotopy groups vanish, for all closed subgroups of G.
In Step 1 through 5, we let C be a G-stably contractible orthogonal G-spectrum,
and we show for successively more general X that C X 
is again G-stably contractible.

Step 1:
We let K be another compact Lie group, W a K-representation
and A a based (G× K)-space.
We define an orthogonal G-spectrum

    A▹_K,W C  =  A_K (^W C)

at an inner product space U as

    (A▹_K,W C)(U)   =  A_K C(U⊕ W)   ;

the K-action on C(U⊕ W) is through the K-action on W;
the G-action on the smash product is diagonally, 
through the G-actions on A and on C.

Now we assume in addition that A is a finite based (G× K)-CW-complex and
the K-action is free (away from the basepoint). 
We claim that then A▹_K,W C is also G-stably contractible.
We argue by the number of equivariant cells, starting with A=∗, in which case
there is nothing to show. Now we assume that the claim holds for A;
we let B be obtained from A by attaching an equivariant cell 
(G× K)/Γ× D^k,
where Γ is a closed subgroup of G× K.
The inclusion A→ B is an h-cofibration of (G× K)-spaces, 
hence the induced morphism
A▹_K,W C → B▹_K,W C
is an h-cofibration of orthogonal G-spectra. 
By the long exact homotopy group sequence 
(Corollary <ref> (i))
it suffices to show the claim for the quotient:

    ( B ▹_K,W C) / ( A ▹_K,W C)       
      (A/B) ▹_K,W C       S^k   (( G× K)/Γ)_+ ▹_K,W C

By the suspension isomorphism we may show that
( (G× K)/Γ)_+ ▹_K,W C 
is G-stably contractible.
Since the K-action on B is free, the stabilizer group Γ 
is the graph of a continuous homomorphism
α:H→ K from some closed subgroup H of G (namely the projection
of Γ to G). So we can rewrite 

    (  (G× K)/Γ)_+ ▹_K,W C     G⋉_H ( ^α^*(W) C)  .

Since C is G-stably contractible, it is also H-stably contractible.
The orthogonal H-spectrum  C S^α^*(W)
is then H-stably contractible
by Proposition <ref> (ii).
Since ^α^*(W)C is _*-isomorphic to 
 C S^α^*(W) (by Proposition <ref> (ii)),
it is H-stably contractible.
Since the induction functor G⋉_H- preserves
_*-isomorphisms (by Corollary <ref>), 
we conclude that (  (G× K)/Γ)_+ ▹_K,W C 
is G-stably contractible.
This finishes the inductive argument, and hence the proof
that the orthogonal G-spectrum A▹_K,W C
is G-stably contractible.

Step 2:
We let G be a compact Lie group, K a closed subgroup of O(m)
and α:G→ W_O(m) K a continuous homomorphism to the Weyl group of K
in O(m).
The represented orthogonal spectrum (^m,-)
has a right O(m)-action through the tautological action on ^m.
The semifree orthogonal spectrum 

    G_m ( O(m)/K_+)  =  (^m,-)/K

has a residual action of the Weyl group W_O(m) K, 
and we restrict scalars along α to obtain the orthogonal G-spectrum 

    α^*((^m,-)/K) .

We claim that then the orthogonal G-spectrum
Cα^*((^m,-)/K) is G-stably contractible.
We show that π_k^H( Cα^*((^m,-)/K) )=0
for k≥ 0 and every closed subgroup H of G; for k<0 we exploit that
π_k^H( Cα^*((^m,-)/K) ) is isomorphic to
π_0^H( (C S^-k)α^*((^m,-)/K) ),
so applying the argument to C S^-k instead of C shows the
vanishing in negative dimensions.
Since the input data is stable under passage to closed subgroups of G
(just restrict α to such a subgroup), it is no loss of generality
to assume H=G.

We can represent every class of π_k^G(Cα^*((^m,-)/K ))
by a based G-map

    f  :   S^V⊕^m+k →  
    (Cα^*((^m,-)/K ))(V⊕^m)

for some G-representation V.
In (<ref>) we exhibited an isomorphism 

    O(V⊕^m)⋉_O(V) C(V)    
    (C(^m,-))(V⊕^m)  .

Under this isomorphism, the K-action on (^m,-)
becomes the right translation K-action on O(V⊕^m),
so passage to K-orbits yields an isomorphism

    O(V⊕^m)/K_+_O(V) C(V)  
    (Cα^*((^m,-)/K))(V⊕^m)  .

The G-action on the right hand side is diagonally from three ingredients:
the G-action on C, the G-action on V, and the G-action
on (^m,-)/K through α. 
So G acts on the left hand side diagonally, also through the actions
on V (by left translation on O(V⊕^m)), on ^m through α,
and on C.

The target of f is thus G-equivariantly homeomorphic to  

    O(V⊕^m)/K _+_O(V)  C(V)  =  ( O(V⊕^m)/K_+ ▹_O(V), V  C)(0)  .

Via this isomorphism, we view f as representing a G-equivariant homotopy class in

    π_m+k^G( Ω^V(O(V⊕^m)/K_+▹_O(V), V  C)) .

The homogeneous space O(V⊕^m)/K has a left G-action by translation
through the composite 

    G → O(V) -⊕^m O(V⊕^m)

and another G-action by restriction of the residual W_O(m)(K)-action 
along α; altogether, the group G acts diagonally.
This space also has a commuting free right
action of O(V) by right translation.
Since O(V⊕^m)/K is a smooth manifold
and the (G× O(V))-action is smooth, 
Illman's theorem <cit.> 
provides a finite (G×  O(V))-CW-structure on O(V⊕^m)/K;
so O(V⊕^m)/K_+ is cofibrant as a based (G× O(V))-space,
with free right O(V)-action.
Since C is G-stably contractible, 
so is O(V⊕^m)/K_+ ▹_O(V), V  C, by Step 1.
The orthogonal G-spectrum
Ω^V( O(V⊕^m)/K_+ ▹_O(V), V  C)
is then G-stably contractible by Proposition <ref> (ii).
In particular, the group (<ref>) is trivial,
so there is a G-representation U such that the composite

    S^U⊕ V⊕^m+k    S^U f  S^U (O(V⊕^m)/K_+) _O(V)  C(V) 
       σ_U,V  O(V⊕^m)/K_+ _O(V)  C(U⊕ V)

is G-equivariantly null-homotopic.
The structure map

    σ_U, V⊕^m :  S^U   (Cα^*((^m,-)/K))(V⊕^m) 
       → (Cα^*((^m,-)/K ))(U⊕ V⊕^m)

of the G-spectrum Cα^*((^m,-)/K ) factors as the composite

    S^U O(V⊕^m)/K_+ _O(V) C(V)
        O(V⊕^m)/K_+ _O(V)σ_U,V
     
    O(V⊕^m)/K_+ _O(V) C(U⊕ V)     → O(U⊕ V⊕^m)/K_+_O(U⊕ V) C(U⊕ V) .

So also the stabilization σ_U,V∘(S^U f)
in the orthogonal G-spectrum Cα^*((^m,-)/K) 
is equivariantly null-homotopic.
Since the stabilization represents the same element as f, this shows that 
π_k^G(Cα^*((^m,-)/K))=0.

Step 3:
We let Γ be a closed subgroup of G× O(m).
We claim that smashing with the orthogonal G-spectrum

    G_m( (G× O(m))/Γ_+)   =  (^m,-)_O(m) (G× O(m))/Γ_+

preserves G-stably contractible orthogonal G-spectra.
To see that we interpret Γ as a `generalized graph'.
We let H≤ G be the projection of Γ onto the first factor.
We let

    K =  { A∈ O(m)  :  (1,A)∈Γ}

be the trace of Γ in O(m). Then Γ is the graph of the continuous 
homomorphism

    α :  H  → W_O(m) K ,   α(h) =  { A∈ O(m) :  (h,A)∈Γ} ,

in the sense that

    Γ =  ⋃_h∈ H {h}×α(h) .

Moreover, the orthogonal G-spectrum 
G_m( (G× O(m))/Γ_+) is isomorphic to

    G⋉_H(α^*((^m,-)/K)) .

So for every orthogonal G-spectrum C, the shearing isomorphism
becomes an isomorphism of  orthogonal G-spectra

    C (G_m ( (G× O(m))/Γ_+) )      
    C ( G⋉_H(α^*((^m,-)/K))) 
         G⋉_H(^G_H(C )α^*((^m,-)/K) ) .

If C is G-stably contractible,
then it is also H-stably contractible;
so ^G_H(C)α^*((^m,-)/K) is  H-stably contractible by Step 2.
Since the induction functor G⋉_H- takes _*-isomorphisms
of orthogonal H-spectra to _*-isomorphisms
of orthogonal G-spectra (Corollary <ref>), this shows that
C G_m((G× O(m))/Γ_+) is G-stably contractible.

Step 4:
We let A be a cofibrant based (G× O(m))-space.
We claim that smashing with the orthogonal G-spectrum G_m A
preserves G-stably contractible orthogonal G-spectra.
A cofibrant based (G× O(m))-space is equivariantly homotopy
equivalent to a based (G× O(m))-CW-complex, so it is no loss of generality
to assume an equivariant CW-structure with skeleton filtration

    ∗ =  A_-1 ⊂ A_0  ⊂ … ⊂ A_n ⊂ … .

We show first, by induction on n, that 
smashing with the orthogonal G-spectrum
G_m A_n preserves G-stably contractible orthogonal G-spectra.
The induction starts with n=-1, where there is nothing to show. 
For n≥ 0 the inclusion
A_n-1→ A_n is an h-cofibration of based (G× O(m))-spaces,
so the induced morphism C G_m A_n-1→ C G_m A_n is an h-cofibration
of orthogonal G-spectra.
By induction and the long exact sequence of equivariant
homotopy groups (Corollary <ref>)
it suffices to show that the quotient 

    ( C G_m A_n ) / ( C G_m A_n-1)    C G_m(A_n / A_n-1)

is G-stably contractible.
Now A_n/A_n-1 is (G× O(m))-equivariantly isomorphic
to a wedge of summands of the form S^n (G× O(m))/Γ_+,
for various closed subgroups Γ of G× O(m).
So the quotient spectrum is isomorphic to the n-fold suspension 
of a wedge of orthogonal G-spectra of the form C G_m((G× O(m))/Γ_+).
These wedge summands are G-stably contractible by Step 3.
Since equivariant homotopy groups take wedges to direct sums
and suspension shifts equivariant homotopy groups,
this proves that C  G_m( A_n/ A_n-1)
is G-stably contractible. This completes the inductive step,
and the proof for all finite-dimensional
based (G× O(m))-CW-complexes.

Since A is the sequential colimit, along h-cofibrations
of based (G× O(m))-spaces, of the skeleta A_n,
the orthogonal G-spectrum
C G_m A is the sequential colimit, along h-cofibrations
of orthogonal G-spectra, of the sequence with terms C G_m A_n. 
Since h-cofibrations are in particular levelwise closed embeddings, 
equivariant homotopy groups commute with such sequential colimits 
(Proposition <ref> (i)), 
so also C G_m A is G-stably trivial.

Step 5:
We let X be a G-flat orthogonal G-spectrum.
We show first, by induction on m, 
that C (^m X) is G-stably contractible,
where ^m  X is the m-skeleton 
in the sense of Construction <ref>.
The induction starts with m=-1, where there is nothing to show. 
For m≥ 0 the morphism
j_m:^m-1 X→^m X is an h-cofibration of orthogonal G-spectra, 
hence so is the induced morphism C j_m:C (^m-1 X)→ C ( ^m X).
By induction and the long exact sequence of equivariant
homotopy groups (Corollary <ref>)
we may show that the quotient 

    (C^m X ) / ( C^m-1 X )    C  (^m X / ^m-1 X)    C  G_m(X(^m)/L_m X)

is G-stably contractible.
Since X is G-flat, X(^m)/L_m X is a cofibrant based (G× O(m))-space,
and the G-equivariant homotopy groups of
C G_m(X(^m)/L_m X) vanish by Step 4.
This finishes the inductive proof that
C (^m X) is G-stably contractible.
The spectrum C X is the sequential colimit,
along h-cofibrations, of the orthogonal G-spectra C (^m  X);
equivariant homotopy groups commute with such colimits, so 
C X is G-stably contractible.

Step 6:
We let X be a G-flat orthogonal G-spectrum and f:A→ B 
a _*-isomorphism of orthogonal G-spectra.
Then the mapping cone C(f) is G-stably contractible
by the long exact homotopy group sequence
(Proposition <ref>).
So the smash product C(f) X is G-stably contractible by Step 5.
Since C(f) X is isomorphic to the mapping cone of f X:A X→ B X,
the morphism f X induces isomorphisms on G-equivariant stable homotopy groups,
again by the long exact homotopy group sequence.




Now we turn to products on equivariant homotopy groups.


Given a compact Lie group G and
two orthogonal G-spectra X and Y, 
we endow the equivariant homotopy groups with 
a pairingsubjectproduct!on global homotopy groups

    × :  π_k^G(X)  × π_l^G (Y)  → π_k+l^G(X Y)  ,

where k and l are integers. We let 

    f : S^U⊕^m+k →  X(U⊕^m)    and  
    g : S^V⊕^n+l →  Y(V⊕^n)

represent classes in π^G_k(X) respectively π^G_l(Y),
for suitable G-representations U and V.
The class [f]×[g] in π_k+l^G(X Y)
is then represented by the composite

    S^U⊕ V⊕^m+n+k+l S^U⊕^m+k S^V⊕^n+l  f g     X(U⊕^m) Y(V⊕^n)
    i_U⊕^m, V⊕^n   (X Y)(U⊕^m⊕ V⊕^n) 
    (X Y)(U⊕τ_^m,V⊕^n)    (X Y)(U⊕ V⊕^m+n) .

The first homeomorphism shuffles the sphere coordinates.
We omit the proof that the class of the
composite only depends on the classes of f and g.





The pairing of equivariant homotopy groups has several expected properties
that we summarize in the next theorem.

 
  Let G be a compact Lie group and X, Y and Z orthogonal G-spectra.
  
  
  * (Biadditivity)
    The product ×:π_k^G(X) ×π_l^G (Y) →π_k+l^G(X Y)
    is biadditive.
  
  * (Unitality)
    The class 1∈π_0^G() represented by the identity of S^0
    is a two-sided unit for the pairing ×.
  
  * (Associativity) For all x∈π_k^G ( X ), y∈π_l^G( Y )
    and z∈π_m^G( Z ) the relation
    
    x×(y× z)  =  (x× y)× z

    holds in π^G_k+l+m(X Y Z).
  
  * (Commutativity) For all x∈π_k^G ( X ) and y∈π_l^G( Y )
    the relation
    
    τ^X,Y_*( x× y )  =  (-1)^kl· ( y× x)

    holds in π^G_k+l( Y X),
    where τ^X,Y:X Y→ Y X is the symmetry isomorphism
    of the smash product.
  
  * (Restriction) For all x∈π_k^G ( X ) and y∈π_l^G( Y )
    and all continuous homomorphisms α:K→ G the relation
    
    α^*(x)×α^*(y) =  α^*(x× y)

    holds in π_k+l^K(α^*(X Y)).
   
  * (Transfer) Let H be a closed subgroup of G.
     For all x∈π_k^G(X) and z∈π_l^H(Y S^L), the relation
     
    x×_H^G(z) =  _H^G(^G_H(x)× z)

     holds in π_k+l^G(X Y), where L=T_e H(G/H)
     is the tangent H-representation.
     For all y∈π_l^H(Y), the relation
     
    x×_H^G(y) =  _H^G(^G_H(x)× y)

     holds in π_k+l^G(X Y).
  


(i)  We deduce the additivity in the first variable from the
general additivity statement in Proposition <ref>.
We consider the two reduced additive functors

    X ⟼ π_k^G(X)     and     
    X⟼ π_k+l^G( X Y)

from the category of orthogonal G-spectra to the category of abelian groups.
Proposition <ref> shows that for every
y∈π_l^G(Y) the natural transformation

    -× y  :  π_k^G(-) → π_k+l^G( - Y)

is additive.
Additivity in the second variable is proved in the same way.
Properties (ii), (iii) and (v) are straightforward consequences
of the definitions.
The sign (-1)^k l in the commutativity relation (iv)
is the degree of the map that interchanges a k-sphere with an l-sphere.
Indeed, if

    f  :  S^U⊕^m+k → X(U⊕^m)     and  
     g  :  S^V⊕^n+l → Y(V⊕^n)

represent classes in π_k^G(X) respectively π_l^G(Y), then 
the left and right vertical composites in the diagram

    @C=20mm@R=6mm
    S^U⊕ V⊕^m+n+k+l[d]_f× g[r]^-τ_U,Vτ_m,n S^k+l   
     S^V⊕ U⊕^n+m+k+l[d]^S^V⊕ U⊕^n+mτ_k,l
    
    (X Y)(U⊕ V⊕^m+n) [d]_τ^X,Y(U⊕ V⊕^m+n)   
     S^V⊕ U⊕^n+m+l+k[d]^g× f
    
    (Y X)(U⊕ V⊕^m+n)[r]_-(Y X)(τ_U,V⊕τ_m,n)   
     (Y X)(V⊕ U⊕^n+m)

represent τ^X,Y_*([f]×[g]), respectively (-1)^k l·([g]×[f]).
Since the two composites differ by conjugation by a G-equivariant linear isometry,
they represent the same class 
by Proposition <ref> (ii).
 
(vi) The following diagram of abelian groups
commutes by naturality of the pairing (<ref>), and because 
restriction from G to H is multiplicative:
   
    @C=6mm@R=6mmπ_k^G(X) ×π_l^G(Y G/H_+) 
         @<-10ex>@/_1pc/[dd]_(.2)×_H^G[r]^-×[d]^×_H^G   π_k+l^G(X Y G/H_+) [d]_^G_H@/^3pc/[dddr]^-_H^G
    π_k^G(X) ×π_l^H(Y G/H_+) 
      [d]^× (Y l)_*[dr]^^G_H×   π_k+l^H(X Y G/H_+)
      @/^2pc/[ddr]_(.8)(X Y l)_*   
    π_k^G(X) ×π_l^H(Y S^L) [dr]_^G_H×   π_k^H(X)×π^H_l(Y G/H_+) [d]^× (Y l)_*[u]_×   
       π_k^H(X) ×π_l^H(Y S^L)       [r]_-×   π_k+l^H(X Y S^L)

Here l:G/H_+→ S^L is the H-equivariant collapse map (<ref>).
The two vertical composites are the respective Wirthmüller isomorphisms
(Theorem <ref>).
Since the external transfer is inverse to the Wirthmüller isomorphism
(up to the effect of the involution S^-:S^L→ S^L),
we can read the diagram backwards and conclude that the upper part of the
following diagram commutes:
   
    @C=10mm@R=6mmπ_k^G(X) ×π_l^H(Y S^L)    [r]^-^G_H×@<-8ex>@/_2pc/[ddd]_(.3)×_H^G[dd]^× (G⋉_H-)_   π_k^H(X) ×π_l^H(Y S^L)  [d]^-×
       π_k+l^H(X Y S^L) [d]^_G⋉_H- @<6ex>@/^2pc/[dd]^(.3)_H^G
    π_k^G(X)×π_l^G(Y G/H_+) [r]_-×[d]^× (Y p)_*   π_k+l^G(X Y G/H _+) [d]_(X Y p)_*
    π_k^G(X)×π_l^G(Y) [r]_-×   π^G_k+l ( X Y)

   Here p:G/H_+→ S^0 is the projection.
   The lower part of the diagram commutes by naturality of the pairing. 
   This proves the first claim about dimension shifting transfers.
   The formula for the degree zero transfers follows by naturality 
   for the inclusion of the origin S^0→ S^L.





An orthogonal ring spectrum
subjectorthogonal ring spectrum 
subjectring spectrum!orthogonal|seeorthogonal ring spectrum 
is a monoid in the category of orthogonal spectra with respect to the smash product.
For a compact Lie group G, an
orthogonal G-ring spectrum
subjectorthogonal G-ring spectrum 
subjectG-ring spectrum@G-ring spectrum!orthogonal|seeorthogonal G-ring spectrum 
is a monoid in the category of orthogonal G-spectra with respect to the smash product.


An orthogonal ring spectrum is thus an orthogonal spectrum R 
equipped with a multiplication morphism
μ:R R→ R and a unit morphism η:→ R such that the associativity and
unit diagrams commute (compare <cit.>).
A morphismsubjectmorphism!of orthogonal ring spectra
of orthogonal ring spectra is a morphism f:R→ S of orthogonal spectra
that satisfies f∘μ^R=μ^S∘(f f) and f∘η^R=η^S.
Via the universal property of the smash product the data contained in the
multiplication morphism can be made more explicit: μ:R R→ R corresponds to
a collection of based continuous maps μ_V,W:R(V) R(W)→ R(V⊕ W)
that together form a bimorphism. The associativity and unit conditions
can also be rephrased in more explicit forms, and then we are requiring
that the multiplication and unit maps make R:→_* into a lax monoidal functor.
Most of the time we will specify the data of an orthogonal ring spectrum
in the explicit bimorphism form.

An orthogonal ring spectrum R (respective orthogonal G-ring spectrum) is 
commutativesubjectorthogonal ring spectrum!commutative if the 
multiplication morphism satisfies μ∘τ_R,R=μ. In the explicit form
this is equivalent to the commutativity of the square

    @C=12mm R(V) R(W) [r]^-μ_V,W[d]_τ_R(V), R(W)   
    R(V⊕ W)[d]^-R(τ_V,W)
    
    R(W) R(V) [r]_-μ_W,V    R(W⊕ V)

for all inner product spaces V and W.
Equivalently, the multiplication and unit maps make R:→_* 
into a lax symmetric monoidal functor.
Commutative orthogonal ring spectra already appear, 
with an extra pointset topological hypothesis and
under the name ℐ_*-prefunctor, in <cit.>.

Since the smash product of orthogonal G-spectra is just
the smash product of the underlying orthogonal spectra, endowed with
the diagonal G-action, an orthogonal G-ring spectrum is nothing
but an orthogonal ring spectrum equipped with a continuous G-action
through homomorphisms of orthogonal ring spectra.
Via the universal property of the smash product,
yet another way to package the data in an 
orthogonal G-ring spectrum is as a continuous lax monoidal functor
from the category  to the category of based G-spaces.




Given a compact Lie group G and an orthogonal G-ring spectrum R, 
we define an internal pairing

    · :  π_k^G (R) ×π_l^G (R)  → π_k+l^G(R)

on the equivariant homotopy groups of R as the composite

    π_k^G (R) ×π_l^G (R)   × π_k+l^G(R R)   μ_*  π_k+l^G(R) .

Theorem <ref> then immediately implies:

 
  Let G be a compact Lie group, R an orthogonal G-ring spectrum
  and H a closed subgroup of G.
  
  
  * The products (<ref>)
    make the abelian groups {π_k^G(R)}_k∈ into a graded ring.
    The multiplicative unit is the class of the unit map S^0→ R(0).
  
  * If the multiplication of R is commutative, then the relation
    
    x· y =   (-1)^kl· y · x

    holds for all classes x∈π_k^G(R) and y∈π^G_l(R). 
 
  * (Restriction) The restriction maps
    ^G_H:π_*^G(R)→π_*^H(R) form a homomorphism of graded rings.
 
  * (Conjugation) For every g∈ G
   the conjugation maps g_⋆:π_*^H^g(R)→π_*^H(R) 
   form a homomorphism of graded rings.subjectconjugation homomorphism!on equivariant homotopy groups  
  
  * (Reciprocity) 
     For all x∈π_k^G(R) and all y∈π_l^H(R), the relation
     
    x·_H^G(y) =  _H^G(^G_H(x)· y)

     holds in π_k+l^G(R).
  



[Norm maps]
A lot more is happening for commutative orthogonal G-ring spectra:
strict commutativity of the multiplication not only makes the
homotopy pairings graded-commutative, but it also gives rise to new operations,
usually called norm mapssubjectnorm map
N_H^G:π_0^H(R)→π_0^G(R) for all closed subgroups
H of finite index in G.
To my knowledge, these norm maps were first defined by
Greenlees and May <cit.>,
and they were later studied for example in <cit.>.

We don't discuss norms maps for commutative orthogonal G-ring spectra here,
but norms maps and power operations will be a major topic 
in the context of ultra-commutative ring spectra.
In Section <ref> we define
power operations on the 0-th equivariant homotopy groups of
an ultra-commutative ring spectrum, and we recall in 
Remark <ref> how to turn the power operations into norm maps.






CHAPTER: GLOBAL STABLE HOMOTOPY THEORY




In this chapter we embark on the investigation of
global stable homotopy theory.
In Section <ref> we specialize the equivariant 
theory of the previous chapter to global stable homotopy types,
which we model by orthogonal spectra (with no additional action of any groups). 
Section <ref> introduces the category of global functors,
the natural home of the collection of equivariant
homotopy groups of a global stable homotopy type (i.e., an orthogonal spectrum).
Global functors play the same role
for global homotopy theory as the category of abelian groups
in ordinary homotopy theory, or the category of G-Mackey functors
for G-equivariant homotopy theory.
Global functors are defined as additive functors on the global Burnside category;
an explicit calculation of the global Burnside category 
provides the link to other notions of global Mackey functors.
In the global context, the pairings on equivariant homotopy groups also
give rise to a symmetric monoidal structure on the global Burnside category,
and to a symmetric monoidal `box product' of global functors.


Section <ref> establishes the global model structure
on the category of orthogonal spectra; more generally,
we consider a global family  and define the
-global model structure, where weak equivalences are tested
on equivariant homotopy groups for all Lie groups in .
The  -global model structure is monoidal with
respect to the smash product of orthogonal spectra,
provided that  is closed under products.
Section <ref> collects aspects of global stable homotopy theory that
refer to the triangulated structure of the global stable homotopy category.
Specific topics are  compact generators, Brown representability,
a t-structure whose heart is the category of global functors,
global Postnikov sections and Eilenberg-Mac Lane spectra of global functors.

The final Section <ref> is a systematic study of the effects
of changing the global family. We show that 
the `forgetful' functor between the global stable homotopy categories
of two nested global families has fully faithful left and right adjoints, 
which are part of a recollement.
We provide characterizations of the global homotopy types in the image of the 
two adjoints; for example,
the right adjoint all the way from the non-equivariant to the
global stable homotopy category models Borel cohomology theories.
We also relate the global homotopy category to the G-equivariant stable homotopy
category for a fixed compact Lie group G;
here the forgetful functor also has both adjoints, but these are no longer
fully faithful if the group is non-trivial.

Also in  Section <ref> we
establish an algebraic model for rational -global
stable homotopy theory, i.e., for rational
global stable homotopy theory based on the global family of finite groups.
Indeed, spectral Morita theory provides a chain of Quillen equivalences 
to the category of chain complexes of rational global functors on finite groups.
On the algebraic side, the abelian category of rational -global functors 
is equivalent to an even simpler category, namely functors from finite groups 
and conjugacy classes of epimorphisms to -vector spaces.
Under the two equivalences, homology groups of chain complexes
correspond to equivariant stable homotopy groups,
respectively geometric fixed point homotopy groups, of spectra.





§ ORTHOGONAL SPECTRA AS GLOBAL HOMOTOPY TYPES



In this section we specialize the equivariant stable homotopy
theory of Chapter <ref> to global stable homotopy types,
which we model by orthogonal spectra (with no additional group action). 
Given an orthogonal spectrum X and a compact Lie group G,
we obtain an orthogonal G-spectrum by letting G act trivially
on the values of X.
We call this the underlying orthogonal G-spectrum of X
and denote it X_G.
subjectunderlying G-spectrum!of an orthogonal spectrum
symbolX_G - underlying G-spectrum of an orthogonal spectrum
To simplify notation we omit the subscript `_G' when we refer
to equivariant homotopy groups, i.e., we simply write π_k^G(X)
instead of π_k^G(X_G).

For global homotopy types (i.e., orthogonal spectra),
the notation related to restriction maps simplifies, and some special features happen.
Indeed, if X is an orthogonal spectrum, then for every continuous homomorphism
α:K→ G we have α^*(X_G)=X_K, because both K and G act trivially.
So for global homotopy types the restriction maps become homomorphisms

    α^*  :  π_*^G(X)  → π_*^K(X) .

In particular, we have these restriction maps when α is an epimorphism;
in that case we refer to α^* as an inflation map.
subjectinflation map!of equivariant point homotopy groups  

If H is a closed subgroup of G and g∈ G, the
conjugation homomorphism c_g:H→ H^g is given by c_g(h)=g^-1h g.
We recall that for an orthogonal G-spectrum Y the conjugation morphism
g_⋆:π_0^H^g(Y)→π_0^H(Y) is defined as
g_⋆=(l_g^Y)_*∘ (c_g)^∗, where l_g^Y:c_g^*(Y)→ Y is left multiplication by g,
compare (<ref>).
If Y=X_G for an orthogonal spectrum X, then l_g^X is the identity;
so for global homotopy types we have

    g_⋆ =  (c_g)^*  :  π_0^H^g(X)  → π_0^H(X) .

A different way to express the significance of the relation (<ref>)
is that whenever Y=X_G underlies a global homotopy type, 
then the action of the group W_G H=(N_G H)/H
on π_0^H(X) discussed in Construction <ref>
factors through an action of the quotient group

    (N_G H) / (H· C_G H) .

The difference is illustrated most drastically for H=e, the trivial subgroup of G.
For a general orthogonal G-spectrum the action of G=W_G e
on π_0^G(Y) is typically non-trivial. If Y=X_G arises from
an orthogonal spectrum, then this G-action is trivial.

[Global homotopy types are split G-spectra]
Obviously, only very special orthogonal G-spectra Y are part of
a `global family', i.e., arise as X_G for an orthogonal spectrum X. 
However, it is not a priori clear what the homotopical
significance of the pointset level condition
that G must act trivially on the values of Y (at trivial representations) is.
Now we formulate obstructions to `being global' in terms 
of the Mackey functor homotopy groups of an orthogonal G-spectrum.subjectMackey functor

The equivariant homotopy groups π_0^G(X)=π_0^G(X_G) of
an orthogonal spectrum X come equipped with restriction maps along arbitrary
continuous group homomorphisms, not necessarily injective. 
This is in contrast to the situation for
a fixed compact Lie group, where one can only restrict to subgroups,
or along conjugation maps by elements of the ambient group,
but where there are no inflation maps.
One obstruction to Y being part of a `global family'
is that the G-Mackey functor structure can be extended to
a `global functor' (in the sense of Definition <ref> below).
In particular, the G-Mackey functor homotopy groups
can be complemented by restriction maps 
along arbitrary group homomorphisms between the subgroups of G. 
So every global functor takes isomorphic
values on a pair of isomorphic subgroups of G;
for G-Mackey functors this is true when the subgroups are conjugate in G, 
but not in general when they are merely abstractly isomorphic.

As an extreme case the global structure includes an inflation map
p_G^*:π_0^e(X)→π_0^G(X) associated to the unique homomorphism p_G:G→ e,
splitting the restriction map π_0^G(X)→π_0^e(X). 
So one obstruction to being global is
that this restriction map from π_0^G(X) to π_0^e(X)
needs to be a split epimorphism.
This is the algebraic shadow of the fact that the G-equivariant homotopy types
that underlie global homotopy types (i.e., are represented by orthogonal spectra)
are `split'subjectsplit G-spectrum in the sense that there
is a morphism from the underlying non-equivariant spectrum to
the genuine G-fixed point spectrum that splits the restriction map.




A morphism f:X→ Y of orthogonal spectra 
is a global equivalencesubjectglobal equivalence!of orthogonal spectra
if the induced map π_k^G(f):π_k^G(X) →π_k^G(Y) is an isomorphism
for all compact Lie groups G and all integers k.



We define the global stable homotopy category symbol   - global stable homotopy categorysubjectglobal stable homotopy category
by localizing the category of orthogonal spectra at the class of
global equivalences.
The global equivalences are the weak equivalences of the
global model structure on the category of orthogonal spectra,
see Theorem <ref> below. 
So the methods of homotopical algebra are available for studying global
equivalences and the associated global homotopy category.
In later sections we will also consider a relative notion of
global equivalence, the `-equivalences', 
based on a global family subjectglobal family
of compact Lie groups. There we require that 
the induced map π_k^G (f): π_k^G (X)→π_k^G(Y) is an isomorphism
for all integers k and all compact Lie groups G that belong to
the global family .


When we specialize Proposition <ref> (ii),
Corollary <ref>,
Proposition <ref>,
Proposition <ref> (ii)
and Proposition <ref>
to the underlying orthogonal G-spectra of orthogonal spectra, we obtain
the following consequences:



  
  
  * For every orthogonal spectrum X the morphism
    
    λ_X :   X S^1  →  X   ,
           its adjoint  λ̃_X  :  X  → Ω X  ,

    the adjunction unit η_X:X→Ω(X S^1) 
    and the adjunction counit ϵ_X:(Ω X) S^1→ X 
    are global equivalences.
  
  * For every finite family of orthogonal spectra
    the canonical morphism from the wedge to the product 
    is a global equivalence.
  
  * For every finite based CW-complex A, the functor _*(A,-)
    preserves global equivalences of orthogonal spectra.
  
  * For every cofibrant based space A, 
    the functor - A preserves global equivalences of orthogonal spectra.
  
  * Let e_n:X_n→ X_n+1 and f_n:Y_n→ Y_n+1 be morphisms 
    of orthogonal spectra that are levelwise closed embeddings, for n≥ 0. 
    Let ψ_n:X_n→ Y_n be global equivalences
    that satisfy ψ_n+1∘ e_n=f_n∘ψ_n for all n≥ 0.
    Then the induced morphism ψ_∞:X_∞→ Y_∞ 
    between the colimits of the sequences is a global equivalence.
  
  * Let f_n:Y_n→ Y_n+1 be global equivalences of orthogonal spectra
   that are levelwise closed embeddings, for n≥ 0. 
   Then the canonical morphism 
   f_∞:Y_0→ Y_∞ to a colimit of the sequence {f_n}_n≥ 0
   is a global equivalence.
  



For later use we record another closure property of global equivalences.


The class of h-cofibrations of orthogonal spectra
that are simultaneously global equivalences
is closed under cobase change, coproducts and sequential compositions.


The class of h-cofibrations is closed under coproducts, cobase change
and composition (finite or sequential), 
compare Corollary <ref> (i).
The class of global equivalences is closed under coproducts
because equivariant homotopy groups take wedges to direct sums 
(Corollary <ref> (i)).
The cobase change of an h-cofibration that is also a global equivalence is 
another global equivalence 
by Corollary <ref> (i).
Every h-cofibration of orthogonal spectra is in particular levelwise
a closed embedding. So the class of h-cofibrations that are also global equivalences
is closed under sequential 
composition by Proposition <ref> (vi).






For an orthogonal spectrum X the 0-th equivariant homotopy groups π_0^G(X)
and the restriction maps between them coincide with the homotopy Rep-functor,
in the sense of (<ref>),
of a certain orthogonal space Ω^∙ X that we now recall.



We introduce the functor

    Ω^∙ :   →

from orthogonal spectra to orthogonal spaces.symbolΩ^∙ - underlying orthogonal space of an orthogonal spectrum
Given an orthogonal spectrum X, the value of 
Ω^∙ X at an inner product space V is

    (Ω^∙ X)(V) =  _*(S^V,X(V)) .

If φ:V→ W is a linear isometric embedding, the induced map

    φ_*  :  (Ω^∙ X)(V) =  _*(S^V,X(V))  → _*(S^W,X(W)) =  (Ω^∙ X)(W)

was defined in (<ref>).
In particular, the orthogonal group O(V) acts on 
(Ω^∙ X)(V)=_*(S^V,X(V)) by conjugation.
If ψ:U→ V is another isometric embedding, then we have
φ_*(ψ_* f)  =  (φψ)_*f.
The assignment (φ,f)↦φ_*f is continuous in both variables,
so we have really defined an orthogonal space Ω^∙ X.
The construction is clearly functorial in the orthogonal spectrum X;
moreover, Ω^∙ has a left adjoint `unreduced suspension spectrum'
functor Σ^∞_+ that we discuss 
in Construction <ref> below.

If G acts on V by linear isometries, then the 
G-fixed subspace of  (Ω^∙ X)(V) is the space
of G-equivariant based maps from S^V to X(V):

    ((Ω^∙ X)(V))^G =  _*^G(S^V,X(V)) .

The path components of this space are precisely the equivariant homotopy
classes of based G-maps, i.e.,

    π_0(((Ω^∙ X)(V))^G) =  π_0 ( _*^G(S^V,X(V)) )  =  [S^V,X(V)]^G .

Passing to colimits over the poset s(_G) gives

    π_0^G(Ω^∙ X) =  π_0^G(X) ,

i.e., the G-equivariant homotopy group of the orthogonal spectrum X
equals the G-equivariant homotopy set 
of the orthogonal space Ω^∙ X.
A direct inspection shows that the spectrum level restriction maps
defined in Construction <ref>subjectrestriction map!of equivariant stable homotopy groups
coincide with the restriction maps for orthogonal spaces
introduced in (<ref>).


[Suspension spectra of orthogonal spaces] 
To every orthogonal space Y we can associate an unreducedsymbolΣ^∞ - suspension spectrum of an orthogonal space
suspension spectrumsubjectsuspension spectrum!of an orthogonal space
 Σ^∞_+Y whose value on an inner product space

    (Σ^∞_+ Y)(V) =  S^V Y(V)_+ ;

here the orthogonal group O(V) acts diagonally and the
structure map

    σ_V,W :  
    S^V (S^W Y(W)_+)  →  S^V⊕ W Y(V⊕ W)_+

is the combination of the canonical homeomorphism S^V S^W S^V⊕ W
and the map Y(i_W):Y(W)→ Y(V⊕ W).
If Y is the constant orthogonal space with value a topological space A,
then Σ^∞_+Y=Σ^∞_+A specializes to the suspension spectrum
of A with a disjoint basepoint added.
The functor

    Σ^∞_+  :   →

is left adjoint to the functor Ω^∙ 
of Construction <ref>.


Let G be a compact Lie group and {V_i}_i≥ 1 
an exhaustive sequencesubjectexhaustive sequence!of representations 
of finite-dimensional G-subrepresentations of the complete universe _G. 
Given an orthogonal space Y, we denote by
_i Y(V_i) the mapping telescope of the sequence of G-spaces

    Y(V_1)  → Y(V_2)  → ⋯ → Y(V_i)  → ⋯ ;

the maps in the sequence are induced by the inclusions,
so they are G-equi­variant, and the telescope inherits
a natural G-action. The canonical maps Y(V_j)→_i Y(V_i)
induce maps of equivariant homotopy classes

    [S^V_j⊕^k, (Σ^∞_+ Y)(V_j)]^G  =     [S^V_j⊕^k,S^V_j Y(V_j)_+]^G 
    →    [S^V_j⊕^k,S^V_j_i Y(V_i)_+]^G →  π_k^G(Σ^∞_+ _i Y(V_i)) ,

where the second map is the canonical one to the colimit.
These maps are compatible with stabilization in the source when we increase j.
Since the exhaustive sequence is cofinal in the poset s(_G),
the colimit over j calculates the k-th equivariant homotopy group
of Σ^∞_+ Y. So altogether, the maps assemble into a natural 
group homomorphism

    π_k^G(Σ^∞_+ Y)  → π_k^G(Σ^∞_+ _i Y(V_i) )  ,

for k≥ 0. For negative values of k, we obtain a similar map
by inserting ^-k in second variable of the above sets of equivariant homotopy
classes.



Let G be a compact Lie group and {V_i}_i≥ 1 an exhaustive sequence 
of G-representations. Then for every orthogonal space Y 
and every integer k the map

    π_k^G(Σ^∞_+ Y)  → π_k^G(Σ^∞_+ _i Y(V_i) )

is an isomorphism.


  The mapping telescope is the colimit, along h-cofibrations,
  of the truncated mapping telescopes _[0,n] Y(V_i).
  So the space S^V_j_i Y(V_i)_+ 
  is the co­limit, along a sequence of closed embeddings,
  of the spaces S^V_j_[0,n] Y(V_i)_+. 
  For every compact based G-space A the canonical map
  
    _n≥ 1  [A, S^V_j_[0,n] Y(V_i)_+ ]^G  → 
      [A, S^V_j_i Y(V_i)_+ ]^G

  is thus bijective. Indeed, every continuous map from the compact spaces A
  and A [0,1]_+ to S^V_j_i Y(V_i)_+ factors through 
  S^V_j_[0,n] Y(V_i) some n, 
  for example by Proposition <ref>.
  The canonical map Y(V_n)→_i Y(V_i)
  factors through Y(V_n)→_[0,n] Y(V_i), and this factorization
  is an equivariant homotopy equivalence. So we can replace 
  _[0,n] Y(V_i) by Y(V_n) and conclude that the map
  
    _n≥ 1  [A, S^V_j Y(V_n)_+ ]^G  → 
      [A, S^V_j_i Y(V_i)_+ ]^G

  is bijective.
  We specialize to A=S^V_j⊕^k and pass to the colimit
  over j by stabilization in source and target. The result is a bijection
  
    _j≥ 1_n≥ 1    [S^V_j⊕^k, S^V_j Y(V_n)_+ ]^G 
       → _j≥ 1    [S^V_j⊕^k, S^V_j_i Y(V_i)_+ ]^G  .

  The source is isomorphic to the diagonal colimit 
  
    _j≥ 1   [S^V_j⊕^k, S^V_j Y(V_j)_+ ]^G

  by cofinality.
  Since the exhaustive sequence is cofinal in the poset s(_G),
  the two colimits over j also calculate the colimits over s(_G),
  and hence the k-th equivariant homotopy group of 
  Σ^∞_+ Y respectively Σ^∞_+ _i Y(V_i).
  This shows the claim for k≥ 0. 
  For negative values of k, the argument is similar: 
  we insert ^-k into the second variable 
  of the sets of equivariant homotopy classes.




The unreduced suspension spectrum functor
Σ^∞_+ takes global equivalences of orthogonal spaces
to global equivalences of orthogonal spectra.   


Let f:X→ Y be a global equivalence of orthogonal spaces and G
a compact Lie group. We choose an exhaustive sequence 
of G-representations {V_i}_i≥ 1. Then the G-map

    _i f(V_i) :  _i X(V_i) → _i Y(V_i)

is a G-weak equivalence
by Proposition <ref> (iii). 
So the map of suspension spectra Σ^∞_+_i f(V_i)
induces isomorphisms on all G-equivariant stable homotopy groups, by
Proposition <ref>.
Since the group π_k^G(Σ^∞_+_i X(V_i))
is naturally isomorphic to the group π_k^G(Σ^∞_+ X)
(by Proposition <ref>),
the morphism of orthogonal suspension 
spectra Σ^∞_+f:Σ^∞_+X→Σ^∞_+ Y
induces an isomorphism on π_*^G.







Now we let Y be an orthogonal space and K a compact Lie group.
We define a map

    σ^K :  π_0^K (Y)  → π_0^K(Σ^∞_+ Y)

as the effect of the adjunction unit Y→Ω^∙(Σ^∞_+ Y)
on the  K-equivariant homotopy set π_0^K, using that
π_0^K(Ω^∙(Σ^∞_+ Y))=π_0^K(Σ^∞_+ Y).
If we unravel the definitions, this comes out as follows: 
if V is a finite-dimensional K-subrepresentation
of the complete K-universe _K and y∈ Y(V)^K a K-fixed point, then
σ^K[y] is represented by the K-map

    S^V - y S^V Y(V)_+  =  (Σ^∞_+ Y)(V)  .

As K varies, the maps σ^K are compatible with restriction
along continuous homomorphisms,
since they arise from a morphism of orthogonal spaces.
By the same argument as for orthogonal K-spectra 
in (<ref>),
transferring from a closed subgroup L to K 
annihilates the action of the Weyl group on π_0^L(Y).

The next proposition is a global analog
of Theorem <ref> (i). 
We call an orthogonal spectrum globally connectivesubjectglobally connective
if its G-equivariant homotopy groups vanish in negative dimensions,
for all compact Lie groups G.

 
Let Y be an orthogonal space. 
Then the suspension spectrum Σ^∞_+ Y is globally connective.
Moreover, for every compact Lie group K the equivariant homotopy group
π_0^K(Σ^∞_+ Y) is a free abelian group with a basis 
given by the elements

    _L^K(σ^L(x))  ,

where L runs through all conjugacy classes of closed subgroups of K
with finite Weyl group and x runs through a set of representatives 
of the W_K L-orbits of the set π_0^L(Y).


We consider the functor on the product poset s(_K)^2 sending
(U,V) to the set [S^V, S^V Y(U)_+]^K.
The diagonal is cofinal in s(_K)^2, so the induced map 

    π_0^K(Σ^∞_+ Y) =  _V∈ s(_G)   [S^V, S^V Y(V)_+]^K 
       → _(V,U)∈ s(_K)^2  [S^V, S^V Y(U)_+]^K

is an isomorphism.
The target can be calculated in two steps, hence the group we are after
is isomorphic to

    _U∈ s(_K)( _V∈ s(_K)  [S^V, S^V Y(U)_+]^K )
     =  _U∈ s(_K)π_0^K(Σ^∞_+ Y(U) ) .

We may thus show that the latter group is free abelian with the specified basis.

Theorem <ref> (i) identifies 
the equivariant homotopy group π_0^K(Σ^∞_+ Y(U) )
as the free abelian group with basis the classes _L^K(σ^L(x)),
where L runs through all conjugacy classes of closed subgroups of K
with finite Weyl group and x runs through a set of representatives 
of the W_K L-orbits of the set π_0(Y(U)^L).
Passage to the colimit over U∈ s(_K) yields the proposition.





We let G be a compact Lie group and  V a  G-representation.
We recall from (<ref>) the tautological class

    u_G,V ∈ π_0^G (_G,V)

in the G-equivariant homotopy set of the semifree orthogonal space _G,V.
The stable tautological classsubjectstable tautological classsubjecttautological class!stable|seestable tautological classsymbole_G,V - stable tautological class in π_0^G(Σ^∞_+ _G,V) is

    e_G,V =  σ^G(u_G,V) ∈ π_0^G(Σ^∞_+ _G,V)  .

Explicitly, e_G,V is the homotopy class of the G-map

    S^V  → S^V ((V,V)/G)_+  =  (Σ^∞_+ _G,V)(V)  ,    
    v ⟼ v (_V· G)  .


In the following corollary we index certain homotopy classes by pairs (L,α)
consisting of a closed subgroup L of a compact Lie group K
and a continuous homomorphism α:L→ G.
The conjugate of (L,α) by a pair (k,g)∈ K× G is the pair
(L^k,c_g∘α∘ c_k^-1) consisting of the
conjugate subgroup L^k and the composite homomorphism

    L^k   c_k^-1  L   α  
    G   c_g  G  .


subjectglobal classifying space  
Let G and K be compact Lie groups and V a faithful G-representation. 
Then the homotopy group π_0^K(Σ^∞_+ _G,V)
is a free abelian group with basis given by the classes

    _L^K(α^*(e_G,V))

as  (L,α) runs over a set of representatives of
all (K× G)-conjugacy classes of pairs consisting of a closed subgroup L of K 
with finite Weyl group and a continuous homomorphism α:L→ G. 

 
The map

    (K,G)  → π_0^K(_G,V) ,   
    [α:K→ G]  ⟼ α^*(u_G,V)

is bijective according to Proposition <ref> (ii).
Proposition <ref> thus says that
π_0^K(Σ^∞_+ _G,V)
is a free abelian group with a basis given by the elements

    _L^K(σ^L(α^*(u_G,V))) =  _L^K(α^*(σ^G(u_G,V))) =  _L^K(α^*(e_G,V))

where L runs through all conjugacy classes of closed subgroups of K 
with finite Weyl group and α runs through a set of representatives of the 
W_KL-orbits of the set (L,G).
The claim follows because (K× G)-conjugacy classes of such pairs
(L,α) biject with pairs 
consisting of a conjugacy class of subgroups (L) and
a W_KL-equivalence class in (L,G).





We discuss a specific example of Corollary <ref>,
with G=A_3 the alternating group on three letters and 
K=Σ_3 the symmetric group on three letters.
The group Σ_3 has four conjugacy classes of subgroups, with representatives
Σ_3, A_3, (12) and e. 
The groups Σ_3, (12) and e admit only trivial homomorphisms to A_3,
whereas the alternating group A_3 also has two automorphisms.
None of the three endomorphisms of A_3 are conjugate,
so the set (A_3,A_3) has three elements.
However, the Weyl group W_Σ_3A_3 has two elements, and its action
realizes the two automorphisms of A_3.
So while π_0^A_3(B_A_3)(A_3,A_3) has three elements,
it only contributes two generators to the stable group
π_0^Σ_3(Σ^∞_+ B_A_3).
A basis for the free abelian group π_0^Σ_3(Σ^∞_+ B_A_3)
is thus given by the classes

    p_Σ_3^*(1)  ,   _A_3^Σ_3(e_A_3)
     ,   _A_3^Σ_3(p_A_3^*(1))
     ,   _(12)^Σ_3(p_(12)^*(1))
       and  _e^Σ_3(p_e^*(1)) .

Here the faithful A_3-representation V is unspecified and 
we write e_A_3=e_A_3,V∈π_0^A_3(Σ^∞_+ B_A_3) 
for the stable tautological class.
Moreover p_H:H→ e denotes the unique homomorphism to the trivial group
and 1=^A_3_e(e_A_3) is the restriction of the class e_A_3 to the
trivial group.



Now we discuss how multiplicative features related to the smash 
product and the homotopy group pairings work out for global homotopy types.
In Construction <ref> we associated
an orthogonal space Ω^∙ X to every orthogonal spectrum X.
This functor is compatible with the smash product of orthogonal
spectra and the box product of orthogonal spaces, 
in the sense of a lax symmetric monoidal transformation

    ( Ω^∙ X)⊠( Ω^∙ Y)  → Ω^∙ ( X Y)  .

This morphism is associated to a bimorphism from (Ω^∙ X,Ω^∙ Y) to
Ω^∙ ( X Y) with (V,W)-component the composite

    _*(S^V,X(V)) ×_*(S^W,Y(W))       _*(S^V⊕ W,X(V) Y(W))  
    _*(S^V⊕ W, i_V,W)    _*(S^V⊕ W,(X Y)(V⊕ W))  .

The morphism is unital, associative and symmetric. 
Finally, the homotopy group pairing (<ref>) 
for k=l=0 coincides with the composite

    π_0^G(X)  ×π_0^G (Y)  =  π_0^G(Ω^∙ X)  ×π_0^G (Ω^∙ Y)   ×     π_0^G(Ω^∙ X ⊠Ω^∙ Y) 
    (<ref>)    π_0^G(Ω^∙(X Y))   =  π_0^G(X Y)  .




[Orthogonal ring spectra and orthogonal monoid spaces]
 
The monoidal morphism (<ref>) 
for the functor Ω^∙ is unital, associative and symmetric. 
In particular the orthogonal space Ω^∙ R
associated to an orthogonal ring spectrum becomes an orthogonal monoid space
via the composite

    ( Ω^∙ R)⊠( Ω^∙ R)  → Ω^∙ ( R R)  Ω^∙μ Ω^∙ R ,

and this passage preserves commutativity of multiplications.
The bimorphism corresponding to the induced product on Ω^∙ R 
thus has as (V,W)-component the composite

    _*(S^V, R(V)) ×_*(S^W, R(W))   -  -      _*(S^V⊕ W, R(V) R(W)) 
    _*(S^V⊕ W, μ_V,W)    _*(S^V⊕ W, R(V⊕ W))  .


The suspension spectrum functor (see Construction <ref>)
takes the box product of orthogonal spaces to the smash product of orthogonal spectra.
In more detail: for all inner product spaces V and W the maps

    (Σ^∞_+ X )(V)(Σ^∞_+ Y )(W)  =     ( S^V X(V)_+ )  (S^W Y(W)_+)
        S^V⊕ W  (X(V)× Y(W))_+ S^V⊕ W  i_X,Y
       S^V⊕ W (X⊠ Y)(V⊕ W)_+  =  
    (Σ^∞_+ (X⊠ Y))(V⊕ W)

form a bimorphism, so they correspond to a morphism of orthogonal spectra

    (Σ^∞_+ X) (Σ^∞_+ Y)  → Σ^∞_+ (X⊠ Y)  .




For all orthogonal spaces X and Y 
the morphism (<ref>) is an isomorphism. Together with the 
unique isomorphism Σ^∞_+∗ 
this makes Σ^∞_+ into a strong symmetric monoidal functor
from the category of orthogonal spaces to the category of orthogonal spectra.  


We consider the composite

    X⊠ Y    η_X⊠η_Y Ω^∙( Σ^∞_+ X)⊠Ω^∙( Σ^∞_+ Y)(<ref>) Ω^∙( (Σ^∞_+ X) (Σ^∞_+ Y)) ,

where η:→Ω^∙∘Σ^∞_+ is the adjunction unit.
This composite is adjoint to a morphism of orthogonal spectra
Σ^∞_+(X⊠ Y)→ (Σ^∞_+ X) (Σ^∞_+ Y).
We omit the verification that this morphism is indeed inverse to 
the morphism (<ref>).




[Orthogonal ring spectra from orthogonal monoid spaces]
The suspension spectrum Σ^∞_+ M 
of an orthogonal monoid space M
becomes an orthogonal ring spectrum via the multiplication map

    (Σ^∞_+ M )(Σ^∞_+ M ) [](<ref>) Σ^∞_+ ( M⊠ M )
     Σ^∞_+μ_M Σ^∞_+ M  .

If the multiplication of M is commutative, then so is the resulting
multiplication on Σ^∞_+ M.
The functor pair (Σ^∞_+,Ω^∙) is again an adjoint
pair when viewed as functors between the categories of 
orthogonal monoid spaces and orthogonal ring spectra. 

This construction includes spherical monoid ring spectra:subjectmonoid ring spectrum 
if M is a topological monoid, then the constant orthogonal space
with value M inherits an associative and unital product from M.
The suspension spectrum
of such a constant multiplicative functor is the monoid ring spectrum.




In Construction <ref>
we introduced pairings on the equivariant homotopy groups
of orthogonal G-spectra.
Now we consider two orthogonal spectra X and Y.
In the global context, the equivariant homotopy groups also support inflation maps,
which we can use to define another pairing

    ⊠ : π_k^G(X)  × π_l^K (Y)  → π_k+l^G× K(X Y)  .

Here G and K are compact Lie groups and k,l∈.
We define this pairing as the composite

    π_k^G(X) ×π_l^K (Y) p_G^*× p_K^* π_k^G× K(X) ×π_l^G× K (Y)  ×  π_k+l^G× K(X Y) ,

where p_G:G× K→ G and p_K:G× K→ K
are the projections to the two factors.
Theorem <ref>
and the additivity of inflation maps imply 
that this pairing is biadditive, and that it satisfies
certain associativity, commutativity and restriction properties.
We do take the time to spell out the most important
properties of these pairings in the next theorem.

As G and K vary through all compact Lie groups,
the ⊠-pairings form a bimorphism of global functors 
in the sense of Construction <ref> below.
Moreover, the passage from the `diagonal' pairings
to the `external' pairings (<ref>) can be reversed
by taking K=G and restricting to the diagonal;
suitably formalized, diagonal and external pairings contain the same
amount of information. We refer 
to Remark <ref> below for more details.





  Let G,K and L be compact Lie groups and X, Y and Z orthogonal spectra.
  
  
  * (Biadditivity)
    The product ⊠:π_k^G(X) ×π_l^K (Y) →π_k+l^G× K(X Y)
    is biadditive.
  
  * (Unitality)
    Let 1∈π_0^e() denote the class represented by the identity of S^0.
    The product is unital in the sense that 1⊠ x=x=x⊠ 1 
    under the identifications X=X=X and e× G G G× e.
  
  * (Associativity) For all x∈π_k^G ( X ), y∈π_l^K( Y )
    and z∈π_m^L( Z ) the relation
    
    x⊠(y⊠ z)  =  (x⊠ y)⊠ z

    holds in π^G× K× L_k+l+m(X Y Z).
  
  * (Commutativity) For all x∈π_k^G ( X ) and y∈π_l^K( Y )
    the relation
    
    τ^X,Y_*( x⊠ y )  =  (-1)^kl·τ_G,K^*( y⊠ x)

    holds in π^G× K_l+k( Y X),
    where τ^X,Y:X Y→ Y X is the symmetry isomorphism
    of the smash product and τ_G,K:G× K → K× G
    interchanges the factors.
  
  * (Restriction) For all x∈π_k^G ( X ) and y∈π_l^K( Y )
    and all continuous homomorphisms α:G̅→ G
    and β:K̅→ K the relation
    
    α^*(x)⊠β^*(y) =  (α⊠β)^*(x⊠ y)

    holds in π_k+l^G̅×K̅(X Y).
   
  * (Transfer) For all closed subgroups H≤ G and L≤ K the square 
    
    @C=15mmπ_k^H(X) ×π^L_l(Y) 
          [r]^-⊠[d]__H^G×_L^K   π^H× L_k+l (X Y) [d]^_H× L^G× K
    π^G_k (X) ×π^K_l(Y) [r]_-⊠   π^G× K_k+l (X Y)

    commutes.
  


Parts (i) through (v) follow from the 
respective parts of Theorem <ref> 
by naturality.
For part (vi) we start with two special cases, namely L=K respectively H=G.
The two proofs are analogous, so we only treat the case L=K:

    _H× K^G× K(x⊠ y)    =  _H× K^G× K( p_H^*(x)× p_K^*(y)) 
     =  _H× K^G× K(p_H^*(x)×^G× K_H× K(p_K^*(y)))
       =  _H× K^G× K(p_H^*(x))× p_K^*(y) 
    =  p_G^*(_H^G(x))× p_K^*(y) =_H^G(x)⊠ y

We slightly abuse notation by writing p_K for both the projections
of H× K and of G× K to K.
The third equation is the reciprocity
relation of Theorem <ref> (vi). 
The fourth equation is the compatibility of transfers with inflations
(Proposition <ref> (ii)).

The general case is now obtained by combining the two special cases:

    _H^G(x)⊠_L^K(y)     =  _H× K^G× K(x⊠_L^K(y))
       =  _H× K^G× K(_H× L^H× K(x⊠ y)) =  _H× L^G× K(x⊠ y)





For all orthogonal spectra (i.e., all global homotopy types), 
the collection of equivariant homotopy groups {π_0^G (X) }_G
come with restriction and transfer maps, and this data together forms
a `global functor', compare Definition <ref> below. 
The geometric fixed point homotopy groups have fewer natural operations,
and they do not allow restriction to subgroups.
However, geometric fixed points still have inflation maps, i.e., 
restriction maps along epimorphisms.
Indeed, in Construction <ref>
we defined inflation maps on geometric fixed point homotopy groups,
associated to a continuous epimorphism α:K→ G between compact Lie groups.
When X is an orthogonal spectrum, representing a global homotopy type,
then α^*(X_G)=X_K, and the inflation maps become homomorphismssubjectinflation map!of geometric fixed point homotopy groups  

    α^*  :  Φ^G_k (X)  → Φ^K_k (X)  .

These inflation maps between the geometric fixed point homotopy groups
are clearly natural in the orthogonal spectrum.



[Semifree orthogonal spectra]
Given a compact Lie group G and a G-representation V, the functor

    _G,V :   → G_*

that sends an orthogonal spectrum X to the based G-space X(V)
has a left adjointsubjectsemifree orthogonal spectrumsubjectorthogonal spectrum!semifreesymbolF_G,V - semifree orthogonal spectrum generated by (G,V)

    F_G,V :  G_*  →  .

The semifree orthogonal spectrum generated by a based G-space A in level V is 

    F_G,V A =  (V,-)_G A ;

the value at an inner product space W is thus given by

    (F_G,VA)(W)  =  (V,W)_G A .

We note that F_G,V A consists of a single point in all levels below
the dimension of V.
The `freeness' property of F_G,VA
is a consequence of the enriched Yoneda lemma,subjectYoneda lemma!enriched
see Remark <ref> or <cit.>;
it means that for every orthogonal spectrum X and every based G-map f:A→ X(V) 
there is a unique morphism 
f^♭:F_G,VA→ X of orthogonal spectra such that the composite

    A   -  (V,V)_G A = (F_G,VA)(V)  f^♭(V) X(V)

is f.
Indeed, the map f^♭(W) is the composite

    (V,W)_G A  _G f(V,W)_G X(V)   act  X(W) .






[Semifree spectra as global Thom spectra]
The underlying non-equivariant stable homotopy type of F_G,V 
is the Thom spectrum of the negative of the bundle

    (V,^∞)×_G V  → (V,^∞)/G  =  B G  ,

the vector bundle over B G associated to the G-representation V. 
So one should think of the semifree orthogonal spectrum F_G,V=F_G,V S^0
as the `global Thom spectrum' associated to a `virtual global vector bundle',
namely the negative of the vector bundle over B_G associated to the
G-representation V. 

The special case G=O(m) of the orthogonal group
with V=ν_m, i.e., ^m with the tautological O(m)-action,
will feature prominently in the rank filtration of the global
Thom spectrum  in Section <ref>.
Non-equivariantly, F_O(m),ν_m is the Thom spectrum 
of the negative of the tautological m-plane bundle
over the Grassmannian G r_m(^∞); 
the traditional notation for this Thom spectrum is M T O(m)
or simply M T (m).
Indeed,
 
    F_O(m),ν_m(^m+n)  =  (^m,^m+n)/O(m)

is the Thom space of the orthogonal complement of
the tautological m-plane bundle over (^m,^m+n)/O(m)=G r_m(^m+n),
and this is precisely the (m+n)-th space of M T O(m), see
for example <cit.>.
Similarly, the non-equivariant homotopy type
underlying F_S O(m),ν_m is an oriented version of M T O(m),
which is usually denoted M T S O(m) or sometimes M T(m)^+.



[Smash products of semifree orthogonal spectra]
The smash product of two semifree orthogonal spectra is again
a semifree orthogonal spectrum.subjectsmash product!of semifree spectra 
In more detail, we consider


  * two compact Lie groups G and K, 

  * a G-representation V and  a K-representation W, and 

  * a based G-space A and a based K-space B.

Then V⊕ W is a (G× K)-representation and
A B is a (G× K)-space via

    (g,k)· (v,w) =  (g v,k w)    respectively  
    (g,k)· (a b) =  g a k b  .

We claim that the smash product
(F_G,VA) (F_K,WB) is canonically isomorphic
to the semifree orthogonal spectrum generated by the (G× K)-space A B
in level V⊕ W. 
Indeed, a morphism

    (F_G,VA) (F_K,WB) →  F_G× K,V⊕ W(A B)

is obtained by the universal property (<ref>)
from the bimorphism with (U,U')-component

    (F_G,VA)(U) (F_K,WB)(U')  =  
    ((V,U)   _G A)((W,U')_K B) 
     ⊕     (V⊕ W,U⊕ U')_G× K(A B) 
     
    =     ((F_G× K,V⊕ W)(A B))(U⊕ U')  .

In the other direction, a morphism 
F_G× K,V⊕ W(A B)→ F_G,VA F_G,WB 
is freely generated by the (G× K)-map

    A B  → (F_G,VA)(V)  (F_K,WB)(W)  i_V,W 
     (F_G,VA F_K,WB)(V⊕ W)  .

These two maps are inverse to each other.




In Proposition <ref> (ii) 
we have seen that for every compact Lie group G,
every G-representation V and every faithful G-representation W 
the restriction morphism of orthogonal spaces
ρ_V,W/G: _G,V⊕ W→_G,W is a global equivalence.
One consequence is that the semifree orthogonal space _G.W 
has a well-defined unstable global homotopy type, independent of which faithful
G-representation is used. 
Another consequence is that the induced morphism

    Σ^∞_+ ρ_V,W/G : Σ^∞_+ _G,V⊕ W → Σ^∞_+ _G,W

of suspension spectra is a global equivalence of orthogonal spectra,
by Corollary <ref>.
For an inner product space W, the untwisting homeomorphisms (<ref>)
descend to homeomorphisms on G-orbit spaces

    (V,W)_G S^V   S^W(V,W)/G_+ .

As W varies, these form the `untwisting isomorphism'
of orthogonal spectrasubjectuntwisting homeomorphism

    F_G,VS^V   Σ^∞_+ _G,V .

So suspension spectra of semifree orthogonal spaces are semifree orthogonal spectra.
We will now prove a generalization of the fact that Σ^∞_+ ρ_V,W/G
is a global equivalence for these global Thom spectra.
Given G-representations V and W,
we define a restriction morphism of orthogonal spectra

    symbolλ_G,V,W - fundamental global equivalence of orthogonal spectraλ_G,V,W :  F_G,V⊕ WS^V →  F_G,W

as the adjoint of the based G-map

    S^V  → (W,V⊕ W)/G  =   F_G,W(V⊕ W)  ,   
    v  ⟼  ((v,0) , i)· G  ,

where i:W→ V⊕ W is the embedding of the second summand.
The value of λ_G,V,W at an inner product space U is then

    λ_G,V,W(U) :  (V⊕ W, U)_G S^V    →  (W,U)/G 
    
    [(u,φ) v]        ⟼  (u+φ(v),φ∘ i)· G .






Let G be a compact Lie  group, V a G-representation
and W a faithful G-representation. Then the morphism

    λ_G,V,W :  F_G,V⊕ WS^V  → F_G,W

is a global equivalence of orthogonal spectra.


To simplify the notation we abbreviate the restriction morphism of orthogonal spaces
to

    ρ =  ρ_V,W :  (V⊕ W,-) → (W,-) .

We let K be another compact Lie group and U∈ s(_K) 
a finite-dimensional K-subrepresentation of the complete K-universe _K.
In a first step we produce a K-representation U'∈ s(_K) 
with U⊆ U' and
a continuous (K× G)-equivariant map 

    h : (W,U)  → (V⊕ W,U')
 
such that the lower right triangle in the diagram

    @C=15mm(V⊕ W,U) [r]^i_*[d]_ρ(U)   (V⊕ W,U') [d]^ρ(U')
    (W,U) [r]_i_*@–>[ur]^(.4)h    (W,U')

commutes, and the upper left triangle
commutes up to (K× G)-equivariant fiberwise homotopy over (W,U'),
where i:U→ U' is the inclusion.

Since G acts faithfully on W (and hence on V⊕ W),
both (W,_K) and (V⊕ W,_K) are universal spaces
for the same family of subgroups of K× G,
namely the family (K;G) of graph subgroups,subjectgraph subgroup
compare Proposition <ref> (i). 
If Γ is the graph of
a continuous homomorphism α:L→ G defined on some closed subgroup L of K,
then the Γ-fixed points of (W,_K) are given by

    (W,_K)^Γ =  ^L(α^* W,^K_L( _K))  ,

the space of L-equivariant linear isometric embeddings from α^* W.
The same is true for V⊕ W instead of W, and so the
Γ-fixed point map ρ(_K)^Γ:(V⊕ W,_K)^Γ→(W,_K)^Γ
is the restriction map

    ^L(α^* V⊕α^* W,^K_L(_K)) → ^L(α^* W,^K_L(_K))

to the summand α^* W. This map is a locally trivial fiber bundle,
hence a Serre fibration. We conclude that the restriction map
ρ(_K):(V⊕ W,_K) →(W,_K) is both
a (K× G)-weak equivalence and a (K× G)-fibration.

Since (W,U) is cofibrant as a (K× G)-space 
(by Proposition <ref> (ii)), 
the (K× G)-map i_*:(W,U)→(W,_K)  
thus admits a (K× G)-equivariant lift 
h:(W,U)→(V⊕ W,_K)  such that ρ(_K)∘ h=i_*.
Since the space (W,U) is compact and 
(V⊕ W,_K) is the filtered union of the closed subspaces
(V⊕ W, U') for U'∈ s(_K), the lift h
lands in the subspace (V⊕ W, U') for suitably large U'∈ s(_K),
and we may assume that U⊆ U'.

The two maps 

    h∘ρ(U) ,  i_*  :  (V⊕ W,U)  → (V⊕ W, U')

become equal after applying ρ(U'):(V⊕ W, U') →(V, U'),
hence the composites with i_*:(V⊕ W,U') →(V⊕ W, _K)
become equal after applying the (K× G)-equivariant 
acyclic fibration ρ(_K):(V⊕ W, _K) →(W, _K).
Since (V⊕ W,U) is also (K× G)-cofibrant, there
is a fiberwise (K× G)-equivariant homotopy between 
h∘ρ(U) and i_* in (V⊕ W,_K).
Again by compactness, the homotopy has image in 
(V⊕ W, U”) for suitably large U”∈ s(_K).
So after increasing U', if necessary, we have proved the claim
subsumed in the diagram (<ref>).


Now we lift the data produced in the first step to the Thom spaces
of the orthogonal complement bundles. The diagram (<ref>) 
is covered by morphisms of (K× G)-vector bundles:

    @C=12mm
    (U'-U) ×ξ(V⊕ W,U)× V 
     [r]^-i̅[d]_(U'-U)×ρ̅(U)   ξ(V⊕ W,U')× V [d]^ρ̅(U')
    
    (U'-U) ×ξ(W,U) [r]_-i̅@–>[ur]^(.4)h̅   ξ(W,U')

The maps on the total spaces of the bundles are defined as follows:
the right vertical map is

    ρ̅(U')  :  ξ(V⊕ W,U')× V → ξ(W,U') ,    
    ((u',φ),  v) ⟼ ( u' +φ(v), φ|_W) .

The map ρ̅(U) is defined in the same way.
The lower horizontal map is 

    i̅ :  (U'-U)×ξ(W,U)
     → ξ(W,U') ,    
    (u',(u,φ)) ⟼ (u'+u, φ) .

The upper horizontal map is defined in the same way, but with V⊕ W 
instead of W and multiplied by the identity of V.
These four outer morphisms in (<ref>) 
are all fiberwise linear isomorphisms; so each of these four bundle
maps expresses the source bundle as a pullback of the target bundle.
In particular, the square

    ξ(V⊕ W, U')× V [r]^-ρ̅(U')[d]    ξ(W,U') [d]
    (V⊕ W, U') [r]_-ρ(U')   (W,U')

is a pullback;  so the composite

    (U'-U)×ξ(W,U)  → (W,U)  h  (V⊕ W,U')

and the map of total spaces i̅:(U'-U)×ξ(W,U)→ξ(W,U')
assemble into a map 

    h̅ :   (U'-U)×ξ(W,U)  → ξ(V⊕ W,U')

that covers h and is a fiberwise linear isomorphism.

In (<ref>) (as in (<ref>))
the outer square and the lower right triangle commute,
but the upper left triangle does not commute.
We will now show that the upper left triangle commutes up to homotopy
of (K× G)-equivariant bundle maps.
For this purpose we let

    H  :  (V⊕ W,U) × [0,1]  → (V⊕ W, U')

be a  (K× G)-equivariant homotopy from the
map i_* to h∘ρ(U), such that
ρ(U')∘ H:(V⊕ W,U) × [0,1] →(W, U')
is the constant homotopy from ρ(U')∘ i_* = i_*∘ρ(U) to itself.
Again because the square (<ref>) is a pullback, the composite

    ξ(V⊕ W,U)× V × (U'-U) × [0,1] → (V⊕ W,U) × [0,1]  H  (V⊕ W, U')

and the map of total spaces 

    (U'-U) ×ξ(V⊕ W,U)× V×  [0,1]    proj 
     (U'-U)×ξ(V⊕ W,U)× V  
       ρ̅(U')∘i̅=i̅∘ρ̅(U) ξ(W,U')

assemble into a map 

    H̅ :   (U'-U) ×ξ(V⊕ W,U) × V ×  [0,1] 
     → ξ(V⊕ W,U')× V

that covers the homotopy H. This lift H̅ is a 
(K× G)-equivariant homotopy of vector bundle morphisms,
and for every t∈ [0,1], the relation

    ρ̅(U') ∘H̅(-,t)     =  ρ̅(U')∘i̅ 
    =  ρ̅(U')∘ (h̅∘ρ̅(U))

holds by definition of H̅.
For t=0 this shows that H̅ starts with 
i̅:(U'-U)×ξ(V⊕ W,U) × V  →ξ(V⊕ W,U')× V; for  t=1 this shows that H̅ 
ends with h̅∘ρ̅(U), one more time 
because (<ref>) is a pullback.
We conclude that H̅ makes the upper left triangle in (<ref>) 
commute up to equivariant homotopy of vector bundle maps.

Passing to Thom spaces in (<ref>) 
gives a diagram of (K× G)-equivariant based maps:

    @C=20mm
    S^U'-U(V⊕ W,U) S^V
     [r]^σ_U,U'-U S^V[d]_S^U'-Uλ_V,W(U)   (V⊕ W,U') S^V [d]^λ_V,W(U')
    
    S^U'-U(W,U) [r]_σ_U,U'-U@–>[ur]^(.4)h̅   (W,U')

Again, the lower right triangle commutes,
and the upper left triangle commutes up to  (K× G)-equivariant 
based homotopy.
We pass to G-orbit spaces and obtain a diagram of based K-spaces

    @C=20mm
    S^U'-U (F_G,V ⊕ W S^V)(U)
    [r]^σ_U,U'-U[d]_S^U'-Uλ_G,V,W(U)    
    (F_G,V ⊕ W S^V)(U') [d]^λ_G,V,W(U')
    
    S^U'-U F_G,W(U) [r]_σ_U,U'-U[ur]^(.4)h̅/G    F_G,W(U')

whose lower right triangle commutes, and whose  upper left triangle
commutes up to K-equivariant based homotopy.
Since we had started with an arbitrary K-subrepresentation U∈ s(_K),
this implies that for every based K-space A 
and G-representation U̅ the map on colimits

    _U∈ s(_K)    [S^U A, (F_G,V ⊕ W S^V)(U⊕U̅)]^K 
       →  _U∈ s(_K)  [S^U A, F_G,W(U⊕U̅) ]^K

induced by the morphism λ_G,V,W is bijective. 
For A=S^k and U̅=0 this shows that π_k^G(λ_G,V,W)
is an isomorphism.
For A=S^0 and U̅=^k this shows that π_-k^G(λ_G,V,W)
is an isomorphism.
So λ_G,V,W is a global equivalence.







§ GLOBAL FUNCTORS


This section is devoted to the category of global functors,
the natural home of the collection of equivariant
homotopy groups of a global stable homotopy type.
The category of global functors is a symmetric monoidal abelian category
with enough injectives and projectives that plays the same role
for global homotopy theory that is played by the category of abelian groups
in ordinary homotopy theory, or by the category of G-Mackey functors
for G-equivariant homotopy theory.

We introduce  global functors in Definition <ref> 
as additive functors on the global Burnside category, 
the category of natural operations between equivariant stable homotopy groups.
The abstract definition ensures that equivariant homotopy groups
of orthogonal spectra are tautologically global functors.
A key result is Theorem <ref> 
that describes explicit bases of the morphism groups of the 
global Burnside category in terms of transfers and restriction operations.
This calculation is the key to comparing our notion of global functors
to other kinds of global Mackey functors, 
as well as for all concrete calculations with global functors.
Example <ref> lists interesting
examples of global functors:
the Burnside ring global functor,
represented global functors, constant global functors,
the representation ring global functor, and Borel type global functors.
Many more examples of global functors are discussed 
in the remaining sections of this book.

An abstract way to motivate the appearance of global functors is as follows.
One can consider the globally connective (respectively globally coconnective)
orthogonal spectra, i.e., those where all equivariant homotopy groups
vanish in negative dimensions (respectively in positive dimensions).
Then the full subcategories of globally connective 
respectively globally coconnective spectra
define a non-degenerate t-structure on the triangulated global stable homotopy category,
and the heart of this t-structure is (equivalent to) 
the abelian category of global functors; we refer the reader to
Theorem <ref> below for details.

In the global context, the external pairing (<ref>)
of equivariant homotopy groups gives rise to a symmetric monoidal structure 
on the global Burnside category,
see Theorem <ref>. Hence the abelian category of
global functors can also be endowed with a Day type convolution product,
the box product of global functors, see Construction <ref>.



subjectBurnside category|(

[Burnside category]
We define the pre-additive Burnside categorysubjectBurnside category .symbol - Burnside category
The objects of  are all compact Lie groups;
morphisms from a group G to K are defined as

    (G,K)  =  (π_0^G,π_0^K)  ,

the set of natural transformations of functors, from orthogonal spectra to sets,
between the equivariant homotopy group functors π_0^G and π_0^K. 
Composition in the category  is composition of natural transformations.


It is not a priori clear that the natural transformations 
from π_0^G to π_0^K form a set (as opposed to a proper class),
but this follows from Proposition <ref> below. 
The Burnside category  is skeletally small: isomorphic compact Lie groups
are also isomorphic as objects in the category , and every compact Lie
group is isomorphic to a closed subgroup of an orthogonal group O(n).
The functor π_0^K is abelian group valued,
so the set (G,K) is an abelian group under objectwise addition of transformations.
Proposition <ref>, 
applied to the category of orthogonal spectra,
shows that set valued natural transformations between 
the two reduced additive functors π_0^G and π_0^K are automatically
additive. So composition in the Burnside category is additive in each variable, 
and  is indeed a pre-additive category.




 
A global functorsubjectglobal functor is an additive
functor from the Burnside category  to the category of abelian groups.
A morphism of global functors is a natural transformation.
We write  for the category of global functors.
symbol   - category of global functors 
symbol   - category of abelian groups

We discuss various explicit examples of interesting global functors 
in Example <ref>.



The definition of the Burnside category  is made so that
the collection of equivariant homotopy groups of an orthogonal spectrum 
is tautologically a global functor.
Explicitly, the homotopy group global functor _0(X)
of an orthogonal spectrum X is defined on objects by

    _0(X)(G) =  π_0^G (X)

and on morphisms by evaluating natural transformations at X.
It is less obvious that conversely every global functor is isomorphic
to the global functor _0(X) of some orthogonal spectrum X;
we refer the reader to Remark <ref> 
below for the construction of Eilenberg-Mac Lane spectra from global functors.



As a category of additive functors out of a skeletally small 
pre-additive category, the category  of global functors has some
immediate properties that we collect in the following proposition.


The category  of global functors is a Grothendieck abelian
category with enough injectives and projectives.


Any category of additive functors out of a skeletally small additive category
is Grothendieck abelian with objectwise notion of exactness,
see <cit.> and <cit.>.
A set of projective generators is given by the represented global functors
(G,-) where G runs through a set of representatives of the
isomorphism classes of compact Lie groups, see <cit.>.
A set of injective cogenerators is given similarly by the global functors

    ((-,K),/)  :   →  .
 


As we shall explain in Construction <ref> below, 
the category  has a closed symmetric monoidal product 
 that arises as a convolution product for a certain symmetric monoidal structure
on the Burnside category .




Our definition of the Burnside category is made so that 
every orthogonal spectrum X gives rise to a homotopy group global functor
without further ado, but it is not clear from the definition how to describe
the morphism groups of  explicitly. Our next aim 
is to show that each morphism group (G,K) is 
a free abelian group with an explicit basis given by certain composites 
of a restriction and a transfer morphism.
This calculation has two ingredients: 
We identify natural transformations
from π_0^G to π_0^K with the group π_0^K(Σ^∞_+ B_G),
and then we exploit the explicit calculation of the latter group 
in Corollary <ref>. 


subjectglobal classifying space   
Let G and K be compact Lie groups and V a faithful G-representation.
Then evaluation at the stable tautological class (<ref>)
is a bijection

    (G,K) =  ^(π_0^G,π_0^K) 
     → π_0^K(Σ^∞_+ _G,V)  ,   τ ⟼ τ(e_G,V)

to the 0-th K-equivariant homotopy group 
of the orthogonal spectrum Σ^∞_+ _G,V.
In other words, the morphism  of global functors

    (G,-) → _0(Σ^∞_+ _G,V)

classified by the stable tautological class e_G,V is an isomorphism.


  We apply the representability result 
  of Proposition <ref>
  to the category of orthogonal spectra and
  the adjoint functor pair (Σ^∞_+,Ω^∙). 
  If G is a compact Lie group, V a G-representation and W
  a faithful G-representation, then the restriction morphism
  ρ_G,V,W:_G,V⊕ W→_G,W is a global equivalence
  of orthogonal spaces (Proposition <ref>). 
  So the induced morphism of unreduced suspension spectra
  Σ^∞_+ρ_G,V,W is a global equivalence
  by Corollary <ref>;
  in particular, the morphism of -functors
  _0(Σ^∞_+ρ_G,V,W) is an isomorphism. So 
  Proposition <ref> applies and shows that
  evaluation at the tautological class is bijective. This proves the claim.





Now we name a basis of the group (G,K).
For a pair (L,α) consisting of a closed subgroup L of K 
and a continuous group homomorphism α:L→ G we define

    [L,α] =  _L^K∘α^* ∈ (G,K)  ,

the natural transformation whose value at X is the composite

    π_0^G (X)   α^*  π_0^L(X)   _L^K  π_0^K(X)

of restriction along α with transfer from L to K.

If L has infinite index in its normalizer, then the transfer map
_L^K, and hence also the element [L,α], 
is zero by Example <ref>.  
The conjugate of (L,α) by a pair 
(k,g)∈ K× G of group elements is the pair
(L^k,c_g∘α∘ c_k^-1) consisting of the
conjugate subgroup L^k and the composite homomorphism

    L^k   c_k^-1  L   α  
    G   c_g  G  .

Since inner automorphisms induce the identity on equivariant homotopy groups
(compare Proposition <ref>), 

    _L^k^K ∘ (c_g∘α∘ c_k^-1)^* =  _L^k^K ∘ k_⋆^-1∘α^* ∘ g_⋆ =  
    k_⋆^-1∘_L^K ∘α^* ∘ g_⋆ =  _L^K ∘α^*  .

So the transformation [L,α] only depends on
the conjugacy class of (L,α), i.e.,

    [ L^k, c_g∘α∘ c_k^-1 ] =   [L,α]
       in (G,K).


 
Let K and G be compact Lie groups. The morphism group (G,K) 
in the Burnside category is a free abelian group 
with basis the transformations [L,α],
where (L,α) runs over all (K× G)-conjugacy classes of pairs consisting of


  * a closed subgroup L≤ K whose Weyl group W_K L is finite, and

  * a continuous group homomorphism α:L→ G.
  


We let V be any faithful G-representation.
By Proposition <ref> the composite

    {[L,α] |  |W_KL| <∞, α:L→ G} → (π_0^G,π_0^K) 
        π_0^K(Σ^∞_+ _G,V)

is an isomorphism, where the first  map takes [L,α] to _L^K∘α^*
and the second map is evaluation at the
stable tautological class e_G,V.
The evaluation map is an isomorphism by 
Proposition <ref>, so the first map
is an isomorphism, as claimed.  




Theorem <ref> amounts to a complete calculation
of the Burnside category, because we know how to
express the composite of two operations, each given in
the basis of Theorem <ref>, as a sum of basis elements.
Indeed, restrictions are contravariantly functorial and transfers are transitive,
and we also know how to expand a transfer followed by a restriction:
every group homomorphism is the composite of an epimorphism and a subgroup
inclusion. Inflations commute with transfers according to
Proposition <ref>, and the restriction of a transfer
can be rewritten via the double coset formula
as in Theorem <ref>.

Theorem <ref> 
tell us what data is necessary to specify a global functor M:→. 
For this, one needs to give the values M(G) at all compact Lie groups G,
restriction maps α^*:M(G)→ M(L) for all continuous group homomorphisms
α:L→ G and transfer maps _L^K:M(L)→ M(K) for all
closed subgroup inclusions L≤ K.
This data has to satisfy the same kind of relations that the 
restriction and transfer maps for equivariant homotopy groups satisfy,
namely:


  * the restriction maps are contravariantly functorial;

  * inner automorphisms induce the identity;

  * transfers are transitive and _K^K is the identity;

  * the transfer _L^K is zero if the Weyl group W_K L is infinite;

  * transfer along an inclusion H≤ G 
interacts with inflation along an epimorphism α:K→ G according to

    α^*∘_H^G  =  _L^K ∘ (α|_L)^*  :  M(H) → M(K)  ,

where L=α^-1(H);

  * for all pairs of closed subgroups H and K of G,
the double coset formula holds, see Theorem <ref>.

The hypothesis that inner automorphisms act as the identity
implies that the restriction map α^* only 
depends on the homotopy class of α.
Indeed, suppose that α,α':K→ G 
are homotopic through continuous group homomorphisms. 
Then α and α' belong to the same path component of the space (K,G)
of continuous homomorphisms, so they are conjugate by an element of G,
compare Proposition <ref>.


This explicit description allows us to relate our notion of global functor
to other `global' versions of Mackey functors.
For example, our category of global functors is equivalent to the
category of functors with regular Mackey structure
in the sense of Symonds <cit.>.
Our global functors are not equivalent to the global Mackey functors
in the sense of tom Dieck <cit.>;
indeed, in the indexing category Ω for tom Dieck's global Mackey functors
the group _Ω(G,K) has a -basis indexed by (G× K)-conjugacy classes
[L,α] where the Weyl group W_K L is allowed to be infinite.
As we shall explain in Remark <ref> below, 
global functors defined on finite groups
are equivalent to inflation functors in the sense of Webb <cit.>
and to `global (∅,∞)-Mackey functors' in 
the sense of Lewis <cit.>. 



[Sphere spectrum]subjectsphere spectrumsymbol - sphere spectrum
The sphere spectrum  is given by (V)= S^V,
the one-point compactification of the inner product space V.
The orthogonal group acts as the one-point compactification
of its action on V. The structure map
σ_V,W:S^V S^W→ S^V⊕ W is the canonical homeomorphism.
The equivariant homotopy groups of the sphere spectrum are
the equivariant stable stems.
The sphere spectrum is the suspension spectrum of the
constant one-point orthogonal space _e,0,

    Σ^∞_+ _e,0 .

The trivial representation is faithful as a representation of the trivial group,
so _e,0=B_e is a global classifying space for the trivial group.
The class  1∈π_0()
represented by the identity of S^0 is the stable tautological class e_e,0
(compare (<ref>)).
By Corollary <ref> the group π_0^K()
is a free abelian group with basis the classes _L^K(p_L^*(1))
where L runs over all conjugacy classes of closed subgroups of K 
with finite Weyl group, and where p_L:L→ e is the unique homomorphism.
For finite groups, this is originally due
to Segal <cit.>, and for general compact Lie groups 
to tom Dieck, as a corollary to his splitting theorem 
(see Satz 2 and Satz 3 of <cit.>).
By Proposition <ref>,
the action on the unit 1∈π_0() is an isomorphism of global functors

    =  (e,-)  → _0 (  )

from the Burnside ring global functor  to the 0-th homotopy
global functor of the sphere spectrum. 



 
(i) The Burnside ring global functorsubjectBurnside ring global functorsymbol - Burnside ring global functor
is the represented global functor
=(e,-) of morphisms out of the trivial group e.  
By Theorem <ref>, the value
(K)=(e,K) at a compact Lie group K is a free abelian group
with basis the set of conjugacy classes of closed subgroups L≤ K 
with finite Weyl group.
When K is finite, then the Weyl group condition is
vacuous and (K) is canonically isomorphic to the Burnside ring of K,
by sending the operation [L,p_L]=_L^K∘ p_L^*∈(K)
to the class of the K-set K/L (where p_L:L→ e is the unique homomorphism). 
As we discussed in Example <ref>, the Burnside ring 
global functor  is realized by the sphere spectrum .
More generally, the represented functors (G,-) are other examples 
of global functors, and we have seen in Proposition <ref>  
that the represented global functor (G,-) is
realized by the suspension spectrum of the global classifying
space B_G.subjectglobal classifying space 


(ii) In <cit.>,
tom Dieck gives a very different construction of the Burnside ring global functor.
We let G be  a compact Lie group.
A G-ENR is a G-space equivariantly homeomorphic
to a G-retract of a G-invariant open subset of a finite-dimensional
G-representation. 
The acronym `ENR' stands for euclidean neighborhood retract. 
Examples of G-ENRs are smooth compact G-manifolds and finite G-CW-complexes.

Tom Dieck calls two compact G-ENRs X and Y equivalent
if for every closed subgroup H of G the Euler characteristics
of the H-fixed point spaces X^H and Y^H coincide;
here Euler characteristics are taken with respect to compactly supported
Alexander-Spanier cohomology, and there is some work involved in showing
that a compact G-ENR has a well-defined Euler characteristic.
Then A(G) is defined as the set of equivalence classes of compact G-ENRs.
The set A(G) is naturally a commutative ring, with addition induced by
disjoint union and multiplication induced by cartesian product of G-ENRs.

Tom Dieck shows in <cit.>, 
that A(G) is a free abelian group with basis the classes of the homogeneous spaces
G/H, where H runs over the conjugacy classes of closed subgroups with
finite Weyl group. Moreover, the class of a general compact G-ENR X
is expressed in terms of this basis by the formula

    [X] =  ∑_(H) χ^AS(G X_(H))· [G/H] ;

the sum is over conjugacy classes of closed subgroups, 
X_(H) is the orbit type subspace, and χ^AS
is the Euler characteristic based on Alexander-Spanier cohomology.
A compact G-ENR has only finitely many orbit types, so the sum is in fact finite.
Restriction of scalars along a continuous homomorphism α:K→ G
induces a ring homomorphism α^*:A(G)→ A(K).
If H is a closed subgroup of G, then extension of scalars –
sending an H-ENR Y to G×_H Y –
induces an additive transfer map _H^G:A(H)→ A(G)
that satisfies reciprocity with respect to restriction from G to H.
By explicit comparison of bases, the homomorphisms

    A(G) → (G)  ,    [G/H] ⟼ _H^G∘ p_H^*

define an isomorphism of global functors.

(iii)
Given an abelian group M, the constant global functor M
subjectglobal functor!constant 
is given by M(G)=M and all restriction maps
are identity maps. The transfer _H^G:M(H)→M(G)
is multiplication by the Euler characteristic of the homogeneous space G/H. 
In particular, if H has finite index in G, then
_H^G is multiplication by the index [G:H].
In this example, the double coset formula
is a special case of the Euler characteristic formula (<ref>),
namely for the K-manifold B=G/H. 



There is a well-known point set level model 
of an Eilenberg-Mac Lane spectrum MsubjectEilenberg-Mac Lane spectrum!of an abelian group
that we recall in Construction <ref> below.
However, contrary to what one may expect,
the 0-th homotopy group global functor _0( M) is not
isomorphic to the constant global functor M.
More precisely, the restriction map p_G^*: π_0^e ( M)→π_0^G( M)
is an isomorphism for finite groups G, but not for general compact Lie groups,
see Example <ref>.



(iv)
The unitary representation ring global functorsubjectrepresentation ring!unitary
 assigns to a compact Lie group G the 
unitary representation ring (G), i.e., the Grothendieck
group of finite-dimensional complex G-representations,
with product induced by tensor product of representations.
The restriction maps α^*:(G)→(K)
are induced by restriction of representations along 
a continuous homomorphism α:K→ G.
The transfer maps _H^G:(H)→(G) along a closed subgroup inclusion
H≤ G are given by the smooth induction 
of Segal <cit.>.
If H is a subgroup of finite index of G, then this induction sends the class 
of an H-representation to the induced G-representation ^H(G,V);
in general, induction may send actual representations to virtual representations.
In the generality of compact Lie groups, 
the double coset formula for  was proved by Snaith <cit.>.
We look more closely at the representation ring global functor
in Example <ref>,
and we show in Theorem <ref> that  
is realized by the global K-theory spectrum 
(see Construction <ref>).


(v)
Given any generalized cohomology theory E (in the non-equivariant sense),
we can define a global functor E  by setting

    E(G)  =  E^0(B G)  ,

the 0-th E-cohomology of a classifying space of the group G.
The contravariant functoriality in group homomorphisms α:K→ G
comes from the covariant functoriality of the classifying space construction.
The transfer map for a subgroup inclusion H≤ G 
comes from the stable transfer map (i.e., Becker-Gottlieb transfer)subjectBecker-Gottlieb transfersubjecttransfer!Becker-Gottlieb|seeBecker-Gottlieb transfer 

    Σ^∞_+ B G  → Σ^∞_+ B H  .

The double coset formula was proved 
in this context by Feshbach <cit.>.
We will show in Proposition <ref>
that the global functor E is realized by an orthogonal spectrum b E,
the `global Borel theory' associated to E.





subjectMackey functor 
If we fix a finite group G and let H run through all subgroups of G, 
then the collection of H-equivariant homotopy groups
π_0^H(X) of an orthogonal spectrum X forms a
G-Mackey functor (see Definition <ref>),
with respect to restriction to subgroups, conjugation and transfer maps.
As we already discussed in Remark <ref>,
not all G-Mackey functors arise this way.
To illustrate this we compare Mackey functors 
for the group C_3={1,τ,τ^2} with three elements
to additive functors on the full subcategory of  spanned by
the group e and C_3. Generating operations can be displayed as
follows:

    @R=1mm@!C=4mm                     
    
     F(C_3)  [rr]_      
    F(e) @<.9ex>@/^1pc/[ll]^p^*@/_1pc/[ll]_       F(C_3)[rr]_      
    F(e)  @/_1pc/[ll]_
    @<2ex>@(dl,ul)[uu]^α^*                  @<-2ex>@(dr,ur)[uu]_τ
             
       global functor on C_3 and e            C_3-Mackey functor

Here =^C_3_e and =_e^C_3 are the restriction and
transfer maps that are present in both cases.
A global functor also comes with inflation maps along the epimorphism 
p:C_3→ e and along the automorphism α:C_3→ C_3 with α(τ)=τ^2, 
and the relations are

    ∘ p^* =       and    ∘ =  3·

as well as α^*∘α^*=,
α^*∘ p^*=p^*, ∘α^*= and
α^*∘=.
In contrast, C_3-Mackey functors have an additional action of C_3
(the Weyl group of e in C_3) on F(e), and this action satisfies the relation

    ∘ =   + τ + τ^2 .





We will now use the exterior homotopy group pairings (<ref>)

    ⊠ :  π_0^G(X)  × π_0^K (Y)  → π_0^G× K(X Y)

to define a biadditive functor

    × :  × →

that is given on objects by the product of Lie groups,
and that extends to a symmetric monoidal structure on the Burnside category . 


Let G, G', K and K' be compact Lie groups.
Given operations τ∈(G,K) and ψ∈(G',K'), 
there is a unique operation

    τ×ψ ∈ (G× G',  K× K')

with the following property:
for all orthogonal spectra X and Y and all classes x∈π_0^G(X)
and y∈π_0^G'(Y) the relation

    (τ×ψ)(x⊠ y) =  τ(x)⊠ψ(y)

holds in π_0^K× K'(X Y).


We choose a faithful G-representation V and a faithful G'-representation V',
which have associated stable tautological 
classes (<ref>)subjectstable tautological class

    e_G,V ∈ π_0^G (Σ^∞_+ _G,V) 
         and      e_G',V' ∈ π_0^G'(Σ^∞_+ _G',V')  .

Combining the isomorphism (<ref>)
with the one from Example <ref> shows that
the orthogonal spectrum Σ^∞_+ _G,VΣ^∞_+ _G',V'
is isomorphic to Σ^∞_+ _G× G',V⊕ V' in a way that matches the class
e_G,V⊠ e_G',V' with the class e_G× G',V⊕ V'.
Proposition <ref> then shows that the pair
(Σ^∞_+ _G,VΣ^∞_+ _G',V',e_G,V⊠ e_G',V') 
represents the functor π_0^G× G'.
There is thus a unique operation τ×ψ∈(G× G',K× K')
that satisfies

    (τ×ψ)(e_G,V⊠ e_G',V') =  τ(e_G,V)⊠ψ(e_G',V')

in π_0^K× K'(Σ^∞_+ _G,VΣ^∞_+ _G',V').

The relation (<ref>) 
is a special case of (<ref>),
and it remains to show that the operation τ×ψ
satisfies the relation (<ref>)
in complete generality. As we already argued in the proof 
of Proposition <ref>, there is a G-representation W
and a morphism of orthogonal spectra

    f  :  Σ^∞_+ _G,V⊕ W → X

that satisfies f_*(e_G,V⊕ W)=x.
Similarly, there is a G'-representation W'
and a morphism of orthogonal spectra

    f'  :  Σ^∞_+ _G',V'⊕ W' → Y

that satisfies f'_*(e_G',V'⊕ W')=y. The morphism

    Σ^∞_+ρ_G,V,W :  Σ^∞_+_G,V⊕ W → Σ^∞_+_G,V   satisfies  
     (Σ^∞_+ρ_G,V,W)(e_G,V⊕ W) =  e_G,V ,

and similarly for the triple (G',V',W').
Naturality then yields

    (Σ^∞_+ρ_G,V,WΣ^∞_+   ρ_G',V',W')_*( (τ×ψ)(e_G,V⊕ W⊠ e_G',V'⊕ W'))
       =  (τ×ψ)( (Σ^∞_+ρ_G,V,W)_*(e_G,V⊕ W)⊠ 
    (Σ^∞_+ρ_G',V',W')_*( e_G',V'⊕ W'))
       =  (τ×ψ)( e_G,V⊠  e_G',V')
    
    _(<ref>)   =  τ( e_G,V)⊠ψ(e_G',V')
       =  τ((Σ^∞_+ρ_G,V,W)_*(e_G,V⊕ W))⊠ψ((Σ^∞_+ρ_G',V',W')_*( e_G',V'⊕ W'))
       =  
    (Σ^∞_+ρ_G,V,WΣ^∞_+ρ_G',V',W')_*
    (τ(e_G,V⊕ W)⊠ψ(e_G',V'⊕ W')) .

The morphism 
Σ^∞_+ρ_G,V,WΣ^∞_+ρ_G',V',W'
is isomorphic to Σ^∞_+ρ_G× G',V⊕ V',W⊕ W'.
The morphism ρ_G× G',V⊕ V',W⊕ W' is a global equivalence of
orthogonal spaces (by Proposition <ref> (ii)),
and so the morphism 
Σ^∞_+ρ_G,V,WΣ^∞_+ρ_G',V',W'
is a global equivalence of orthogonal spectra 
(by Corollary <ref>).
So in particular it induces an isomorphism on π_0^K× K', 
and we can conclude that

    (τ×ψ)(e_G,V⊕ W⊠ e_G',V'⊕ W') =  τ(e_G,V⊕ W)⊠ψ(e_G',V'⊕ W')  .

Now the relation (<ref>) follows by simple naturality:

    (τ×ψ)(x⊠ y)    =  (τ×ψ)(f_*(e_G,V⊕ W)⊠ f'_*(e_G',V'⊕ W'))
       =  
     (τ×ψ)((f f')_*(e_G,V⊕ W⊠ e_G',V'⊕ W'))
       =  
    (f f')_*((τ×ψ)(e_G,V⊕ W⊠ e_G',V'⊕ W'))
       =  
    (f f')_*(τ(e_G,V⊕ W)⊠ψ(e_G',V'⊕ W'))
       =  
    f_*(τ(e_G,V⊕ W))⊠ f'_*(ψ(e_G',V'⊕ W'))
       =  τ(f_*(e_G,V⊕ W))⊠ψ(f'_*(e_G',V'⊕ W'))=  τ(x)⊠ψ(y)





 We calculate the product of two generating operations in the Burnside category.
We recall that for a pair (L,α) consisting of a closed subgroup L of K 
and a continuous homomorphism α:L→ G we defined

    [L,α] =  _L^K∘α^*  ∈ (G,K)  .

By Theorem <ref>, a certain subset
of these operations forms a basis of the abelian group (G,K). 
Using parts (v) and (vi) of Theorem <ref> we deduce

    [L× L',α×α'](x⊠ y)    =  _L× L'^K× K'( (α×α')^*(x⊠ y))
       =  _L× L'^K× K'( α^*(x)⊠ (α')^*(y))
       =  _L^K( α^*(x))⊠_L'^K'((α')^*(y))=   [L,α](x)⊠ [L',α'](y) .

So the operation [L× L',α×α'] has the property
that characterizes the operation [L,α]× [L',α'].
Hence the monoidal product in  satisfies

    [L,α]× [L'α'] =  [L× L',α×α'] .



Now we are ready for the monoidal structure of the Burnside category.


 
Let G,G',K and K' be compact Lie groups.
  
  
  * The map
    
    × :  (G,K)×(G',K') → (G× G',K× K')

    is biadditive.
  
  * As the Lie groups vary, the maps of (i)
    form a functor ×:×→.
  
  * The restriction operations along the group isomorphisms
    
    a_G,G',G” :  G× (G'× G”)       (G× G')× G” ,
    τ_G,G' :  G× G'   G'×  G      respectively   
        G× e  G  e× G

    make the functor × into a symmetric monoidal structure on
    the global Burnside category.
  


(i) We show additivity in the first variable, the other case being analogous. 
The relation

    ((τ×ψ)+(τ'×ψ))(x⊠ y)
        =   (τ×ψ)(x⊠ y)+(τ'×ψ)(x⊠ y)
       =   (τ(x)⊠ψ(y))+(τ'(x)⊠ψ(y))  
       =   (τ(x)+τ'(x))⊠ψ(y)   =    (τ+τ')(x)⊠ψ(y)

shows that the operation (τ×ψ')+(τ'×ψ)
has the property that characterizes the operation 
(τ +τ')×ψ. So

    (τ+τ')×ψ =  (τ×ψ)+(τ'×ψ) .


(ii) The relation

    ((τ'×ψ')∘(τ×ψ))(x⊠ y)    =  (τ'×ψ')( (τ×ψ)(x⊠ y))  
       =  (τ'×ψ')( τ(x)⊠ψ(y) )  
       = τ'(τ(x))⊠ψ'(ψ(y))  =  (τ'∘τ)(x)⊠ (ψ'∘ψ)(y)

shows that the operation (τ'×ψ')∘(τ×ψ)
has the property that characterizes the operation 
(τ'∘τ)×(ψ'∘ψ). So

    (τ'∘τ)×(ψ'∘ψ) =  
    (τ'×ψ')∘(τ×ψ) .

A similar (but even shorter) argument shows that _G×_G'=_G× G'.

(iii) We start with naturality of the associativity isomorphism.
We consider operations τ∈(G,K), ψ∈(G',K') 
and κ∈(G”,K”). The relation

    (τ×(ψ×κ))(a_G,G',G”^*((x⊠ y)⊠ z))     =  ((τ×(ψ×κ))(x⊠ (y⊠ z))) 
       =  τ(x)⊠(ψ(y)⊠κ(z))
       =  a_K,K',K”^*( (τ(x)⊠ψ(y))⊠κ(z))
       =  a_K,K',K”^*(((τ×ψ)×κ)((x⊠ y)⊠ z))

shows that the operation 
(a_K,K',K”^*)^-1∘(τ×(ψ×κ))∘ a_G,G',G”^*
has the property that characterizes the operation 
(τ×ψ)×κ. So

    (τ×(ψ×κ))∘ a_G,G',G”^*  =  
    a_K,K',K”^*∘((τ×ψ)×κ) .

The arguments for naturality of the unit and symmetry isomorphisms
are similar, and we omit them.

The unit, associativity (pentagon) and symmetry (hexagon)
coherence relations in  follow from the corresponding coherence relations
for the product of groups, and the fact that passage from
group homomorphisms to restriction operations is functorial.





The full subcategory _ of the Burnside category  spanned 
by finite groups has a different, more algebraic description, as we shall now
recall. This alternative description is in terms of `bisets',
and is often taken as the definition in algebraic treatments of global functors
(which are then sometimes called `biset functors').subjectbiset functor
The category of `global functors on finite groups',
i.e., additive functors from _ to abelian groups, is thus equivalent to
the category of `inflation functors' in the sense of Webb <cit.>
and to the `global (∅,∞)-Mackey functors' in 
the sense of Lewis <cit.>. 

We define the additive biset category ^c. 
The objects of ^c are all finite groups. The abelian group ^c(G,K) 
of morphisms from a group G to K is the Grothendieck
group of finite K-G-bisets where the right G-action is free.
In the special case G=e of the trivial group as source we
obtain ^c(e,K), the Burnside ring of finite K-sets.
Composition

    ∘ :  ^c(K,L) ×^c(G,K)  → ^c(G,L)

is induced by the balanced product over K, i.e., it is the
biadditive extension of 

    (S,T)  ⟼ S×_K T  .
 
Here S has a left L-action and a commuting free right K-action,
whereas T has a left K-action and a commuting free right G-action.
The balanced product S×_K T than inherits a left L-action from S
and a free right G-action from T.
Since the balanced product is associative up to isomorphism,
this defines a pre-additive category.
So the category ^c is the `group completion' of the category ^+_, 
the restriction of the effective Burnside category ^+ 
(compare Construction <ref>) to finite groups.


We define additive maps

    Ψ_G,K :  (G,K) → ^c(G,K)

that form an additive equivalence of categories (restricted to finite groups).
The map Ψ_G,K sends a basis element [L,α] to the
class of the K-G-biset

    K×_(L,α) G  =  K× G/(kl,g)∼(k,α(l)g)

whose right G-action is free.
Every transitive G-free K-G-biset is isomorphic to one of this form, 
and K×_(L,α) G is isomorphic, as a K-G-biset,
to K×_(L',α') G if and only if (L,α) is conjugate to
(L',α'). So the map Ψ_G,K sends the basis of
(G,K) of Theorem <ref> 
to a basis of ^c(G,K), and it is thus an isomorphism.

We claim that the maps Ψ_G,K form a functor as G and K vary through
all finite groups; this then shows that Ψ is an additive equivalence
of categories from the full subcategory of _ to ^c.
The functoriality boils down to the fact that in both categories
restriction, inflation and transfer interact with each other in exactly the
same way. We omit the details.

The restriction of the monoidal structure on the Burnside category
to finite groups has an interpretation in terms of the cartesian product of
bisets: under the equivalence of categories Ψ:_^c, 
it corresponds to the monoidal structure

    ^c(G,K)×^c(G',K') → ^c(G× G', K× K') , 
     ([S], [S'])  ⟼ [S× S'] .

Here S is a right free K-G-biset
and S' is a right free K'-G'-biset; the cartesian product
S× S' is then a right free (K× K')-(G× G')-biset.
Indeed, the equivalence Ψ:_^c 
sends the basis element [L,α]∈(G,K) 
to the class of the biset K×_(L,α) G.
So the equivalence is monoidal because 

    (K×_(L,α) G)× (K'×_(L',α') G')
       and   (K× K')×_(L× L',α×α') G× G'

are isomorphic as  (K× K')-(G× G')-bisets.


subjectBurnside category|)

[Box product of global functors]
Since global functors are additive functors on the Burnside category ,
the symmetric monoidal product on  gives rise to a
symmetric monoidal convolution product on the category of global functors.
This is a special case of the general construction of Day <cit.> that
we review in Appendix <ref>.
We now make this convolution product more explicit.
We denote by  the pre-additive category whose objects
are pairs of compact Lie groups, and with morphism groups

    ()((G,G'),(K,K'))  =  (G,K) (G',K')  .

We let F, F' and F” be global functors.
We denote by F F':→ the objectwise
tensor product given on objects by

    (F F')(G,G') =  F(G) F'(G')  .

A bimorphism is a natural transformation 

    F F' → F”∘×

of additive functors on the category .subjectbimorphism!of global functors
Since the morphism groups in the Burnside category are
generated by transfers and restrictions, this means more explicitly
that a bimorphism is a collection of group homomorphisms

    b_G,G' :  F(G)  F'(G')  → F”(G× G')

for all compact Lie groups G and G', such that for all continuous homomorphisms
α:K→ G and α':K'→ G' 
and for all closed subgroups H≤ G and H'≤ G' the  diagram

    @C=15mm@R=10mm 
    F(H)  F'(H') [r]^-_H^G_H'^G'[d]_b_H,H'   
    F(G)  F'(G') [r]^-α^* (α')^*[d]_b_G,G'   
    F(K)  F'(K') [d]^b_K,K'
    
    F”(H× H') [r]_-_H× H'^G× G'   
    F”(G× G') [r]_-(α×α')^*     F”(K× K')

commutes.
Here we exploited that the generating operations multiply as 
described in (<ref>).
Equivalently: for every compact Lie group G the maps {b_G,G'}_G'
form a morphism of global functors F(G) F'(-)→ F”(G×-)
and for every compact Lie group G' the maps {b_G,G'}_G
form a morphism of global functors F(-) F'(G')→ F”(-× G').

A box productsubjectbox product!of global functors|(symbol - box product of global functors 
of F and F' is a universal example of a global functor
with a bimorphism from F and F'.
More precisely, a box product 
is a pair (F F',i) consisting of a global functor F F'
and a universal bimorphism i,
i.e., such that for every global functor F” the map

    (F F',F”)  → Bimor((F,F'),F”)  ,    f⟼ f i

is bijective. Box products exist by the general theory
(see Proposition <ref>), 
and they are unique up to preferred isomorphism
(see Remark <ref>). 
The universal property guarantees
that given any collection of choices of box product F F'
for all pairs of global functors, F F' extends to
an additive functor in both variables.
Moreover, there are preferred associativity and commutativity isomorphisms
that make the box product into a symmetric monoidal structure on the category
of global functors, with the Burnside ring global functor  
as a strict unit object, compare Theorem <ref>.
The box product of representable global functors is again representable,
by Remark <ref>.





We claim that the box product is right exact in both variables. To see this we 
recall from Remark <ref> that the values 
of a Day type convolution product can be described as an enriched coend. 
In our present situation this says that
F M is a cokernel of a certain homomorphism of global functors

    d :  ⊕_H,H',G,G'(H× H',-)(G,H)   (G',H') F(G) M(G') 
    →    ⊕_G,G'(G× G',-) F(G) M(G')   .

The left sum is indexed over all quadruples (H,H',G,G') in a set
of representatives of isomorphism classes of compact Lie groups,
the right sum is indexed over pairs (G,G') of such groups.
The map d is the difference of two homomorphisms; one of them
sums the tensor products of 

    (H× H',-)(G,H)(G',H') → (G× G',-) 
    ,  φττ'  ⟼ φ∘(τ×τ')

and the identity on F(G) M(G').
The other map sums the tensor product of the identity of 
the global functor (H× H',-)
and the action maps (G,H) F(G)→ F(H)
respectively  (G',H') M(G')→ M(H').
Cokernels of global functors are calculated objectwise, so
the value (F M)(K) is a cokernel of the morphism of abelian groups
that we obtain by plugging K into the free variable above.
Theorem <ref> 
describes explicit free generators for the morphism groups
in the Burnside category; using this, the value (F M)(K) 
can be expanded into a cokernel of a morphism between two huge sums
of tensor products of values of F and M.

Now we consider a short exact sequence of global functors 

    0 → M → M' → M”→ 0  .

This gives rise to a commutative diagram
of global functors

    @C=5mm@R=5mm⊕(H× H',-)(G,H)(G',H')  F(G) M(G')[r][d]    ⊕(G× G',-) F(G) M(G') [d] 
    ⊕(H× H',-)(G,H)(G',H') F(G)  M'(G')[r][d]    ⊕(G× G',-) F(G) M'(G') [d] 
    ⊕(H× H',-)(G,H)(G',H') F(G)  M”(G')[r][d]    ⊕(G× G',-) F(G) M”(G') [d] 
    
    0     0

with exact columns. The induced sequence of horizontal cokernels

    F M → F M' → F M”→ 0

is thus also exact.



While the box product of global functors shares many properties with
the tensor product of modules over a commutative ring,
the constructions differ fundamentally in one aspect:
projectives are not generally flat in the category of global functors. 
In other words, for most projective global functors P,
the functor - P does not send monomorphisms to monomorphisms. 
This kind of phenomenon has been analyzed in
great detail by Lewis in <cit.>; 
Lewis' notation for the category ^c is ℬ_*(∅,∞),
our -global functors are his `global (∅,∞)-Mackey functors'
and the category of -global functors is denoted 𝔐_*(∅,∞).
Theorem 6.10 of <cit.> shows that 
the representable functor _C_p is not flat, 
where C_p is a cyclic group of prime order p.

subjectbox product!of global functors|)




We now remark that bimorphisms of global functors can be identified
with another kind of structure that we call `diagonal products'.



Let X,Y and Z be global functors. A diagonal product
is a natural transformation X Y→ Z of Rep-functors
to abelian groups that satisfies reciprocity,
where X Y is the objectwise tensor product.


More explicitly, a diagonal product consists of additive maps

    ν_G  :  X(G) Y(G)  → Z(G)

for every compact Lie group G that are natural for restriction along
continuous homomorphisms and satisfy the reciprocity relation

    ^G_H (ν_H(x ^G_H(y)))  =  ν_G(^G_H(x)  y)

for all closed subgroups H of G and all classes x∈ X(H) and y∈ Y(G), 
and similarly in the other variable.


Any bimorphism μ:(X, Y)→ Z gives rise to a diagonal product
as follows. For a group G we define ν_G as the composite

    X(G)⊗ Y(G)  μ_G,G Z(G× G) Δ^*_G Z(G)

where Δ_G:G→ G× G is the diagonal. For a group homomorphism
α:K→ G we have Δ_G∘α=(α×α)∘Δ_K,
so the following diagram commutes:

    @C=12mm
    X(G)⊗ Y(G) [r]^-μ_G,G[d]_α^*α^*   
    Z(G× G)[r]^-Δ_G^*[d]^(α×α)^*    
    Z(G)[d]^α^*
    
    X(K)⊗ Y(K) [r]_-μ_K,K   
    Z(K× K)[r]_-Δ_K^*    Z(K)

Since there is only one double coset for the left Δ_G-action
and the right (H× G)-action on G× G, the double coset formula becomes 

    Δ_G^*∘_H× G^G× G =  _H^G ∘Δ_H^*∘^H× G_H× H .

We conclude that

    ^G_H(ν_H(x^G_H(y)))     =  _H^G(Δ_H^*(^H× G_H× H(μ_H,G(x y)))) 
       =  Δ_G^*(_H× G^G× G(μ_H,G(x y))) 
       =  Δ_G^*(μ_G,G(^G_H(x) y))  =  ν_G(^G_H(x) y) ,

the reciprocity relation for the diagonal product ν.
The reciprocity in the other variable is similar.

Conversely, given a diagonal product ν, we define a bimorphism 
with (G,K)-component as the composite

    X(G) Y(K)   p_G^* p_K^*  X(G× K) Y(G× K)  ν_G× K  Z(G× K) ,

where p_G:G× K→ G and p_K:G× K→ K are the projections.
If the diagonal product ν was defined from an external product μ as above, then

    ν_G× K∘ (p_G^* p_K^*)    =  Δ_G× K^* ∘μ_G× K,G× K∘ (p_G^* p_K^*)
       =  Δ_G× K^*∘ (p_G× p_K)^*∘μ_G,K =  μ_G,K

because the composite (p_G× p_K)∘Δ_G× K
is the identity. So the external product can be recovered from
the diagonal product.

Given homomorphisms α:G→ G' and
β:K→ K', we have p_G'∘(α×β)=α∘ p_G and
p_K'∘(α×β)=β∘ p_K, so the left part of
the diagram

    @C=12mm
     X(G) Y(K) [r]^-p_G^* p_K^*[d]_α^*β^*   
     X(G× K) Y(G× K) [r]^-ν_G× K[d]^(α×β)^*(α×β)^*   
     Z(G× K)[d]^(α×β)^*
     
     X(G') Y(K') [r]_-p_G'^* p_K'^*   
     X(G'× K') Y(G'× K') [r]_-ν_G'× K'   
     Z(G'× K')

commutes. The right part commutes by naturality of the diagonal
product ν.

For naturality with respect to transfers we let H be a closed subgroup of G,
and we consider classes x∈ X(H) and y∈ Y(K). Then

    _H× K^G× K(μ_H,K(x y))     =  _H× K^G× K(ν_H× K(p_H^*(x)^G× K_H× K(p_K^*(y)))) 
       = ν_G× K(_H× K^G× K(p_H^*(x)) p_K^*(y)) 
       =ν_G× K(p_G^*(_H^G(x)) p_K^*(y)) =  μ_G,K(_H^G(x) y)  .

The second equality is reciprocity, the third is compatibility of transfer 
and inflation.
The argument for transfer naturality in the K-variable is similar.








§ GLOBAL MODEL STRUCTURES FOR ORTHOGONAL SPECTRA




In this section we establish the strong level and global model structures
on the category of orthogonal spectra. Many arguments are parallel
to the unstable analogs in  Section <ref>,
so there is a certain amount of repetition. 
The main model structure of interest for us is
the global model structure,
see Theorem <ref>.
The weak equivalences in this model structure are the global equivalences
and the cofibrations are the flat cofibrations.
More generally, we consider a global family  and define the
-global model structure,
see Theorem <ref> below,
with weak equivalences the -equivalences, 
i.e., those morphisms inducing isomorphisms
of G-equivariant homotopy groups for all G in .
Proposition <ref> shows that the global model structure
is monoidal with respect to the smash product of orthogonal spectra;
more generally, the -global model structure is monoidal,
provided that  is closed under products.





As we explained in more detail in Construction <ref>,
there is a functorial way to write an orthogonal spectrum X as 
a sequential colimit of spectra which 
are made from the information below a fixed level, the skeleta 

    ^m X =  l_m(X^≤ m)  ,

the extension of the restriction of X to _≤ m.
The skeleton comes with a natural morphism i_m:^m X→ X, the counit of
the adjunction (l_m,(-)^≤ m).
The value i_m(V):(^m X)(V)→ X(V) is an isomorphism 
for all inner product spaces V of dimension at most m.
The word `filtration' should be used with caution
because the morphism i_m need not be injective.subjectskeleton!of an orthogonal spectrum
The m-th latching spacesubjectlatching space!of an orthogonal spectrum
of X is the based O(m)-space L_m X  =  (^m-1 X)(^m);
it comes with a natural based O(m)-equivariant map

    ν_m=i_m-1(^m) : L_m X → X(^m)  ,

the m-th latching map. 


  We let G be a compact Lie group and V a G-representation of
  dimension n.
  Then the semifree orthogonal spectrum (<ref>)
  F_G,V A generated by a based G-space A in level V 
  is `purely n-dimensional' in the following sense.
  The space (F_G,VA)_m is trivial for m < n, and hence the 
  latching space L_m (F_G,VA) is trivial for m≤ n.
  For m>n the latching map ν_m:L_m(F_G,VA) → (F_G,VA)(^m) 
  is an isomorphism. So the skeleton ^m (F_G,VA) is trivial for m<n
  and ^m (F_G,VA)=F_G,VA is the entire spectrum for m ≥ n.


Proposition <ref> 
is a fairly general construction of level model structures.
We specialize the general recipe to the category of orthogonal spectra.
For a morphism f:X→ Y  of orthogonal spectra and m≥ 0
we have a commutative square of O(m)-spaces:

    @C=12mm L_m X[r]^L_m f[d]_ν_m^X     L_m Y [d]^ν_m^Y
    
    X(^m)[r]_f(^m)    Y(^m)

We thus get a natural morphism of based O(m)-spaces

    ν_m f =  f(^m)∪ν_m^Y  :  X(^m)∪_L_m XL_m Y  → Y(^m)  .



  A morphism f:X→ Y of orthogonal spectra is
  a flat cofibrationsubjectflat cofibration!of orthogonal spectra 
  if the latching morphism
  ν_m f:X(^m)∪_L_m XL_m Y→ Y(^m) is an O(m)-cofibration  
  for all m≥ 0.
  An orthogonal spectrum Y is flatsubjectflat!orthogonal spectrumsubjectorthogonal spectrum!flat if the morphism from the trivial spectrum to it
  is a flat cofibration, i.e., for every m≥ 0 the latching map ν_m:L_m Y → Y(^m)
  is an O(m)-cofibration.


Flatness as just defined is thus a special case of `G-flatness'
in the sense of Definition <ref>, for G a trivial compact Lie group.



We let  be a global familysubjectglobal family 
in the sense of Definition <ref>,
i.e., a non-empty class of compact Lie groups
that is closed under isomorphisms, closed subgroups and quotient groups.
As in the unstable situation 
in Section <ref>,
we now develop the -level model structure
on the category of orthogonal spectra,
in which the -level equivalences are the weak equivalences.
This model structure has a `global' (or `stable') version, 
see Theorem <ref> below. 

We recall that ∩ G denotes the family of those closed subgroups
of a compact Lie group G that belong to the global family .
Moreover, (m)=∩ O(m) is the family of those closed subgroups
of the orthogonal group O(m) that belong to the global family .


 Let  be a global family.
  A morphism f:X→ Y of orthogonal spectra is
  
  
  * an -level equivalencesubjectF-level equivalence@-level equivalence!of orthogonal spectra 
    if the map f(^m):X(^m)→ Y(^m) is an (m)-equivalence for all m≥ 0;
  
  * an -level fibrationsubjectF-level fibration@-level fibration!of orthogonal spectra 
    if the map f(^m):X(^m)→ Y(^m) is an (m)-fibration for all m≥ 0;
  
  * an -cofibration if the latching morphism
    ν_m f:X(^m)∪_L_m XL_m Y→ Y(^m) is an (m)-cofibration 
    or all m≥ 0.



In other words, f:X→ Y is an -level equivalence 
(respectively -level fibration) if for every m≥ 0 
and every subgroup H of O(m) that belongs to the family  
the map f(^m)^H:X(^m)^H→ Y(^m)^H is a weak equivalence
(respectively Serre fibration).

We let G be any group from the family  and V a faithful G-representation
of dimension m.
We let α:^m→ V be a linear isometry and
define a homomorphism c_α:G→ O(m) by `conjugation by α',
i.e., we set 

    (c_α(g))(x)  =  α^-1(g·α(x))

for g∈ G and x∈^m.
We restrict the O(m)-action on X(^m) to a G-action
along the homomorphism c_α. Then the map

    X(α) :  c_α^*( X(^m) ) → X(V)

is a G-equivariant homeomorphism, natural in X;
it restricts to a natural homeomorphism from X(^m)^G̅ to X(V)^G,
where G̅=c_α(G) is the image of c_α. This implies:

 
Let  be a global family and f:X→ Y a morphism of orthogonal spectra.


  * The morphism f is an -level equivalence
  if and only if for every compact Lie group G 
  and every faithful G-representation V the map
  f(V):X(V)→ Y(V) is an (∩ G)-equivalence.
  
  * The morphism f is an -level fibration
    if and only if for every compact Lie group G 
    and every faithful G-representation V the map
    f(V):X(V)→ Y(V) is an (∩ G)-fibration.






Now we are ready to establish the -level model structure.


  Let  be a global family.
     The -level equivalences, -level fibrations 
    and -cofibrations form a model structure,
    the -level model structure,subjectF-level model structure@-level model structure!for orthogonal spectra
    on the category of orthogonal spectra.
   The  -level model structure is topological and cofibrantly generated.


The first part is a special case of 
Proposition <ref>, in the following way.
We let (m) be the (m)-projective
model structure on the category of based O(m)-spaces,
i.e., the based version of the model structure of 
Proposition  <ref>.
With respect to these choices of model structures (m),
the classes of level equivalences, level fibrations and cofibrations
in the sense of Proposition <ref> 
become the -level equivalences, -level fibrations and -cofibrations.

The consistency condition (see Definition <ref>)
becomes the following condition:
for all m,n≥ 0 and every acyclic cofibration i:A→ B in 
the (m)-projective model structure on based O(m)-spaces,
every cobase change, in the category of based O(m+n)-spaces, of the map 

    (^m,^m+n)_O(m) i  : (^m,^m+n)_O(m) A  → (^m,^m+n)_O(m)B

is an (m+n)-weak equivalence.
We show a stronger statement, namely that the functor

    (^m,^m+n)_O(m) -  :  O(m)_*  → O(m+n)_*

takes acyclic cofibrations in 
the (m)-projective model structure 
to acyclic cofibrations in the projective model structure
on the category of based O(m+n)-spaces
(i.e., the -projective model structure, where  is
the family of all closed subgroups of O(m+n)). 
Since the functor under consideration
is a left adjoint, it suffices to prove the claim for the generating
acyclic cofibrations, i.e., the maps

    ( O(m)/H × j_k)_+

for all H∈(m) and all k≥ 0,
where j_k:D^k×{0}→ D^k× [0,1] is the inclusion.
Up to isomorphism, the functor takes this map to

    O(m+n)⋉_H× O(n) S^n  (j_k)_+

where H acts trivially on S^n.
Since the projective model structure on the category of based O(m+n)-spaces
is topological, it suffices to show that 
O(m+n)⋉_H× O(n) S^n is cofibrant in this model structure.
Since S^n is O(n)-equivariantly homeomorphic to the reduced mapping cone
of the map O(n)/O(n-1)_+→ S^0, it suffices to show that
the two O(m+n)-spaces

    O(m+n)×_H× O(n)(O(n)/O(n-1))      and    
     O(m+n)/(H× O(n))

are cofibrant in the unbased sense.
Both of these O(m+n)-spaces are homogeneous spaces, hence O(m+n)-cofibrant.

We describe explicit sets of generating cofibrations 
and generating acyclic cofibrations for the  -level model structure.
As before we denote by

    G_m  =  F_O(m),^m :  O(m)_*  →

 the left adjoint to the evaluation functor X↦ X(^m),
i.e., the semifree functor (<ref>)
indexed by the tautological O(m)-representation.
We let I_ be the set of all morphisms G_mi for m≥ 0 and for i in the set
of generating cofibrations for the (m)-projective model
structure on the category of O(m)-spaces specified 
in (<ref>).
Then the set I_ detects the acyclic fibrations 
in the  -level model structure 
by Proposition <ref> (iii). 
Similarly, we let J_ be the set of all morphisms G_m j 
for m≥ 0 and for j in the set
of generating acyclic  cofibrations for the (m)-projective model
structure on the category of O(m)-spaces specified 
in (<ref>).
Again by Proposition <ref> (iii),
J_ detects the fibrations in the  -level model structure. 
The -level model structure is topological 
by Proposition <ref>,
where we take  as the set of semifree orthogonal spectra F_H,^m
for all m≥ 0 and all H∈(m), and =∅
as the empty set.



When =e is the minimal global family consisting of all trivial groups,
then the e-level equivalences (respectively e-level fibrations) 
are the level equivalences (respectively level fibrations) of
orthogonal spectra in the sense of Definition 6.1 of <cit.>.
Hence the e-cofibrations are the `q-cofibrations' 
in the sense of <cit.>.
For the minimal global family, the  e-level model structure 
thus specializes to the level model structure of <cit.>.

For easier reference we spell out the case = of the maximal global family 
of all compact Lie groups. In this case (m) is the family of all closed
subgroups of O(m), and the -cofibrations specialize to the flat cofibrations. 
We introduce special names for the -level equivalences
and the -level fibrations,
analogous to the unstable situation 
in Section <ref>.


A morphism f:X→ Y of orthogonal spectra is
a strong level equivalencesubjectstrong level equivalence!of orthogonal spectrasubjectlevel equivalence!strong|seestrong level equivalence 
(respectively strong level fibration)subjectstrong level fibration!of orthogonal spectra 
if for every m≥ 0 the map f(^m):X(^m)→ Y(^m) 
is an O(m)-weak equivalence (respectively an O(m)-fibration).




For the global family =, Proposition <ref> 
specializes to:


    The strong level equivalences, strong level fibrations 
    and flat cofibrations form a model structure,
    the strong level model structure,subjectstrong level model structure!for orthogonal spectra
    on the category of orthogonal spectra.
    The strong level model structure is topological and cofibrantly generated. 




Now we introduce and discuss an important class of orthogonal spectra.


 
An orthogonal spectrum X is a 
global Ω-spectrumsubjectglobal Omega-spectrum@global Ω-spectrum
if for every compact Lie group G, every faithful G-representation W
and an arbitrary G-representation V the adjoint structure map 

    σ̃_V,W :  X(W) → _*(S^V,X(V⊕ W))

is a G-weak equivalence.


The global Ω-spectra will turn out to be
the fibrant objects in the global model structure on orthogonal spectra, see
Theorem <ref> below.
This means that global Ω-spectra abound, because every
orthogonal spectrum admits a global equivalence to a global Ω-spectrum.



Global Ω-spectra are a very rich kind of structure,
because they encode compatible equivariant
infinite loop spaces for all compact Lie groups at once. 
For a global Ω-spectrum X and a compact Lie group G 
the associated orthogonal G-spectrum  X_G is 
`eventually an Ω-G-spectrum' in the sense that
the Ω-G-spectrum condition of <cit.>
holds for all `sufficiently large' (i.e., faithful) G-representations.
However, if G is a non-trivial group,
then the associated orthogonal G-spectrum X_G is in general not
an Ω-G-spectrum since there is no control over
the G-homotopy type of the values at non-faithful representations.  

For every compact Lie group G and every faithful G-representation W, 
the G-space 

    X[G]  = Ω^W X(W)

is a `genuine' equivariant infinite loop space, i.e., deloopable in
the direction of every representation.
Indeed, for every G-representation V, the composite

    X[G] =  Ω^W X(W) Ω^W(σ̃_V,W) Ω^W (Ω^V X(V⊕ W))  Ω^V (Ω^W X(V⊕ W))

is a G-weak equivalence, so the global Ω-spectrum X provides 
a V-deloop Ω^W X(V⊕ W) of X[G].
The G-space X[G] is also independent, up to G-weak equivalence,
of the choice of faithful G-representation. Indeed, if W̅ is another
faithful G-representation, then the G-maps

    Ω^W X(W) Ω^W (σ̃_W',W)   Ω^W(Ω^W' X(W'⊕ W)) 
        Ω^W'(Ω^W,X(W⊕ W'))  Ω^W'(σ̃_W,W')Ω^W' X(W')

are G-weak equivalences.

As G varies, the equivariant infinite loop spaces X[G] are 
closely related to each other. For example, if H is a closed subgroup of G, 
then any faithful G-representation is also faithful as an H-representation.
So X[H] is H-weakly equivalent to the restriction of the 
G-equivariant infinite loop space X[G].




 
Let X be a global Ω-spectrum.
Specialized to the trivial group, the condition in Definition <ref> 
says that X is in particular a non-equivariant Ω-spectrum 
in the sense that the adjoint structure map 
σ̃_,W:X(W)→Ω X(⊕ W) is a weak equivalence of 
(non-equivariant) spaces for every inner product space W.

If X is a global Ω-spectrum, then so is the shifted
spectrum X and the function spectrum _*(A,X)
for every cofibrant based space A.
Indeed, if W is a faithful G-representation, then 
W⊕, the sum with a trivial 1-dimensional representation,
is also faithful. So the adjoint structure map

    σ̃_V,W⊕^X  :  X(W⊕)  → Ω^V(X(V⊕ W⊕))

is a G-weak equivalence for every G-representation V.
But this map is also the adjoint structure map

    σ̃_V,W^ X :  ( X)(W)  → Ω^V (( X)(V⊕ W))

of the shifted spectrum.
The argument for mapping spectra is similar.



subjectglobal fibration!of orthogonal spectra
    A morphism f:X→ Y of orthogonal spectra is a global fibration
    if it is a strong level fibration
    and for every compact Lie group G,
    every G-representation V and every faithful G-representation W 
    the square 
    
    @C=13mm X(W)^G [d]_   f(W)^G[r]^-(σ̃_V,W)^G   _*^G(S^V, X(V⊕ W)) [d]^_*^G(S^V,f(V⊕ W))
    
              Y(W)^G [r]_-(σ̃_V,W)^G   _*^G(S^V, Y(V⊕ W))

    is homotopy cartesian.  


We state a useful criterion for checking when a morphism is a global fibration.
We recall from Construction <ref>
the homotopy fiber F(f) of a morphism f:X→ Y of orthogonal spectra 
along with the natural map i:Ω Y→ F(f).
We apply this for the shifted morphism
f: X→ Ysubjectshift!of an orthogonal spectrum
and precompose with the morphism λ̃_Y,
the adjoint to λ_Y:Y S^1→ Y defined in (<ref>).
We denote by ξ(f) the resulting composite

    Y   λ̃_Y  Ω (  Y)  i   F( f)   .

We note that the orthogonal spectrum F(_ X) has a preferred
contraction, so part (ii) of the next proposition is a way to make precise that
the sequence 

    X   f  Y   ξ(f)  F( f)

is a `strong level homotopy fiber sequence'.


Let f:X→ Y be a strong level fibration of orthogonal spectra. 
Then the following two conditions are equivalent.


  * The morphism f is a global fibration.

  * The strict fiber of f is a global Ω-spectrum and
the commutative square

    @C=15mm 
    X [d]_f [r]^-ξ(_X)    F(_ X) [d] 
    
    Y [r]_-ξ(f)     F( f)

is homotopy cartesian in the strong level model structure.



(i)⟹(ii)
Since the square (<ref>) 
is homotopy cartesian and the vertical maps are Serre fibrations,
the induced map on strict fibers (f^-1(*)(W))^G→_*^G(S^V,f^-1(*)(V⊕ W))
is a weak equivalence. So the strict fiber is a global Ω-spectrum.

The square in (ii) factors as the composite of two commutative squares:

    X [d]_f [r]^-λ̃_X   Ω X  [d]^Ω f[r]^-i     F(_ X) [d] 
    
      Y [r]_-λ̃_Y   Ω Y [r]_i     F( f )

Specializing the global fibration property (<ref>)
for W= with trivial G-action shows that the left square is 
homotopy cartesian in the strong level model structure.
For every continuous based map g:A→ B the square

    Ω A [r]^-i [d]_Ω g    F(_A)[d] 
    Ω B [r]_-i     F(g)

is homotopy cartesian; applying this to  g=(( f)(V))^G
for all G-representa­tions V shows that 
the right square is homotopy cartesian  in the strong level model structure.
So the composite square is homotopy cartesian in the strong level model structure.

(ii)⟹(i)
We let G be a compact Lie group, V a G-representation,
and W a faithful G-representation. We consider the commutative diagram

    @C=10mm X(W)^G [d]_f(W)^G[r]^-(σ̃_V,W)^G   _*^G(S^V, X(V⊕ W)) [d]^_*^G(S^V,f(V⊕ W))
    
          Y(W)^G [d]_(ξ(f)(W))^G[r]_-(σ̃_V,W)^G   _*^G(S^V, Y(V⊕ W)) [d]^_*^G(S^V,ξ(f)(V⊕ W))
    
              F(f)(W⊕)^G [r]^-∼_-(σ̃_V,W)^G   _*^G(S^V, F(f)(V⊕ W⊕))

Since f is a strong level fibration, the natural morphism from
the strict fiber to the homotopy fiber is a strong level equivalence.
So the lower horizontal map is a weak equivalence because the strict fiber,
and hence the homotopy fiber, is a global Ω-spectrum.
Moreover, the two vertical columns are homotopy fiber sequences by hypothesis,
so the upper square is homotopy cartesian, both with respect to the strong
level model structure.





  Let  be a global family. 
  
  
  * A morphism f:X→ Y
    of orthogonal spectra is an -equivalencesubjectF-equivalence@-equivalence!of orthogonal spectra
    if the induced map π_k^G (f) : π_k^G (X) →π_k^G (Y) is an isomorphism
    for all G in  and all integers k.
  
  * A morphism f:X→ Y of orthogonal spectra is an -global fibration
    if it is an -level fibration
    and for every compact Lie group G in ,
    every G-representation VsubjectF-global fibration@-global fibration!of orthogonal spectra 
    and every faithful G-representation W the square (<ref>) 
        is homotopy cartesian.  
      
  * An orthogonal spectrum X is an 
        -Ω-spectrumsubjectF-Omega-spectrum@-Ω-spectrum
if for every compact Lie group G in , every G-representation V
and every faithful G-representation W the adjoint structure map 

    σ̃_V,W :  X(W) → _*(S^V,X(V⊕ W))

is a G-weak equivalence.





When = is the maximal global family of all compact Lie groups,
then an -equivalence is just a global equivalence
in the sense of Definition <ref>,
and the -global fibrations are the global fibrations in the
sense of Definition <ref>.
Also, an -Ω-spectrum is the same as a global Ω-spectrum 
in the sense of Definition <ref>.
When =e is the minimal global family of all trivial groups,
then the e-equivalences are just the traditional
non-equivariant stable equivalences of orthogonal spectra,
also known as `π_*-isomorphisms'.
The e-Ω-spectra are just the traditional non-equivariant Ω-spectra.

The following proposition collects various useful relations
between -equi­valences, -level equivalences and -Ω-spectra.


  Let  be a global family.
  
  
  * Every -level equivalence of orthogonal spectra is an -equivalence.
  
  * Let X be an -Ω-spectrum. Then for every G in , 
    every faithful G-representation V and every k≥ 0 the stabilization map
    
    [S^V⊕^k, X(V)]^G  → π_k^G(X)  ,    [f] ⟼ f

    is bijective.
  
  * Let X be an -Ω-spectrum such that π^G_k (X)=0
    for every integer k and all G in . Then for every group G 
    in  and every faithful G-representation V the space
    X(V) is G-weakly contractible.
  
  * Every -equivalence between -Ω-spectra
    is an -level equivalence.
  
  * Every -equivalence
    that is also an -global fibration is an -level equivalence.
  


(i) We let f:X→ Y be an -level equivalence, and we need to show that the map
π_k^G(f): π_k^G (X)→π_k^G(Y) is an isomorphism for all integers k and
all G in . We start with the case k=0.
We let G be a group from  and V a finite-dimensional
faithful G-subrepresentation of the complete G-universe _G.
By Proposition <ref> (i) 
the map f(V):X(V)→ Y(V) is a G-weak equivalence.
Since the representation sphere S^V
can be given a G-CW-structure, the induced map on G-homotopy classes

    [S^V,f(V)]^G  :  [S^V,X(V)]^G →  [S^V,Y(V)]^G
 
is bijective. The faithful G-representations are cofinal in
the poset s(_G), so taking the colimit over V∈ s(_G) shows that 
π_0^G(f):π_0^G(X)→π_0^G (Y) is an isomorphism.
For k>0 we exploit that π_k^G(X) is naturally isomorphic
to π_0^G(Ω^k X) and the k-fold loop
of an -level equivalence is again an -level equivalence.
For k<0 we exploit that π_k^G(X) is naturally
isomorphic to π_0^G(^-k X) and that every shift
of an -level equivalence is again an -level equivalence.

(ii) We may assume that V is a G-invariant subspace
of the complete G-universe _G used to define π_k^G(X).
Since V is faithful, so is every G-represen­tation that contains V.
So the directed system whose colimit is  π_k^G(X)
consists of isomorphisms `above V'. Hence the canonical map

    [S^V⊕^k, X(V)]^G  → π_k^G(X)

is bijective.

(iii) We adapt an argument of Lewis-May-Steinberger <cit.> to our context.
Every -Ω-spectrum is in particular a non-equivariant Ω-spectrum;
every non-equivariant Ω-spectrum with trivial homotopy groups
is levelwise weakly contractible, so this takes care of trivial groups.

Now we let G be a non-trivial group in .
We argue by a nested induction over the `size' of G:
we induct over the dimension of G and, for fixed dimension, 
over the cardinality of the finite set of path components of G.
Every proper closed subgroup H of G either has strictly smaller dimension than G,
or the same dimension but fewer path components.
So we know by induction that the fixed point space X(V)^H is weakly contractible 
for every proper closed subgroup H of G.
So it remains to analyze the G-fixed points of X(V).

We let W=V-V^G be the orthogonal complement of the fixed subspace V^G
of V. Then W is a faithful G-representation with trivial fixed points.
The cofiber sequence of G-CW-complexes

    S(W)_+  → D(W)_+  → S^W

induces a fiber sequence of equivariant mapping spaces

    _*^G(S^W,X(V))  → ^G(D(W),X(V))  →^G(S(W),X(V))  .

Since W^G=0, the G-fixed points S(W)^G are empty,
so any G-CW-structure on S(W) uses only equivariant cells of the
form G/H× D^n for proper subgroups H of G.
But for proper subgroups H, all the fixed point spaces X(V)^H 
are weakly contractible by induction. 
Hence the space ^G(S(W),X(V)) is weakly contractible.
Since the unit disc D(W) is equivariantly contractible,
the space ^G(D(W),X(V)) is homotopy equivalent to X(V)^G,
and we conclude that evaluation at the fixed point 0∈ W is a weak equivalence

    ^G_*(S^W,X(V))  ≃ X(V)^G .

We have X(V)=X(W⊕ V^G)=(^V^GX)(W).
The stabilization map

    π_k( ^G_*(S^W,X(V)))    = [S^W⊕^k, (^V^GX)(W)]^G
       → π_k^G(^V^GX) π_k-(V^G)^G (X)

is bijective for all k≥ 0, by part (ii). Since we assumed that the
G-equivariant homotopy groups of X vanish, the space _*^G(S^W, X(V))
is path connected and has vanishing homotopy groups, i.e., 
it is weakly contractible.
So X(V)^G is weakly contractible, and this completes the proof of (iii).

(iv) Let f:X→ Y be an -equivalence 
between -Ω-spectra.
We let F denote the homotopy fiber of f, and we let G be a group from . 
For every G-representation V the G-space F(V) is the homotopy fiber
of f(V):X(V)→ Y(V). So F is again an -Ω-spectrum.
The long exact sequence of homotopy groups 
(see Proposition <ref>)
implies that π^H_*(F)=0 
for all H in .

If G acts faithfully on V, then by the -Ω-spectrum property, 
the space X(V) is G-weakly equivalent
to Ω X(V⊕) and similarly for Y.
So the map f(V) is G-weakly equivalent to 

    Ω f(V⊕)  :  Ω X(V⊕)
     → Ω X(V⊕) .

Hence we have a homotopy fiber sequence of G-spaces

    X(V)  f(V)  Y(V)  → F(V⊕)  .

Since F is an -Ω-spectrum with vanishing equivariant
homotopy groups for groups in , the space F(V⊕) 
is G-weakly contractible by part (iii). 
So f(V) is a G-weak equivalence. 
The morphism f is then an -level equivalence 
by the criterion of Proposition <ref> (i). 

(v) We let f:X→ Y be an -equivalence and an -global fibration.
Then the strict fiber f^-1(∗) of f is an -Ω-spectrum
with trivial G-equivariant homotopy groups for all G∈.
So f^-1(∗) is -level equivalent to the trivial spectrum by
part (iii). Since f is an -level fibration, 
the embedding f^-1(∗)→ F(f) of the strict fiber into the homotopy fiber 
is an -level equivalence, so the  homotopy fiber F(f)
is -level equivalent to the trivial spectrum.
The homotopy cartesian square of Proposition <ref> (ii)
then shows that f is an -level equivalence.





Our next result, in fact the main result of this section,
is the -global model structure.
This model structure is cofibrantly generated, and we spell out
an explicit set of generating cofibrations and generating acyclic cofibrations.
The set I_ was defined in 
the proof of Proposition <ref> as the set of morphisms G_m i 
for m≥ 0 and for i in the set of generating cofibrations 
for the (m)-projective model structure on the category of O(m)-spaces specified 
in (<ref>).
Similarly, the set J_ is the set of morphisms G_m j 
for m≥ 0 and for j in the set of generating acyclic cofibrations 
for the (m)-projective model structure on the category of O(m)-spaces specified 
in (<ref>).

Given any compact Lie group G and G-representations V and W
we recall from (<ref>) the morphism

    λ_G,V,W :  F_G,V⊕ WS^V  → F_G,W .

If the representation W is faithful, 
then this morphism is a global equivalence 
by Theorem <ref>. 
We set

    K_ =  ⋃_G,V,W(λ_G,V,W)  ,

the set of all pushout products of sphere inclusions i_m:∂ D^m→ D^m
with the mapping cylinder inclusions of the morphisms λ_G,V,W
(compare Construction <ref>);
here the union is over a set of representatives
of the isomorphism classes of triples (G,V,W) consisting of
a compact Lie group G in the family , a G-representation V 
and a faithful G-representation W.

 
  Let  be a global family.
  
  
  * The -equivalences, -global fibrations and -cofibrations 
    form a model structure on the category of orthogonal spectra, 
    the -global model structure.subjectF-global model structure@-global model structure!for orthogonal spectrasubjectglobal model structure!-
  
  * The fibrant objects in the  -global model structure 
    are the -Ω-spectra.subjectF-Omega-spectrum@-Ω-spectrum
  
  * A morphism of orthogonal spectra is:
    
    
  * an acyclic fibration in the  -global model structure
      if and only if it has the right lifting property 
      with respect to the set I_; 
    
  * a fibration in the  -global model structure
      if and only if it has the right lifting property 
      with respect to the set J_∪ K_.  
    
  
  * The -global model structure is cofibrantly generated, 
    proper and topological. 
  
  * The adjoint functor pair
    
    Σ^∞_+  :   @<.4ex>[r]      :  Ω^∙@<.4ex>[l]

    is a Quillen pair for the two -global model structures
    on orthogonal spaces and orthogonal spectra.
  


The category of orthogonal spectra is complete and cocomplete 
(MC1), the -equivalences satisfy the 2-out-of-3 property (MC2)
and the classes of -equivalences, -global fibrations and 
-cofibrations are closed under retracts (MC3).
The -level model structure (Proposition <ref>)
shows that every morphism of orthogonal spectra
can be factored as f∘ i for an -cofibration i
followed by an -level equivalence f that is also an -level fibration.
For every G in , every G-representation V 
and every faithful G-representation W,
both vertical maps in the commutative square (<ref>) 
are then weak equivalences, so the square is homotopy cartesian.
The morphism f is thus an -global fibration and an -equivalence,
so this provides one of the factorizations as required by MC5.

The morphism λ_G,V,W represents the map

    (σ̃_V,W)^G : X(W)^G → _*^G(S^V,X(V⊕ W))^G ;

by Proposition <ref>, 
the right lifting property with respect to the union J_∪ K_ thus
characterizes the -global fibrations. We apply the small object argument 
(see for example <cit.> or <cit.>)
to the  set J_∪ K_.
All morphisms in J_ are flat cofibrations and -level equivalences;
-level equivalences are -equivalences by Proposition <ref> (i).
Since F_G,V⊕ WS^V and F_G,W are flat, the morphisms in K_
are also flat cofibrations, and they are -equivalences
because the morphisms λ_G,V,W are.
The small object argument provides a functorial factorization
of every morphism φ:X→ Y of orthogonal spectra
as a composite

    X   i   W   q   Y

where i is a sequential composition of cobase changes of coproducts
of morphisms in J_∪ K_, 
and q has the right lifting property with respect 
to J_∪ K_; in particular, the morphism q 
is an -global fibration.
All morphisms in J_∪ K_ are -equivalences and -cofibrations,
hence also h-cofibrations 
(by Proposition <ref> 
applied to the strong level model structure).
By Corollary <ref>
(or rather its modification for -equivalences),
the class of h-co­fibrations that are simultaneously -equivalences 
is closed under coproducts, cobase change and sequential composition.
So the morphism i is an -cofibra­tion and an -equivalence.

Now we show the lifting properties MC4. 
By Proposition <ref> (v)
a morphism that is both an -equivalence and an -global fibration
is an -level equivalence, and hence an acyclic fibration
in the -level model structure. So every morphism that is
simultaneously an -equivalence  and an -global fibration has the
right lifting property with respect to -cofibrations.
Now we let j:A→ B be an -cofibration that is also an -equivalence and 
we show that it has the left lifting property with respect 
to -global fibrations.
We factor j=q∘ i, 
via the small object argument for J_∪ K_,
where i:A→ W is a (J_∪ K_)-cell complex
and q:W→ B is an -global fibration.
Then q is an -equivalence since j and i are,
so q is an acyclic fibration in the -level model structure, by the above.
Since j is an -cofibration, a lifting in

    A [r]^-i [d]_j     W [d]^q_(.6)∼
    
    B @=[r] @..>[ur]     B

exists. Thus j is a retract of the morphism i that has the left lifting property
with respect to all -global fibrations.
But then j itself has this lifting property.
This finishes the verification of the model category axioms.
Alongside we have also specified sets of generating flat cofibrations I_
and generating acyclic cofibrations J_∪ K_.
Sources and targets of all morphisms in these sets are small with
respect to sequential colimits of flat cofibrations. So the 
 -model structure is cofibrantly generated.

Left properness of the -global model structure follows from
Corollary <ref> (i) and the fact that
-cofibrations are h-cofibrations.
Right properness follows from
Corollary <ref> (ii) and the fact that
-global fibrations are in particular -level fibrations.

The -global model structure is topological by 
Proposition <ref>;
here we take  to be the set of semifree orthogonal spectra G_m(O(m)/H)
for all m≥ 0 and all H∈(m),
and we take  as the set of mapping cylinder inclusions c(λ_G,V,W)
of the morphisms λ_G,V,W indexed as in the set K_.

Part (v) is straightforward from the definitions.


In the case =e of the minimal global family of trivial groups,
the e-equivalences are the (non-equivariant) stable equivalences
of orthogonal spectra, and the  e-global model structure
coincides with the stable model structure established by Mandell, May, Shipley and the
author in <cit.>.
For easier reference we spell out the special case =
for the maximal family of all compact Lie groups,
resulting in the global model structure
on the category of orthogonal spectra.


 
    The global equivalences, global fibrations and flat cofibrations 
    form a model structure,
    the global model structuresubjectglobal model structure!for orthogonal spectra
    on the category of orthogonal spectra. 
    The fibrant objects in the global model structure are the global Ω-spectra.
    The global model structure is proper, topological and cofibrantly generated.
  


The same proof as in the unstable situation 
in Corollary <ref>
applies to prove the following characterization of -equivalences.


Let f:A→ B be a morphism of orthogonal spectra
and  a global family.
Then the following conditions are equivalent.


  * The morphism f is an -equivalence.subjectF-equivalence@-equivalence!of orthogonal spectra

  * The morphism can be written as f=w_2∘ w_1 for an
  -level equivalence w_2 and a global equivalence w_1.

  * For some (hence any) -cofibrant approximation
  f^c:A^c→ B^c in the  -level model structure
and every -Ω-spectrum Y the induced map

    [f^c,Y]  :  [B^c,Y]  → [A^c,Y]

on homotopy classes of morphisms is bijective.




[Mixed global model structures]
Cole's `mixing theorem' for model structures <cit.>  
allows to construct many more -model structures 
on the category of orthogonal spectra.
We consider two global families such that ⊆. 
Then every -equivalence is an -equivalence and every 
fibration in the  -global model structure
is a fibration in the  -global model structure.
By Cole's theorem <cit.> 
the -equivalences and the fibrations of the
 -global model structure
are part of a model structure,
the -mixed -global model structuresubjectmixed global model structuresubjectglobal model structure!mixed
on the category of orthogonal spectra.
By <cit.> 
the cofibrations in the -mixed -global model structure 
are precisely the retracts of all composites
h∘ g in which g is an -cofibration and h is simultaneously
an -equivalence and an -cofibration.
In particular, an orthogonal spectrum is cofibrant in the
-mixed -global model structure if it is -cofibrant and
-equivalent to an -cofibrant 
orthogonal spectrum <cit.>.
The -mixed -global model structure is again proper 
(Propositions 4.1 and 4.2 of <cit.>).

When =e is the minimal family of trivial groups,
this provides infinitely many -mixed model structures on
the category of orthogonal spectra that are all Quillen equivalent.
In the extreme case, the -mixed e-model structure
is the -model structure of Stolz <cit.>.




[-global homotopy theory via symmetric spectra]
We denote by symbol - global family of finite groups
the global family of finite groups.subjectglobal family!of finite groups 
The -global stable homotopy theory
has another very natural model, namely the category of 
symmetric spectrasubjectsymmetric spectrum
in the sense of Hovey, Shipley and Smith <cit.>.
In <cit.> 
Hausmann has established a global model structure 
on the category of symmetric spectra, 
and he showed that the forgetful functor is a right Quillen equivalence
from the category of orthogonal spectra with the -global model structure
to the category of symmetric spectra with the global model structure.
Symmetric spectra cannot model global homotopy types for all compact Lie groups,
basically because compact Lie groups of positive dimensions do not have
any faithful permutation representations.



One of the points of model category structures is that they facilitate
the analysis of and constructions in the homotopy category (which only depends
on the class of weak equivalences). An example of this is the existence
and constructions of products and coproducts in a homotopy category.
We take the time to make this explicit for the -global stable homotopy category,
for which we write _.


 
Let  be a global family.


  * The coproduct (i.e., wedge) of every family  of orthogonal spectra
is also a coproduct in the homotopy category _. 

  * Let {X_i}_i∈ I be a family of orthogonal spectra such that the
canonical map

    π_k^G(∏_i∈ I  X_i) → ∏_i∈ I π_k^G(X_i)

is an isomorphism for every compact Lie group G in  and every integer k.
Then the product ∏_i∈ I  X_i of orthogonal spectra
is also a product of {X_i}_i∈ I 
in the homotopy category _. 

In particular, the -global stable homotopy category _ 
has all set indexed coproducts and products.


(i) Since the -equivalences are part of the -global model structure,
general model category theory guarantees that coproducts in _ can be constructed
by taking the pointset level coproduct of -cofibrant approximations,
see for example <cit.>.
Since equivariant homotopy groups take wedges of orthogonal spectra
to direct sums (Corollary <ref> (i)),
any wedge of -equivalences is again an -equivalence.
So the pointset level wedge maps by 
an -equivalence to the wedge of the -cofibrant approximations,
and these two are isomorphic in _.

(ii) This part is essentially dual to (i), with the caveat that
infinite products of -equivalences need not be -equivalences in general.
To construct a product in _ of the given family,
the abstract recipe is to choose -equivalences f_i:X_i→ X_i^
to -Ω-spectra, and then form the pointset level
product of the replacements.
For all groups G in  we consider the commutative square:

    π_k^G(∏_i∈ I X_i)[r][d]_(∏ f_i)_*   ∏_i∈ Iπ_k^G(X_i) [d]^∏ (f_i)_*
    π_k^G(∏_i∈ I X^_i)[r]    ∏_i∈ Iπ_k^G(X^_i)

The upper map is an isomorphism by hypothesis, and the right map
is an isomorphism since each f_i is an -equivalence.
The lower map is also an isomorphism: since all X_i^ 
are -Ω-spectra, the colimit system

    π_k^G( ∏_i∈ I  X^_i)    = _V∈ s(_G) [ S^V⊕^k, ∏_i∈ I  X_i^(V) ]^G 
       = _V∈ s(_G) ( ∏_i∈ I  [ S^V⊕^k,  X_i^(V) ]^G)

consists of isomorphisms starting at any faithful G-representation V.
Since faithful representations are cofinal in the poset s(_G),
in this particular situation the colimit commutes with the product.
Altogether we can conclude that in our situation the morphism

    ∏ f_i  :  ∏_i∈ I  X_i → ∏_i∈ I   X^_i

is an -equivalence.
Since the right hand side is a product in _ of the family {X_i}_i∈ I,
so is the left hand side.




We turn to the interaction of the smash product of orthogonal spectra
with the level and global model structures.
Given two morphisms f:A→ B and g:X→ Y of orthogonal spectra,
the pushout product morphismsubjectpushout product is defined as

    f g = (f Y)∪(B g)  :  
    A Y∪_A XB X  → B Y .

We introduce another piece of notation that is convenient for
the discussion of the monoidal properties: when I and I' are two sets
of morphisms of orthogonal spectra, then we define

    I ⊏ I'

to mean that every morphism in I is isomorphic to a morphism in I'.
Moreover, we write I I' for the set of pushout product morphisms
f g for all f∈ I and all g∈ I'.
If  and  are global families, then we denote by
× the smallest global family that contains all
groups of the form G× K for G∈ and K∈.

The sets I_ and J_ of generating cofibrations 
respectively generating acyclic cofibrations 
for the -level model structure
were defined in the proof of Proposition <ref>.
They consist of the morphisms

    F_H,^m (i_k)_+
         respectively    
    F_H,^m (j_k)_+

for all k≥ 0, all m≥ 0 and all compact Lie groups H in (m).
Here we used that the orthogonal spectrum F_G,V( G/H_+)
is isomorphic to F_H,V.
The set K_ of acyclic cofibrations for the -global model structure
was defined in (<ref>).


 
Let  and  be two global families.
Then the following relations hold for the sets of generating
cofibrations and acyclic cofibrations:

    I_   I_ ⊏ I_× ,  
      I_   J_ ⊏ J_×   and  
      I_    K_ ⊏ K_× .



  We start with two key observations concerning 
  the generating cofibrations i_k:∂ D^k→ D^k
  and the generating acyclic cofibrations j_k: D^k×{0}→ D^k× [0,1]
  for the model structure of spaces: the pushout product i_k i_m
  of two sphere inclusions is homeomorphic to the map i_k+m;
  similarly, the pushout product i_k j_m is homeomorphic to the map j_k+m.

  The first relation   I_ I_⊏ I_×
  is then a consequence of the compatibilities between  
  smash products and the isomorphism (<ref>) 
  for the smash product of two semifree orthogonal spectra:
  
    ( F_G,V (i_k)_+) ( F_K,W (i_m)_+)   
      ( F_G,V F_K,W) (i_k i_m)_+  
      F_G× K,V⊕ W (i_k+m)_+

  The second relation I_ J_⊏ J_×
  is proved in the same way:
  
    ( F_G,V (i_k)_+) ( F_K,W (j_m)_+)   
      ( F_G,V F_K,W) (i_k j_m)_+   
      F_G× K,V⊕ W (j_k+m)_+

  For the third relation we recall that c_K,U,W is the mapping cylinder inclusion 
  of the global equivalence
  
    λ_K,U,W :  F_K,U⊕ W S^U  → F_K,W .

  The claim I_ K_⊏ K_× then follows from
  
    ( F_G,V (i_k)_+) ( c_K,U,W (i_m)_+)      
      ( F_G,V c_K,U,W) (i_k i_m)_+ 
          c_G× K,V⊕ U, W (i_k+m)_+ .



Certain pushout product properties are now formal consequences.
In the special case where ==e=×
are the trivial global families, part (iii) of the previous proposition 
specializes to Proposition 12.6 of <cit.>. 


 
  Let  and  be two global families.
  
  
  * The pushout product of an -cofibration with an -cofibration
    is an (×)-cofibration.  
  
  * The pushout product of a flat cofibration 
    with an -cofibration that is also an -equivalence
    is a flat cofibration and global equivalence.
  
  * Let  be a multiplicative global family,subjectglobal family!multiplicative|seemultiplicative global familysubjectmultiplicative global family
    i.e., × =. Then the  -global model structure  
    satisfies the pushout product property with respect to the smash product
    of orthogonal spectra.
  

 
  (i) It suffices to show the claim for a set of generating cofibrations,
  where it follows from the relation  I_ I_⊏ I_×
  established in Proposition <ref>.

  (ii) Again it suffices to check the pushout product of   
  any generating flat cofibration with a generating acyclic cofibration 
  for the -global model structure.
  For generators, the claim follows from the relation
  
    I_( J_∪ K_)     =  (I_ J_)∪(I_ K_) 
       ⊏ J_×∪ K_× =      J_∪ K_

  established in Proposition <ref>.

  (iii) The pushout product of two -cofibrations is an -cofibration by part (i)
  and the hypothesis that  is multiplicative.
  Since -cofibrations are in particu­lar flat cofibrations, 
  the pushout product of two -cofibrations 
  one of which is also an -equivalence
  is another -equivalence by part (ii).



The sphere spectrum  is the unit object for the smash product 
of orthogonal spectra, and it is `free', i.e., e-cofibrant.
Thus  is cofibrant in the  
-global model structure for every global family .
So if  is multiplicative, then with respect to the smash product, 
the -global model structure is a 
symmetric monoidal model category 
in the sense of <cit.>.
A corollary is that the
homotopy category _, i.e., the localization of the category
of orthogonal spectra at the class of -equivalences,
inherits a closed symmetric monoidal structure, 
compare <cit.>. The 
derived smash productsubjectderived smash productsubjectsmash product!derived|seederived smash productsymbol^_ - derived smash product 
 
    ^_ :  _×_ → _ ,

i.e., the induced symmetric monoidal product on _,
is any total left derived functor of the smash product.

 
For every multiplicative global family , the -global 
homotopy category _symbol  _ - global stable homotopy category with respect to the global family 
is closed symmetric monoidal under the derived smash product (<ref>). 

The value of the derived smash product 
at a pair (X,Y) of orthogonal spectra can be calculated as

    X^_ Y  =  X^c  Y^c ,

where  X^c→ X and Y^c→ Y are
cofibrant replacements in the -global model structure,
i.e., -equivalences with -cofibrant sources.
By the following 'flatness theorem',
it actually suffices to `resolve' only one of the factors,
and it is enough to require the source of the `resolution'
to be flat (as opposed to being -cofibrant).






subjectflatness theorem!for orthogonal spectra 
Let  be a global family.


  * Smashing with a flat orthogonal spectrum preserves -equivalences.

  * Smashing with any orthogonal spectrum preserves -equivalences
between flat orthogonal spectra.



(i) We let G be a compact Lie group.
Every cofibration of O(m)-spaces is also a cofibration 
of (G× O(m))-spaces when we let G act trivially.
So if we let G act trivially on a flat orthogonal spectrum X, 
then it is G-flat in the sense of Definition <ref>.
Now we let f:Y→ Z be an -equivalence of orthogonal spectra.
If G belongs to the global family , then f is in particular
a _*-isomorphism of orthogonal G-spectra with respect
to the trivial G-actions. So X f:X Y→ X Z 
is a _*-isomorphism of orthogonal G-spectra by
Theorem <ref>, because X is G-flat.
This proves that X f is again an -equivalence.

(ii) 
We let f:X→ Y be an -equivalence between flat orthogonal spectra.
We choose a global equivalence φ:A^♭→ A with flat source
and consider the commutative square:

    A^♭ X[r]^-φ X[d]_A^♭ f   
    A X[d]^A f
    
    A^♭ Y[r]_-φ Y    A Y

The two horizontal morphisms are global equivalences by part (i),
because X and Y are flat. The left vertical morphism is an -equivalence
by part (i), because A^♭ is flat. 
So A f is also an -equivalence.




Now we prove another important relationship between
the global model structures and the smash product, 
the monoid axiom <cit.>.
As in the unstable situation in Proposition <ref>
we only discuss the weaker form of the monoid axiom with
sequential (as opposed to more general transfinite) compositions.

subjectmonoid axiom!for the smash product of orthogonal spectra
We let  be a global family.
For every flat cofibration i:A→ B that is also an -equivalence
and every orthogonal spectrum Y the morphism

    i Y  :  A Y  → B Y

is an h-cofibration and an -equivalence.subjecth-cofibration
Moreover, the class of h-cofibrations that are also -equivalences
is closed under cobase change, coproducts and sequential compositions.


Given Theorem <ref>, this is a standard argument, 
similar to the proofs of the monoid axiom in the non-equivariant
or the G-equivariant context, compare 
<cit.>, <cit.>,
<cit.> or <cit.>.
Every flat cofibration is an h-cofibration 
(Corollary <ref> (iii)), 
and h-cofibra­tions are closed under smashing with any orthogonal spectrum, 
so i Y is an h-cofibration.
Since i is a h-cofibration and -equivalence, its cokernel B/A
is -stably contractible by the long exact homotopy group sequence
(Corollary <ref>).
But B/A is also flat as a cokernel of a flat cofibration,
so the spectrum (B/A) Y is -stably contractible by 
Theorem <ref>. 
Since i Y is an h-cofibration with cokernel isomorphic to (B/A) Y, 
the long exact homotopy group sequence then shows that i Y is an -equivalence.

The proof that the class of h-cofibrations that are also -equivalences
is closed under cobase change, coproducts and sequential compositions 
is the same as for for global equivalences
in Corollary <ref>.




Every  -cofibration is in particular a flat cofibration.
So the monoid axiom implies 
the monoid axiom in the  -global model structure.
If the global family  is closed under products,
Theorem <cit.> applies
to the  -global model structure and shows the following lifting results.
The additional claims in part (i) about the behavior of
the forgetful functor on the cofibrations are proved 
as in the unstable analog in Corollary <ref>. 



 
Let R be an orthogonal ring spectrum and  
  a multiplicative global family.subjectmultiplicative global family
  
  
  * The category of R-modules admits the -global model structuresubjectF-global model structure@-global model structure!for R-modules  
    in which a morphism is an equivalence (respectively fibration)
    if the underlying morphism of orthogonal spectra is an -equivalence
    (respectively -global fibration).
    This model structure is cofibrantly generated.
    Every cofibration in this -global model structure 
    is an h-cofibration of underlying orthogonal spectra.
    If the underlying orthogonal spectrum of R is -cofibrant,
    then every cofibration of R-modules is an -cofibration of underlying
    orthogonal spectra.
  
  * If R is commutative, then with respect to _R 
    the -global model structure of R-modules is a monoidal model category 
    that satisfies the monoid axiom.
  
  * If R is commutative, then the category of R-algebras 
    admits the -global model structuresubjectF-global model structure@-global model structure!for R-algebras  
    in which a morphism is an equivalence (respectively fibration)
    if the underlying morphism of orthogonal spectra is an -equivalence
    (respectively -global fibration).
    Every cofibrant R-algebra is also cofibrant as an R-module.
  


Strictly speaking, Theorem 4.1 of <cit.>
does not apply verbatim to the -global model structure
because the hypothesis that every object is small (with respect to some regular cardinal)
is not satisfied. However, in our situation the sources of the generating cofibrations
and generating acyclic cofibrations are small with respect to
sequential compositions of h-cofibrations, and this suffices to
run the countable small object argument
(compare also Remark 2.4 of <cit.>).

 
Let R be an orthogonal ring spectrum and N a right R-module
that is cofibrant in the -global model structure 
of Corollary  <ref> (i). 
Then for every global family ,
the functor N_R - takes -equivalences of left R-modules
to -equivalences of orthogonal spectra.


The argument is completely parallel to the unstable
precursor in Proposition <ref>. 
We call a right R-module N homotopical
if the functor N_R - takes -equivalences of left R-modules
to -equivalences of orthogonal spectra.  
Since the -global model structure on the category of right R-modules
is obtained by lifting the global model structure of orthogonal spectra
along the free and forgetful adjoint functor pair, every cofibrant right R-module
is a retract of an R-module that arises as the colimit of a sequence

    ∗ = M_0  → M_1  →… → M_k  → …

in which each M_k is obtained from M_k-1 as a pushout

    A_k R [r]^-f_k R[d]     
     B_k  R [d]    
    
    M_k-1[r]     M_k

for some flat cofibration f_k:A_k→ B_k between flat orthogonal spectra.
For example, f_k can be chosen as a wedge of morphisms 
in the set I_ of generating flat cofibrations.
We show by induction on k that each module M_k is homotopical.
The induction starts with the trivial R-module, where there is nothing to show.
Now we suppose that M_k-1 is homotopical, and we claim that then M_k
is homotopical as well.
To see this we consider an -equivalence of left R-modules
φ:X→ Y. Then M_k_R φ
is obtained by passing to horizontal pushouts in the following commutative diagram
of orthogonal spectra:

    M_k-1_R X [d]_M_k-1_R φ    
     A_k  X[d]^A_kφ[l][r]^-f_k X    B_k  X[d]^B_kφ
    
    M_k-1_R Y      A_k  Y[l][r]_-f_k Y    B_k  Y

Here we have exploited that (A_k R)_R X
is naturally isomorphic to A_k X.
In the diagram, the left vertical morphism is an -equivalence by hypothesis. 
The middle and right vertical morphisms are 
-equivalences because smash product with a flat orthogonal spectrum
preserves -equivalences (Theorem <ref> (i)).
Moreover, since the morphism f_k is a flat cofibration, it is an h-cofibration
(by Corollary <ref> (iii)),
and so the morphisms f_k X and f_k Y are h-cofibrations.
Corollary <ref> (i)
then implies that the induced morphism on horizontal
pushouts M_k_R φ is again an -equivalence.

Now we let M be a colimit of the sequence (<ref>).
Then M_R X is a colimit of the sequence M_k _R X.
Moreover, since f_k:A_k→ B_k is an h-cofibration, so is the 
morphism f_k R, and hence also its cobase change M_k-1→ M_k.
So the sequence whose colimit is M_R X consists of h-cofibrations,
which are in particular levelwise closed embeddings.
The same is true for M_R Y.
Since each M_k is homotopical and colimits of orthogonal spectra
along closed embeddings are homotopical
(see Proposition <ref> (ii)),
we conclude that the morphism M_Rφ:M_R X→ M_R Y 
is an -equivalence, so that M is homotopical.
Since -equivalences are closed under retracts, 
the class of homotopical R-modules is closed under retracts, and so every
cofibrant right R-module is homotopical.




For later use we record a positive version of 
the global model structure of Theorem <ref>.
This positive variant is the one that can be lifted to the category
of ultra-commutative ring spectra, 
see Theorem <ref> below


 A morphism f:X→ Y of orthogonal spectra is a positive cofibrationsubjectcofibration!positivesubjectpositive cofibration!of orthogonal spectra
if it is a flat cofibration and the map f(0):X(0)→ Y(0) is a homeomorphism.  
An orthogonal spectrum is a positive global Ω-spectrumsubjectpositive global Ω-spectrumsubjectglobal Omega-spectrum@global Ω-spectrum!positive
if for every compact Lie group G, every G-representation V
and every faithful G-representation W with W 0 
the adjoint structure map 

    σ̃_V,W :  X(W) → _*(S^V,X(V⊕ W))

is a G-weak equivalence.


If G is a non-trivial compact Lie group, then any faithful G-representation
is automatically non-trivial. So a positive global Ω-spectrum
is a global Ω-spectrum (in the absolute sense) 
if in addition the adjoint structure map σ̃_0,:X(0)→Ω X() 
is a non-equivariant weak equivalence.



 
  The global equivalences and positive cofibrations are part of 
  a proper, cofibrantly generated, topological model structure,
  the positive global model structuresubjectpositive global model structure!for orthogonal spectrasubjectglobal model structure!positive
  on the category of orthogonal spectra. 
  A morphism f:X→ Y of orthogonal spectra is a fibration 
  in the positive global model structure if and only if
  for every compact Lie group G, every G-representation V
  and every faithful G-representation W with W 0 
  the map f(W)^G:X(W)^G→ Y(W)^G is a Serre fibration and the square
    
    @C=13mm X(W)^G [d]_f(W)^G[r]^-(σ̃_V,W)^G   _*^G(S^V, X(V⊕ W)) [d]^_*^G(S^V,f(V⊕ W))
    
              Y(W)^G [r]_-(σ̃_V,W)^G   _*^G(S^V, Y(V⊕ W))

    is homotopy cartesian.  
    The fibrant objects in the positive global model structure 
    are the positive global Ω-spectra.
    The model structure is monoidal with respect 
    to the smash product of orthogonal spectra.


  We start by establishing a positive strong level model structure.
  A morphism f:X→ Y of orthogonal spectra is a
  positive strong level equivalencesubjectlevel equivalence!positive strong!of orthogonal spectra 
  (respectively positive strong level fibration)subjectlevel fibration!positive strong!of orthogonal spectra
  if for every inner product space V with V 0
  the map f(V):X(V)→ Y(V)  is an O(V)-weak equivalence
  (respectively an O(V)-fibration).
  We claim that the positive strong level equivalences, 
  positive strong level fibrations and positive cofibrations form a model structure
  on the category of orthogonal spectra.

  The proof is another application of the general construction of
  level model structures in Proposition <ref>. 
  Indeed, we let (0) be the degenerate model structure on 
  the category _* of based spaces in which every morphism
  is a weak equivalence and a fibration, but only the isomorphisms
  are cofibrations. 
  For m≥ 1 we let (m) be the projective
  model structure (for the family of all closed subgroups)
  on the category of based O(m)-spaces.
  With respect to these choices of model structures (m),
  the classes of level equivalences, level fibrations and cofibrations
  in the sense of Proposition <ref>
  precisely become the positive strong level equivalences,
  positive strong level fibrations and positive cofibrations.
  The consistency condition (Definition <ref>)
  is now strictly weaker than for the strong level model structure,
  so it holds. 
  The positive strong level model structure is topological 
  by Proposition <ref>,
  where we take  as the set of semifree orthogonal spectra F_H,^m
  for all m≥ 1 and all closed subgroup H of O(m).
  
  We obtain the positive global model structure for orthogonal spectra by `mixing' 
  the positive strong level model structure 
  with the global model structure of Theorem <ref>.
  Every positive strong level equivalence is a global equivalence and every 
  positive cofibration is a flat cofibration.
  The global equivalences and the positive cofibrations
  are part of a model structure by Cole's mixing theorem <cit.>,
  which is our first claim. 
  By <cit.> (or rather its dual formulation),
  an orthogonal spectrum is fibrant in the positive global model structure 
  if it is equivalent in the positive strong level model structure to 
  a global Ω-spectrum; this is equivalent to
  being a positive global Ω-spectrum.
  The proof that the positive global model structure is proper and topological 
  is similar as for the global model structure.
  The proof of the pushout product property is as in the absolute global model
  structure (see Proposition <ref>); the only new ingredient is
  that the class of generators F_G,V with V 0 for the positive cofibrations
  is closed under the smash product of orthogonal spectra.

  Finally, the positive global model structure is cofibrantly generated:
  we can simply take the same sets of generating cofibrations 
  and generating acyclic cofibrations as for the global model structure, 
  except that we  omit all morphisms freely generated in level 0.
  






§ TRIANGULATED GLOBAL STABLE HOMOTOPY CATEGORIES



subjectglobal stable homotopy category|(

As the homotopy category of a stable model structure, 
the global stable homotopy category 
comes with the structure of a triangulated category.
The shift functor is the suspension of orthogonal spectra,
and the distinguished triangles arise from mapping cone sequences.
In this section we collect the aspects of global stable homotopy theory that
are best expressed in terms of the triangulated structure.

More generally, we work in the triangulated -global
stable homotopy category _, where  is any global family. 
Theorem <ref> identifies
a set of compact generators of _, 
namely the suspension spectra of global classifying spaces of the groups in .
A consequence is Brown representability for cohomological
and homological functors out of _, 
compare Corollary <ref>.
Theorem <ref> shows that
the classes of -connective and -coconnective spectra 
form a non-degenerate t-structure on _; 
moreover, taking 0-th equi­variant homotopy groups
is an equivalence from the heart of this t-structure to the
category of -global functors.
Immediate consequences are global Postnikov sections
and the existence of Eilenberg-Mac Lane spectra that realize global functors.
Proposition <ref> establishes another connection between
the smash product and the algebra of global functors: 
for globally connective orthogonal spectra the box product of global functors 
calculates the 0-th homotopy groups of a derived smash product.

The last topic of this section are certain distinguished triangles 
in the global stable homotopy category that 
arise from special representations,
namely when a compact Lie group G acts faithfully and transitively on the unit sphere.
The stabilizer of a unit vector is then a closed subgroup H 
such that G/H is diffeomorphic to a sphere, 
and Theorem <ref> 
exhibits a distinguished triangle that relates the global classifying space
of G to certain semifree orthogonal spectra of G and H.
The main examples of this situation are the tautological 
representations of the groups O(m), S O(m), U(m), S U(m) and S p(m);
these will show up again in Section <ref> 
in the rank filtrations of global Thom spectra.




The -global stable homotopy category _ is the homotopy category
of a stable model structure, so it is naturally a triangulated category,
for example by <cit.> or <cit.>.
The shift functor is modeled by the pointset level suspension of orthogonal
spectra. More precisely, the suspension functor - S^1:→ 
of Construction <ref> preserves -equivalences,
so it descends to a functor on the -global stable homotopy category

    - S^1  :  _ → _

for which we use the same name. 
The distinguished triangles are defined from mapping cone sequences,
i.e., a triangle is distinguished if and only if it is isomorphic, 
in _, to a sequence of the form

    X   f  Y   i  C f   p   X S^1

for some morphism of orthogonal spectra f:X→ Y; here the morphisms i 
and p were defined in (<ref>).



[Shift preserves distinguished triangles]
The shift functor :→ 
of Construction <ref> preserves -equivalences,
so it descends to a functor on the -global stable homotopy category

    :  _ → _

for which we use the same name. Moreover, shifting commutes with smashing
with a based space on the nose, i.e., ( X) A=(X A);
so we can (and will) leave out parentheses in such expressions.
Since the suspension functor on _ is induced by smashing with S^1,
the shift functor commutes with the suspension functor, again on the nose,
both on the pointset level and also on the level of the -global stable homotopy
category. We will now argue that shifting also preserves distinguished 
triangles on the nose; equivalently, the derived shift is an exact
functor of triangulated categories if we equip it with the identity isomorphism
∘(- S^1)=(- S^1)∘.
  
To prove our claim we consider a distinguished triangle in _:

    A   f  B   g  C    h   A S^1

The morphism λ_A:A S^1→ A is a natural global equivalence
by Proposition <ref> (i), so all
vertical morphisms in the following diagram are isomorphisms in _:

    @C=10mm 
     A  S^1 [r]^-f S^1[d]_λ_A   
     B  S^1 [r]^-g S^1[d]_λ_B    
    C S^1 [r]^-- h S^1[d]^λ_C   
    A S^1 S^1 [d]^λ_A S^1
     A [r]_- f    B [r]_- g    C [r]_- h    A S^1

The claim now follows from the following three observations:


  * the suspension functor in any triangulated category 
preserves distinguished triangles up to a sign;
so the upper triangle above is distinguished;

  * the left and middle squares commute by naturality of the λ-morphisms;

  * the right square commutes because the two morphisms
λ_A S^1,λ_A S^1:A S^1 S^1 → A S^1
differ by the twist involution of S^1 S^1; since this involution has degree -1,
we obtain

    ( h)∘λ_C  =  λ_A S^1∘ (h S^1) =  - (λ_A S^1)∘ (h S^1) .




We recall the notion of compactly generated triangulated categories.
Compact generation has strong formal consequences, see Theorem <ref> below.
Theorem <ref> shows that the 
-global stable homotopy category enjoys this special property.




Let  be a triangulated category with infinite sums.
An object C of  is compactsubjectcompact!object in a triangulated category 
if for every family {X_i}_i∈ I of objects the canonical map

    ⊕_i∈ I (C,  X_i)  → (C, ⊕_i∈ I  X_i)

is an isomorphism. 
A set  of objects of  is called a set of
weak generatorssubjectgenerator!weak
if the following condition holds: if 
the groups (G[k],X) are trivial for all k∈ and all G∈, 
then X is a zero object.
The triangulated category  is compactly generatedsubjectcompactly generated!triangulated category
if it has sums and a set of compact weak generators.




If G is from a global family , then the functor π_0^G:→
takes -equivalences to isomorphisms. So the universal property
of a localization provides a unique factorization

    π_0^G : _ →

through the -global stable homotopy category. We will abuse notation
and use the same symbol for the equivariant homotopy group functor
on the category of orthogonal spectra and for its `derived'
functor defined on _. This abuse of notation is mostly harmless,
but there is one point where it can create confusion, namely in the context of
infinite products; 
we refer the reader to Remark <ref> for this issue.

We recall from Definition <ref> 
that the global classifying space B_Gsubjectglobal classifying space 
of a compact Lie group G is the semifree orthogonal space
_G,V=(V,-)/G, for some faithful G-representation V.
The choice of faithful representation is omitted from the notation 
because the global homotopy type of B_G does not depend on it.
The suspension spectrum of B_ G comes with a stable tautological class

    e_G  =   e_G,V∈ π_0^G(Σ^∞_+ B_ G)

defined in (<ref>).
subjectstable tautological class

In the proof of the next theorem
we will start using the shorthand notationsymbolX,Y_ -  morphism group in the -global stable homotopy category

    X,Y _ =  _(X,Y)

for the abelian group of morphisms
in the triangulated -global stable homotopy category.


Let  be a global family and G a compact Lie group in .


  * The pair (Σ^∞_+ B_G, e_G) represents the functor
π_0^G:_→(sets).

  * The orthogonal spectrum Σ^∞_+ B_G 
is a compact object of the -global stable homotopy category _.

  * As G varies through a set of representatives
of isomorphism classes of groups in , the spectra
Σ^∞_+ B_G form a set of weak generators for 
the -global stable homotopy category _. 

In particular, the -global stable homotopy category _ 
is compactly generated.


(i) We need to show that for every orthogonal spectrum X the map

    Σ^∞_+ B_G, X _ → π_0^G (X) ,    
    f⟼ f_*(e_G)

is bijective. Since both sides take -equivalences in X to bijections,
we can assume that X is an -Ω-spectrum, and hence fibrant
in the -global model structure.
For G in the family , 
the orthogonal spectrum Σ^∞_+ B_G is -cofi­brant.
So the localization functor induces a bijection

    (Σ^∞_+ B_G, X) / homotopy → Σ^∞_+ B_G, X _
 
from the set of homotopy classes of morphisms of orthogonal spectra
to the set of morphisms in _.

We let V be the faithful G-representation that is implicit in the
definition of the global classifying space B_G. 
By the freeness property of B_ G=_G,V, morphisms from 
Σ^∞_+ B_G to X biject with based G-maps S^V→ X(V),
and similarly for homotopies.
The composite

    [ S^V, X(V) ]^G     Σ^∞_+ B_G, X _ f↦ f_*(e_G) π_0^G (X)

is the stabilization map, and hence bijective by 
Proposition <ref> (ii).
Since the left map and the composite are bijective, so is the evaluation
map at the stable tautological class.

(ii)
By Proposition <ref> 
the wedge of any family of orthogonal spectra is a coproduct in _. 
The vertical maps in the commutative square

    ⊕_i∈ IΣ^∞_+ B_G,  X_i _[r][d]    Σ^∞_+ B_G, ⊕_i∈ IX_i _[d] 
    ⊕_i∈ Iπ^G_0(X_i) [r]    π_0^G( ⋁_i∈ IX_i)

are evaluation at the stable tautological class,
and hence isomorphisms by part (i).
The lower horizontal map is an isomorphism by 
Corollary <ref> (i),
hence so is the upper horizontal map.
This shows that Σ^∞_+ B_G is compact 
as an object of the triangulated category _.

(iii) If X is an orthogonal spectrum such that the group
Σ^∞_+ B_G[k],  X is trivial for every G in 
and all integers k, 
then X is -equivalent to the trivial orthogonal spectrum by part (i);
so X is a zero object in _. This proves that the
spectra Σ^∞_+ B_G form a set of weak generators 
for _ as G varies over .






A covariant functor E from a triangulated category  
to the category of abelian groups is called
homologicalsubjecthomological functor if 
for every distinguished
triangle (f,g,h) in  the sequence of abelian groups

    E(A)  E(f) E(B)  E(g) E(C)   E(h) E(A[1])

is exact.
A contravariant functor E from  to the category of abelian groups 
is called cohomologicalsubjectcohomological functor if 
for every distinguished
triangle (f,g,h) in  the sequence of abelian groups

    E(A[1])  E(h) E(C)  E(g) E(B)   E(f) E(A)

is exact.


Let  be a compactly generated triangulated category.


  * Every cohomological functor ^→ that takes sums in  
to products of abelian groups is representable, 
i.e., isomorphic to the functor (-,X) for some object X of .

  * Every homological functor → that takes products in  
to products of abelian groups is representable, 
i.e., isomorphic to the functor (Y,-) for some object Y of .

  * An exact functor F:→ to another triangulated category
  has a right adjoint if and only if it takes sums in  to
  sums in .

  * An exact functor F:→ to another triangulated category
  has a left adjoint if and only if it takes products in  to
  products in .




A proof of part (i) of this form of Brown representability can be found in
 <cit.>
or <cit.>.
A proof of part (ii) of this form of Brown representability can be found 
in <cit.>
or <cit.>.
Part (iii) is a formal consequence of part (i):
if F preserves sums, then for every object X of  the functor

    (F(-), X)  :  ^ →

is cohomological and takes sums to products.
Hence the functor is representable by an object R X in 
and an isomorphism

    (A, R X)   (F A, X)  ,

natural in A.
Once this representing data is chosen, the assignment X↦ R X extends 
canonically to a functor R:→ that is right adjoint to F.
In much the same way, part (iv) is a formal consequence of part (ii).



We let  be a triangulated category with sums.
A localizing subcategorysubjectlocalizing subcategory 
of  is a full subcategory  which is closed under sums and under 
extensions in the following sense: if

    A   f   B   g  C   h  A[1]

is a distinguished triangle in 
such that two of the objects A, B or C 
belong to , then so does the third.
A set of compact objects is a set of weak generators in the sense
of Definition <ref> if and only if the smallest
localizing subcategory containing the set is all of ,
see for example <cit.>.
Hence Theorem <ref> entitles us to the 
following corollary.



Let  be a global family and  a triangulated category.


  * Every localizing subcategory of the -global stable homotopy category 
that contains the spectrum Σ^∞_+ B_G for every group G of 
is all of _.

  * Every cohomological functor on _ 
that takes sums to products is representable.  

  * Every homological functor on _ that takes products
to products is representable. 

  * An exact functor F:_→  has a right adjoint if 
  and only if it preserves sums.

  * An exact functor F:_→  has a left adjoint 
  if and only if it preserves products.





[Equivariant homotopy groups of infinite products]subjectequivariant homotopy group!of a product
In Corollary <ref> (ii) 
we showed that 
for every compact Lie group G
the functor π_0^G:→ preserves finite products.
However, it is not true that π_0^G, as a functor on the category of 
orthogonal spectra, preserves infinite products in general.

On the other hand, the `derived' functor π_0^G:→ is representable
by the spectrum Σ^∞_+ B_G, so it preserves infinite products.
This is no contradiction because
an infinite product of orthogonal spectra is not in general
a product in the global homotopy category. To calculate a
product in  of a family {X_i}_i∈ I of orthogonal spectra,
one has to choose stable equivalences f_i:X_i→ X_i^f
to global Ω-spectra.
For an infinite indexing set, the morphism

    ∏_i∈ I  f_i :  ∏_i∈ I  X_i → ∏_i∈ I  X_i^f

may fail to be a global equivalence, and then the target,
but not the source, of this map is a product in  
of the family {X_i}_i∈ I.
So when considering infinite products it is important to be aware 
of our abuse of notation and to remember that the symbol π_0^G
has two different meanings.





The preferred set of compact generators {Σ^∞_+ B_G}
of the global stable homotopy category has another special property,
it is `positive' in the following sense: for all compact Lie groups G and K
and all n>0 the group

    Σ^∞_+ B_ G, Σ^∞_+ B_ K[n]  π_0^G(Σ^∞_+ B_ K[n]) 
      π_-n^G(Σ^∞_+ B_ K)

is trivial because every orthogonal suspension spectrum
is globally connective (see Proposition <ref>). 
A set of positive compact generators in this sense has strong implications,
as we shall now explain. 

A t-structure as introduced by 
Beilinson, Bernstein and Deligne in <cit.>
axiomatizes the situation in the derived category of an abelian category
given by cochain complexes whose cohomology vanishes 
in positive respectively negative dimensions.
We are mainly interested in spectra, where a homological
(as opposed to co homological) grading is more common.
So we adapt the definition of a t-structure to homological notation.


A t-structuresubjectt-structure on a triangulated category 
is a pair (_≥ 0,_≤ 0) of full subcategories satisfying the
following three conditions, where _≥ n= _≥ 0[n] 
and _≤ n=_≤ 0[n].


  * For all X∈_≥ 0 and all Y∈_≤ -1 we have (X,Y)=0.

  * _≥ 0⊂_≥ -1 and _≤ 0⊃_≤ -1. 

  * For every object X of  there is a distinguished triangle

    A  → X  → B  → A[1]

such that A∈_≥ 0 and B∈_≤ -1.

The t-structure is non-degenerate
if every object in the intersection ⋂_n∈_≥ n is a zero object
and every object in the intersection ⋂_n∈_≤ n is a zero object.




Some of the basic results of Beilinson, Bernstein and Deligne about t-struc­tures are
(in our homological notation):


  * the inclusion _≥ n→ has a right adjoint
τ_≥ n:→_≥ n, 
and the inclusion _≤ n→ 
has a left adjoint τ_≤ n:→_≤ n <cit.>;

  * given choices of adjoints as above, then for all
m≤ n there is a preferred natural isomorphism of functors 
between τ_≥ m∘τ_≤ n and τ_≤ n∘τ_≥ m <cit.>;

  * the heartsubjectheart! of a t-structure 

    ℋ =  _≥ 0∩_≤ 0 ,

viewed as a full subcategory of , is an abelian category
and τ_≤ 0∘τ_≥ 0:→ℋ is a homological 
functor <cit.>.




We will now show that the global stable homotopy category
has a preferred t-structure whose heart is equivalent
to the category of global functors.


Let  be a global family.
An orthogonal spectrum X is -connectivesubjectF-connective@-connective
if the homotopy group π_n^G(X) is trivial for 
every group G in  and every n<0.
An orthogonal spectrum X is -coconnectivesubjectF-coconnective@-coconnective
if the homotopy group π_n^G(X) is trivial 
for every group G in  and every n>0.


An -global functorsubjectF-global functor@-global functor
is an additive functor to abelian groups
from the full subcategory of the Burnside category
with objects the global family .
A morphism of -global functors is a natural transformation.
We write _ for the category of -global functors.
symbol  _ - category of -global functors 



  For every global family ,
  the classes of -connective spectra and -coconnective spectra 
  form a non-degenerate t-structure on _.
  The heart of this t-structure 
  consists of those orthogonal spectra X such that π_n^G(X)=0
  for all G∈ and all n 0.
  The functor
  
    _0  :  ℋ → _

  is an equivalence of categories from the heart of the t-structure to the
  category of -global functors.


We deduce this from the more general arguments 
of Beligiannis and Reiten <cit.> 
who systematically investigate torsion pairs 
and t-structures in triangulated categories that are generated by compact objects. 
By Theorem <ref> the set 

    = {Σ^∞_+ B_ G}_[G]∈

is a set of compact weak generators for the triangulated category _,
where [G] indicates that we choose representatives from the
isomorphism classes of compact Lie groups in .

We let  be the class of orthogonal spectra Y such that 

    P[n], Y_ =  0

for all P∈ and all n≥ 0.
The representability result of Theorem <ref> (i)
shows that this is precisely the class of those Y such that the
group π_n^G(Y) vanishes for all G∈ and all n≥ 0.
So [1] is the class of -coconnective orthogonal spectra.
We let  be the `left orthogonal' to , i.e., 
the class of orthogonal spectra X such that 
X,Y_=0 for all Y∈.
Since the objects of  are compact in _
by Theorem <ref> (ii), 
<cit.> shows that 
the pair (,) is a `torsion pair'
in the sense of <cit.>.
This simply means that the pair (,[1]) is a t-structure
in the sense of Definition <ref>, 
see <cit.>.

It remains to show that  coincides with the class of
-connective orthogonal spectra.
This needs the `positivity property' of the set  of compact generators.
Representability (Theorem <ref> (i))
and the suspension isomorphism provide an isomorphism

    Σ^∞_+ B_ G, Σ^∞_+ B_ K[n]_  π_0^G(Σ^∞_+ B_K[n])   π_-n^G(Σ^∞_+ B_K)  .

The spectrum Σ^∞_+ B_K is globally connective by 
Proposition <ref>,
so this latter group vanishes for all n≥ 1 and all G,K∈.
So <cit.>
shows that  coincides with the class of 
those orthogonal spectra X such that
Σ^∞_+ B_ G, X[n]_=0
for all G∈ and n≥ 1.
Since the group Σ^∞_+ B_ G, X[n]_ is isomorphic
to π_-n^G(X), this shows that  is precisely the class of
-connective orthogonal spectra.
The t-structure is non-degenerate because spectra with trivial -equivariant
homotopy groups are zero objects in _.

We denote by () the `endomorphism category' of the set , 
i.e., the full pre-additive subcategory of _ with object set .
By an ()-module we mean an additive functor

    ()^ →

from the opposite category of ().
The tautological functor

    _ → ()

takes an object X to the restriction of the 
contravariant Hom-functor -,X_ to the full subcategory ().
Because  is a set of positive, compact generators 
for the triangulated category _, <cit.>
shows that the restriction of the tautological functor (<ref>)
to the heart is an equivalence of categories

    ℋ    ()  .

So to establish the last claim it suffices to show that () 
is anti-equivalent to the full subcategory of the Burnside category  with
object class , in such a way that the tautological functor 
corresponds to the functor _0.
The equivalence ()^→_ is given by the inclusion on objects, 
and on morphisms by the isomorphisms

    Σ^∞_+ B_ G ,Σ^∞_+ B_ K _  π_0^G(Σ^∞_+ B_ K)   (K,G)

specified in Theorem <ref> (i) respectively
Proposition <ref>.  





[Postnikov sections]subjectPostnikov section!of an orthogonal spectrum
For the standard t-structure on the global stable homotopy category 
(i.e., Theorem <ref> for =)
the truncation functor

    τ_≤ n :   → _≤ n ,

left adjoint to the inclusion, provides `global Postnikov sections':
for every orthogonal spectrum X the spectrum τ_≤ nX
satisfies _k(τ_≤ nX)=0 for k>n and the adjunction unit
X→ X_≤ n induces an isomorphism on the global functor _k
for every k≤ n.


[Eilenberg-Mac Lane spectra]
In the case = of the maximal global family,
Theorem <ref>
in particular provides an Eilenberg-Mac Lane spectrumsubjectEilenberg-Mac Lane spectrum!of a global functorsymbolH M - Eilenberg-Mac Lane spectrum of a global functor 
for every global functor M, i.e., an orthogonal spectrum HM
such that _k(HM)=0 for all k 0
and such that the global functor _0(HM) is isomorphic to M;
moreover, these properties characterize HM up to preferred isomorphism in . 
Indeed, a choice of inverse to the equivalence _0 of
Theorem <ref> , composed with the inclusion of
the heart, provides an Eilenberg-Mac Lane functor

    H  :   →

to the global stable homotopy category.


We let  be a triangulated category with infinite sums and
we let  be a class of objects of . 
We denote by _+ the smallest class of objects of 
that contains , is closed under sums (possibly infinite)
and is closed under cones in the following sense: if

    A → B → C  →  A[1]

is a distinguished triangle such that A and B belong to the class,
then so does C. 
Any non-empty class of objects that is closed under cones 
contains all zero objects (because a zero object
is a cone of any identity morphism) and is closed under suspension 
(because A[1] is a cone of the morphism from A to a zero object).




The class Σ^∞_+ B_ GG compact Lie_+ 
in the triangulated category  
coincides with the class of globally connective spectra.  


The spectra Σ^∞_+ B_ G are globally connective by
Proposition <ref>.
The class of globally connective orthogonal spectra is closed under sums
because equivariant homotopy groups commute with sums 
(Corollary <ref> (i)).
The class of globally connective orthogonal spectra is closed under cones
by the long exact homotopy group sequence of Proposition <ref>. So the class Σ^∞_+ B_ G_+ 
is contained in the class of globally connective spectra.
  
For the converse we choose a set of representatives 
of the isomorphism classes of compact Lie groups and set

    =  { (Σ^∞_+ B_G)[k]}_[G], k≥ 0 ,

indexed by the chosen representatives.
We let X be any orthogonal spectrum.
By induction on n we construct objects A_n in _+
and morphisms i_n:A_n→ A_n+1 and  u_n:A_n→ X such that
u_n+1 i_n=u_n. We start with 

    A_0  =  ⊕_P∈, x∈P,X  P .

Then  A_0 belongs to _+ and the canonical morphism
u_0:A_0→ X (i.e., the morphism x on the summand indexed by x)
induces a surjection P,u_0:P,A_0→P,X
for all P∈.

In the inductive step we suppose that A_n and u_n:A_n→ X
have already been constructed. We define

    D_n  =  ⊕_P∈, x∈P,u_n  P ,

which comes with a tautological morphism 
τ:D_n→ A_n, again given by x on the summand indexed by x.
We choose a distinguished triangle

    D_n  τ  A_n  i_n  A_n+1→ D_n [1]  .

Since D_n and A_n belong to the class _+, we also have A_n+1∈_+.
Since u_nτ=0 (by definition), 
we can choose a morphism u_n+1:A_n+1→ X such that
u_n+1i_n=u_n. This completes the inductive construction.

Now we choose a homotopy colimit (A,{φ_n:A_n→ A}_n)
of the sequence of morphisms i_n:A_n→ A_n+1, i.e.,
a distinguished triangle

    ⊕_n≥ 0  A_n  1- ⊕_n≥ 0  A_n       
    A       ⊕_n≥ 0 A_n [1] .

Since all the objects A_n are in _+, so is A.
Since a homotopy colimit in  is a weak colimit,
we can choose a morphism u:A→ X such that uφ_n=u_n 
for all n≥ 0.
The map

    P,u_0 =  P,u∘P,φ_0 :  P,A_0 → P,X

is surjective for P∈, 
hence so is P,u:P,A→P,X.

We claim that P,u is also injective for all P∈.
We let α:P→ A be a morphism such that u α=0. 
Since P is compact, there is an n≥ 0 and a morphism α':P→ A_n 
such that α=φ_nα'.
Then u_nα'=uφ_nα'=uα=0.
So α' indexes one of the summands of D_n.
Thus α' factors through the tautological 
morphism τ:D_n→ A_n as α'=τα”, and hence 

    α =  φ_nα'  =  φ_n+1i_nτα” =  0

since i_n and τ are consecutive morphisms in a distinguished triangle.
So P,u is also injective, hence bijective.
By the choice of the set  and the natural isomorphism

    (Σ^∞_+ B_G)[k],A  π_k^G(A) ,

this shows that the morphism u:A→ X induces isomorphisms
on all equivariant homotopy groups in non-negative dimensions.

So far the spectrum X was arbitrary. 
If we now assume that X is globally connective,
then the morphism u:A→ X is a global equivalence
(because A is globally connective by the previous paragraph).
So X is isomorphic in  to A, and hence X∈_+.






Given orthogonal spectra X and Y, the external product maps (<ref>)

    ⊠ :  π_0^G (X) π_0^K (Y)  → π_0^G× K(X Y)

form a bimorphism of global functors by Theorem <ref>. 
The box product of global functors was introduced in
Construction <ref>.
The universal property of the box product produces a morphism of
global functors

    _0(X)   _0 (Y)  → _0(X Y)  .

We recall from (<ref>) that 
the symmetric monoidal derived smash product ^ 
on the global stable homotopy category is obtained as the total left derived
functor of the smash product of orthogonal spectra.
When applied to flat replacements of X and Y, 
the morphism (<ref>)
becomes the morphism of the following proposition.

 
For all globally connective spectra X and Y 
the orthogonal spectrum X^ Y is globally connective 
and the natural morphismsubjectsmash product!derivedsubjectbox product!of global functors

    (_0 X)    (_0 Y)  → _0(X^ Y)

is an isomorphism of global functors.

 We fix a compact Lie group K and let  be the class
of globally connective orthogonal spectra X such that 
X^Σ^∞_+ B_K is globally connective and the natural morphism

    _0 X   _0(Σ^∞_+ B_ K)
      → _0(X^Σ^∞_+ B_K )

is an isomorphism of global functors. 
The class  is closed under sums and we claim that it is also closed under cones.
We let

    A  →  B →  C → Σ A

be a distinguished triangle in  such that A and B belong to .
Since A is globally connective the global functor _-1(A) vanishes;
since A belongs to  the global functor 
 _-1(A ^Σ^∞_+ B_ K ) vanishes.
Since -_0(Σ^∞_+ B_ K) is right exact
(by Remark <ref>),
the upper row in the commutative diagram

    @C=5mm 
    (_0 A) _0(Σ^∞_+ B_ K)  [r][d]    
     (_0 B) _0(Σ^∞_+ B_ K) [r][d]    
    (_0 C) _0(Σ^∞_+ B_ K) [r][d]    
    0  
    _0(A^Σ^∞_+ B_ K) [r]    _0(B^Σ^∞_+ B_ K)[r]    _0(C^Σ^∞_+ B_ K)[r]    
    0

is exact. The lower row is exact because smashing with Σ^∞_+ B_K preserves
distinguished triangles. The two left vertical maps are isomorphisms because
A and B belong to the class . So the right vertical map
is an isomorphism and C belongs to  as well.
Moreover, we have 

    _0(Σ^∞_+ B_ G)  _0(Σ^∞_+ B_ K)      (G,-)  (K,-) 
        (G× K,-)   _0(Σ^∞_+ B_ (G× K))

by Proposition <ref> and Remark <ref>,
and 

    (Σ^∞_+ B_G) (Σ^∞_+ B_K)   Σ^∞_+ (B_G⊠ B_ K)  Σ^∞_+ B_(G× K)

by Proposition <ref>, respectively by (<ref>). 
This shows that the class  is closed under sums and cones
and contains the suspension spectra Σ^∞_+ B_G for all
compact Lie groups G.
Proposition <ref>
then shows that  is the class of all globally connective orthogonal spectra.
This proves the proposition in the special case Y=Σ^∞_+ B_ K

Now we perform the same argument in the other variable. 
We fix a globally connective spectrum X and let  denote the class
of globally connective orthogonal spectra Y such that 
X^ Y is globally connective and the natural morphism
of the proposition is an isomorphism of global functors. 
The class  is again closed under sums and cones, by the same
arguments as above. 
Moreover, for every compact Lie group K
the suspension spectrum Σ^∞_+ B_K belongs
to the class  by the previous paragraph. 
Again Proposition <ref>
shows that  is the class of all globally connective orthogonal spectra.





The semifree orthogonal spectrum  F_G,V=(V,-)/G
generated by a G-representation V
was defined in Construction <ref>.
Now we identify the functor represented by F_G,V
in the global stable homotopy category. The element

    (0,_V)· G  ∈ (V,V)/G =  F_G,V(V)

is a G-fixed point, so the G-map

    S^V  → (V,V)/G S^V  =  (F_G,V S^V)(V)  ,   
    v  ⟼ (0,_V)· G  (-v)

represents an equivariant homotopy class

    a_G,V ∈ π_0^G(F_G,V S^V)  .

The _*-isomorphism of orthogonal G-spectra λ^V_E:E S^V→^V E 
was defined in (<ref>);
its value at an inner product space U is
the opposite structure map σ^_U,V:E(U) S^V → E(U⊕ V) = (^V E)(U).
The following lemma is a direct consequence of
Proposition <ref> (i).

 
The morphism λ^V_F_G,V:F_G,V S^V→^V F_G,V
takes the element a_G,V to the class in π_0^G(^V F_G,V) 
that is represented by the G-fixed
point (0,_V)· G of (^V F_G,V)(0).



We emphasize that the representative of a_G,V
involves the involution S^-:S^V→ S^V,
which represents a certain unit in the ring π_0^G() that squares to 1.
Lemma <ref> is our reason 
for choosing this particular normalization.
We could remove the involution from the definition of a_G,V,
thereby changing the class a_G,V into ε_V(a_G,V),
but then the unit would instead appear in other formulas later on.


 
Let G be a compact Lie group and V a faithful G-represen­tation.
Then the pair (F_G,V, a_G,V) represents the functor

    →(sets) ,    E  ⟼ π_0^G(E S^V) .

The semifree orthogonal spectrum F_G,V is a compact object 
of the global stable homotopy category .


We define

    b_G,V =  (λ^V_F_G,V)_*(a_G,V) ∈ π_0^G (^V F_G,V) ;

Lemma <ref> shows that the class b_G,V
is represented by the G-fixed point

    (0,_V)· G  ∈ (V,V)/G =  (^V F_G,V)(0)  .

Since the morphism λ^V_F_G,V is a _*-isomorphism
of orthogonal G-spectra,
we may thus show that the pair (F_G,V, b_G,V) represents the functor

    →(sets) ,    E  ⟼ π_0^G(^V E) ,

i.e., for every orthogonal spectrum E the map

    F_G,V, E  → π_0^G (^V E) ,    
    f⟼ (^V f)_*(b_G,V)

is bijective. Since both sides take global equivalences in E to bijections,
we can assume that E is a global Ω-spectrum.
The orthogonal spectrum F_G,V is flat, and hence cofibrant
in the global model structure.
So the localization functor induces a bijection

    (F_G,V, E) / homotopy →  F_G,V, E
 
from the set of homotopy classes of morphisms of orthogonal spectra
to the set of morphisms in .
By the freeness property, morphisms from F_G,V to E biject 
with G-fixed points of E(V), and homotopies correspond to
paths of G-fixed points.
The composite

    π_0( E(V)^G )      F_G,V, E  f↦ (^V f)_*(b_G,V) π_0^G (^V E)

is the stabilization map of the G-Ω-spectrum ^V E, 
and hence bijective.
Since the left map and the composite are bijective, so is the evaluation
map at the class b_G,V.

Now we prove that F_G,V is a compact object in .
In the commutative square

    ⊕_i∈ I F_G,V,  X_i [r][d]     F_G,V, ⊕_i∈ IX_i [d] 
    ⊕_i∈ Iπ_0^G(X_i S^V) [r]    π_0^G( ⋁_i∈ IX_i S^V)

both vertical maps are evaluation at the class a_G,V,
which are isomorphisms by part (i).
The lower horizontal map is an isomorphism by 
Corollary <ref> (i),
hence so is the upper horizontal map.
This shows that F_G,V is compact as an object of the triangulated category .



Now we construct certain distinguished triangles in the global stable homotopy
category that arise from special representations of compact Lie groups G,
namely when G acts faithfully and transitively on the unit sphere.
Equivalently, we are looking for ways to present spheres as homogeneous spaces.
The distinguished triangles relate two semifree orthogonal spectra for the
compact Lie group G and a semifree orthogonal spectrum for a closed subgroup H 
that occurs as the stabilizer of a unit vector 
in the transitive faithful G-representation. 
The main case we care about is G=O(m)subjectorthogonal group
acting tautologically on ^m with stabilizer group H=O(m-1).
This special case will become relevant in Section <ref>
when we analyze the rank filtration of the global Thom spectrum .
Similarly, we can let G=S O(m)subjectspecial orthogonal group act tautologically
on ^m with stabilizer group H=S O(m-1), and this
shows up in the rank filtration of the global Thom spectrum ; 
here we need m≥ 2, since S O(1) does not act transitively on S().
Other examples are U(m)subjectunitary group or S U(m)subjectspecial unitary group
 (the latter for m≥ 2)
acting on the underlying -vector
space of ^m, with stabilizer groups U(m-1) respectively S U(m-1). 
Similarly, we can consider the tautological representation of S p(m)subjectsymplectic group
on the underlying -vector space of ^m with stabilizer group S p(m-1).
These examples show up in the rank filtrations of the global Thom spectra ,
𝐦𝐒𝐔 and 𝐦𝐒𝐩.
There are also more exotic examples, such as the exceptional 
Lie group G_2,subjectG2@G_2 exceptional Lie group
the group of -algebra automorphism of the octonions 𝕆.
Here we take V as the tautological 7-dimensional representation
on the imaginary octonions, with stabilizer group isomorphic to S U(3).
For a complete list of examples we refer the reader to <cit.>
and the references given therein.



We consider a representation V of a compact Lie group G
such that G acts faithfully and transitively on the unit sphere S(V).
We choose a unit vector v∈ S(V) and let H be the stabilizer group of v.
Then the underlying H-representation of V decomposes as

    V  =  L⊕ (· v)

where L is the orthogonal complement of v.
We use the letter L for this complement because it essentially `is' 
the tangent representation T_e H(G/H).
More precisely, the differential at e H of the smooth G-equivariant embedding

    G/H  → V  ,    g H ⟼ g v

is an H-equivariant linear isomorphism from T_e H(G/H) onto L.

We define a based G-map

    r  :  S^V      → (L,V)/H =   F_H,L(V)
     by    
     r(g· t· v)     = g·( (t^2-1)/t· v,incl)· H ,

where g∈ G and t∈[0,∞].
We let 

    T : Σ^∞_+ B_ G =  Σ^∞_+ _G,V →  F_H,L

denote the adjoint of r.
We define a morphism of orthogonal spectra

    i  :  F_H,L →  F_G,V

as the adjoint of the H-fixed point

    ψ^-1· G ∈ (V,L⊕)/G  =  ( F_G,V)(L)  ,

where the H-equivariant isometry ψ:L⊕ V
is defined as ψ(x,t)= x+ t· v. 
The value of the morphism i at an inner product space W is then the map

    i(W)  :  F_H,L(W) = (L,W)/H
        → (V,W⊕)/G  =   ( F_G,V)(W) 
    
    (w,φ)· H        ⟼     
    ((w,0),(φ⊕)∘ψ^-1)· G  .

We define a morphism of orthogonal spectra 
a:F_G,V→Σ^∞_+ _G,V=Σ^∞_+ B_G
as the adjoint of the G-fixed point

    0 · G ∈  S^V(V,V)/G_+  =  
    (Σ^∞_+ _G,V)(V) .






The morphism λ_X:X S^1→ X 
was defined in (<ref>) and is a global equivalence 
by Proposition <ref> (i).
The stable tautological classsubjectstable tautological class

    e_G  =  e_G,V ∈ π_0^G(Σ^∞_+ B_G )

was defined in (<ref>).
In the next theorem we abuse notation and use the same symbols
for morphisms of orthogonal spectra and their images in the global
stable homotopy category. 



Let G be a compact Lie group and V a G-representation
such that G acts faithfully and transitively on the unit sphere S(V);
let H be the stabilizer group of a unit vector of V.


  * The sequence

    F_G,V  a  Σ^∞_+ B_ G  T  F_H,L -λ_F_G,V^-1∘ i    F_G,V S^1

is a distinguished triangle in the global stable homotopy category.

  * The morphism T satisfies the relation

    T_*(e_G)  =  _H^G(a_H,L)

in the group π_0^G(F_H,L).



We consider the based map q:G/H_+→ S^0 that sends G/H
to the non-basepoint.
We identify the mapping cone of q
with the sphere S^V via the G-equivariant homeomorphism

    h  :  C q  S^V

that is induced by the map

    G/H×[0,1]  → S^V  ,   
    (g  H,x)  ⟼ g · (1-x)/x · v  .

Under this identification the mapping cone inclusion i:S^0→ C q becomes the
inclusion ι:{0,∞}=S^0→ S^V,
and the projection p:C q→ G/H_+ S^1 becomes the map

    p∘ h^-1 :  S^V → G/H_+ S^1  ,  
    g· t· v  ⟼ g H (1-t^2)/t ,

where g∈ G and t∈ [0,∞].
The semifree functor F_G,V takes mapping cone sequences of
based G-spaces to mapping cone sequences of orthogonal spectra, so the sequence

    F_G,V(G/H_+) F_G,V q F_G,V F_G,Vι 
    F_G,VS^V F_G,V(p∘ h^-1) 
    F_G,V( G/H_+ S^1)

is a distinguished triangle in the global stable homotopy category.


We define a morphism Λ:F_G,V( G/H_+ S^1)→ F_H,L
as the adjoint of the G-map

    G/H_+ S^1    → (L,V)/H  =  F_H,L(V) 
    
    g  H  t      ⟼ g · ( - t· v, incl )· H  .

The morphism Λ is the composite of an isomorphism
F_G,V( G/H_+ S^1) F_H,L⊕S^1
and the morphism λ_H,L,:F_H,L⊕S^1→ F_H,L
defined in (<ref>). The morphism
λ_H,L, is a global equivalence
by Theorem <ref>, hence Λ is 
a global equivalence as well. The composite

    Σ^∞_+ B_ G untwist^-1 
    F_G,V S^V F_H,L(p∘ h^-1) F_G,V(G/H_+ S^1)
      Λ  F_H,L

coincides with the morphism T, by direct inspection
of the effects at the inner product space V.

Our next claim is that the following diagram 
of orthogonal spectra commutes up to homotopy:

    @C=13mm 
    F_G,V(G/H_+) S^1[r]^-(F_G,Vq) S^1[d]_Λ    
    F_G,V S^1[d]^λ_F_G,V
    
    F_H,L[r]_-i     F_G,V

The representing property of F_G,V reduces this to evaluating 
the square at V and showing that the two resulting 
G-maps from G/H_+ S^1 to ( F_G,V)(V)
are equivariantly homotopic. Since G-maps out of
G/H correspond to H-fixed points,
this in turn reduces to the claim that the two maps

    S^1  → ( (V,V⊕)/G)^H =  ( ( F_G,V)(V))^H

that send t∈ S^1 to

    ( (-t· v,0), i_0)· G      respectively     
    ( (0,t), i_1)· G

are based homotopic, where i_0,i_1:V→ V⊕ are given by 

    i_0(x) =  (x- x,v· v, x,v) 
       respectively  
    i_1(x) =   (x,0) .

A homotopy that witnesses this is

    S^1× [0,1]     →    ((V,V⊕)/G)^H 
    
    (t,s)     ⟼ 
    ( (-t·cos(π s/2)· v, t·sin(π s/2)),  i_s)· G ,

with

    i_s  :   V → V⊕ , i_s(x) =  (x + ( sin(π s/2)-1)·x,v· v, cos(π s/2)·x,v) .

This shows the claim that the square (<ref>) 
is homotopy commutative.

So altogether we have produced a commutative diagram 
in the global stable homotopy category:

    @C=13mm
    F_G,V@=[d] [r]^-F_G,Vι    F_G,V S^V[d]_untwist^[r]^-F_G,V(p∘ h^-1)   
    F_G,V(G/H_+ S^1) [d]^Λ_∼[r]^--(F_G,Vq S^1)    
    F_G,V[d]^λ_F_G,V_∼ S^1
    
    F_G,V[r]_-a    Σ^∞_+ B_G[r]_-T    
    F_H,L[r]_--i    F_G,V

The upper sequence is obtained by rotating a distinguished triangle,
so it is distinguished.
Since all vertical morphisms are isomorphisms in , we conclude that
the sequence (a,T,-λ^-1_F_G,V∘ i) is a distinguished triangle in .

It remains to analyze the effect of the morphism T on the
stable tautological class e_G.
We consider the wide G-equivariant embedding

    G/H  → V  ,    g  H ⟼ g v .

This embedding was already used to identify the tangent space T_e H(G/H)
at the preferred coset with the subspace L inside V;
the inclusion L→ V corresponds to the differential at e H.
The orthogonal complement of L is precisely the line 
spanned by the vector v, and the subgroup H acts trivially on this complement.

As described more generally in Construction <ref>,
the wide embedding gives rise to a
G-equivariant Thom-Pontryagin collapse map (<ref>)

    c  :  S^V →  G/H_+ S^1  .
 
In the more general context of an arbitrary closed subgroup, 
the target of the collapse map is G⋉_H S^V-L; 
but in our situation H acts trivially on · v=V-L, 
so we can (and will) identify V-L with  via x↦x,v.
The collapse map c sends the origin and all vectors of length at least 2 to
the basepoint; if x∈ V satisfies 0<|x|<2, then 
x=g · t· v for some g∈ G and t∈(0,2).
The collapse map c sends such a vector x to

    g  Ht-1/1-|t-1| ∈ G/H_+ S^1  .


Now we calculate the class _H^G(a_H,L).
We factor the external transfer G⋉_H-:π_0^G(E S^L)→π_0^G(G⋉_H E)
as the composite

    π_0^H(E S^L)  (λ^L_E)_* π_0^H(^L E)  G⊠_H - π_0^G(G⋉_H E) .

Here λ^L_E:E S^L→^L E is the morphism of orthogonal H-spectra
defined in (<ref>);
the second homomorphism is a variation of the external transfer, 
and is defined as follows.
We let U be an H-representation and f:S^U → E(U⊕ L)=(^L E)(U) 
an H-equivariant 
based map that represents a class in π_0^H(^L E).
By enlarging U, if necessary, we can assume that it
is the underlying H-representation of a G-representation.
We denote by f♢ the composite H-map

    S^U S^1  f S^1 E(U⊕ L) S^1 σ^_U⊕ L, E(U⊕ L⊕) []E(U⊕ψ) E(U⊕ V)  .

The class G⊠_Hf in π_0^G(G⋉_H E)
is then represented by the composite G-map

    S^U⊕ V  S^U c     S^U (G/H_+ S^1)  []shear   
    G⋉_H( S^U S^1) 
       G⋉_H(f♢)   
    G⋉_H  E(U⊕ V)  (G⋉_H E)(U⊕ V)  .

The relation G⊠_H((λ^L_E)_*f)=G⋉_H f
is straightforward from the definitions.

By Lemma <ref>, the class
(λ_F_H,L^L)_*(a_H,L)
is represented by the H-fixed point

    (0,_L)· H  ∈ (L,L)/H  =  (^L F_H,L)(0)  .

So the class _H^G(a_H,L)=act_*(G⊠_H((λ^L_F_H,L)_*(a_H,L)))
is represented by the composite G-map

    S^V      c   
    G/H_+  S^1   G⋉_H( (0,_L)H ♢)  G⋉_H ( (L,V)/H)  act (L,V)/H  .

Expanding all the definitions identifies this composite 
as the map

    S^V  → (L,V)/H  ,    
    g· t· v  ⟼ g·(ξ(t)· v,incl)· H ,

for g∈ G and t∈ [0,∞], with

    ξ :  [0,∞] → S^1  ,
       ξ(t) =  t-1/1-|t-1|    if t∈ (0,2), and
        ∗    else.
 
The function ξ is homotopic, relative to {0,∞}, 
to the function sending t to (t^2-1)/t. 
Any relative homotopy between these two functions
induces a based G-equivariant homotopy between the representative
of the class _H^G(a_H,L) and the map r
defined in (<ref>). So r itself is a representative
of the class _H^G(a_H,L).
Since T:Σ^∞_+ B_G→ F_H,L is adjoint to the G-map r,
we deduce the desired relation _H^G(a_H,L)=[r]=T_*(e_G).




Part (ii) of Theorem <ref>
can be interpreted as saying that for global stable homotopy types,
the morphism of orthogonal spectra
T:Σ^∞_+ B_G→ F_H,L represents the dimension shifting transfer
from H to G.
Indeed, the pair (Σ^∞_+ B_G, e_G) represents
the functor π_0^G (by Theorem <ref> (i)), 
and the pair (F_H,L,a_H,L) 
represents the functor π_0^H(- S^L)
(by Theorem <ref>).
Naturality of transfers then promotes the relation T_*(e_G)=_H^G(a_H,L)
to a commutative diagram:

    @C=15mmF_H,L, E [r]^-T,E[d]_f↦ f_*(a_H,L)^   Σ^∞_+ B_G, E [d]^f↦ f_*(e_G)_
    π_0^H(E S^L)[r]_-_H^G    π_0^G( E)
  






§ CHANGE OF FAMILIES


In this section we compare the global stable homotopy categories for two
different global families  and , where we suppose that ⊆. 
Then every -equivalence is also an -equivalence, 
so we get a `forgetful' functor on the homotopy categories

    U = U^_ :  _ → _

from the universal property of localizations.
The global model structures are stable, so the two global homotopy categories
_ and _ have preferred triangulated structures,
and the forgetful functor is canonically an exact functor of
triangulated categories. 
We show in Theorem <ref> that this forgetful functor
has a left and a right adjoint, both fully faithful,
and that this data is part of a recollement of triangulated categories.
If the global families  and  are multiplicative, 
then the smash product of orthogonal spectra can be derived to
symmetric monoidal products on _ and on _ 
(see Corollary <ref>). The forgetful functor
is strongly monoidal with respect to these derived smash products.
Indeed, the derived smash product in _ can be calculated by
flat approximation up to -equivalence;
every -equivalence is also an -equivalence, so these 
flat approximations can also be used to calculate 
the derived smash product in _.
Theorem <ref> below also exhibits symmetric monoidal structures
on the two adjoints of the forgetful functor.

The special case with  the trivial global family
allows a quick calculation of the Picard group of _,
with the possibly disappointing answer that it is free abelian of rank 1, 
generated by the suspension of the global sphere spectrum 
(see Theorem <ref>).
Propositions <ref> and <ref> 
provide characterizations of the global homotopy types in the image of the 
left and right adjoints.
As we explain in Example <ref>,
the `absolute' right adjoint from the non-equivariant to the
full global stable homotopy category models Borel cohomology theories.
Construction <ref> exhibits a particularly nice
lift of the global Borel functor to the pointset level,
i.e., a lax symmetric monoidal endofunctor of the category of orthogonal spectra.

In Theorem <ref> we relate 
the global stable homotopy category to the G-equivariant stable homotopy
category G- for a fixed compact Lie group G.
There is another forgetful functor → G-
which is an exact functor of triangulated categories and
has both a left adjoint and a right adjoint.
Here, however, the adjoints are not fully faithful as soon
as the group G is non-trivial.

The global family  of finite groups is an important example
to which the discussion of this section applies.
We will show that rationally, the associated -global stable homotopy category
admits an algebraic model: Theorem <ref>
provides a chain of Quillen equivalences between
the category of orthogonal spectra with the 
rational -global model structure and the category of chain complexes of
rational global functors on finite groups.
Under this equivalence, the homotopy group global functor for spectra
corresponds to the homology group global functor for complexes.

The algebraic model can be simplified further.
Rational G-Mackey functors for a fixed finite group
naturally split into contributions indexed by conjugacy classes of subgroups,
see Proposition <ref>.subjectMackey functor
While the analogous global context is not semisimple, the
category of rational -global functors is equivalent
to a simpler one, namely contravariant functors 
from the category  of finite groups and
conjugacy classes of epimorphisms to -vector spaces,
by Theorem <ref> below.
So rational -global stable homotopy theory is also modeled
by the complexes of such functors. Under the composite equivalence,
the geometric fixed point homotopy group functors for spectra
corresponds to the homology group -functors for complexes,
see Corollary <ref>.








  Let  and  be two global families such that ⊆. 
  
  
  * The forgetful functor
    
    U  :  _ → _

    has a left adjoint L and a right adjoint R, and both adjoints are fully faithful.
  
  * If the global families  and  are multiplicative,subjectmultiplicative global family  
    then the right adjoint has a preferred lax symmetric monoidal structure
    ( R A ) ^_ ( R B ) → R(  A _^  B ).subjectderived smash product     
  * If the global families  and  are multiplicative, then
    the left  adjoint has a preferred strong symmetric monoidal structure, i.e., 
    a natural isomorphism between L(  A _^  B ) and 
    ( L A ) ^_ ( L B ).
  


By Proposition <ref> 
the categories _ and _ have infinite sums
and infinite products, and the forgetful functor preserves both.

(i) As spelled out in Corollary <ref> (v),
the existence of the left adjoint is a formal consequence
of the fact that _ is compactly generated 
and that the functor U preserves products.
But instead of arguing by hand that U preserves products,
we give an alternative construction 
of the left adjoint by model category theory.
Indeed, it is immediate from the definitions of -equivalences
and -global fibrations that the identity functor is a right Quillen functor 
from the -global to the -global model structure.
So its derived functor has a (derived) left adjoint L:_→_, 
see for example
<cit.> or <cit.>.
Since the right Quillen functor (i.e., the identity) 
preserves all weak equivalences, the adjunction 
unit A→ U(L A) is an isomorphism in the -global homotopy category. 
So the left adjoint is fully faithful.

By Proposition <ref> sums in
both _ and _ are represented by the wedge of orthogonal spectra. 
So the forgetful functor U preserves sums.
As spelled out in Corollary <ref> (iv),
the existence of the right adjoint is then a formal consequence
of the fact that _ is compactly generated. 

The fact that R is fully faithful is also a formal consequence of properties
of the adjoint pair (L,U). As we saw above,
the adjunction unit η_A:A→ U(L A) is an isomorphism
for every object A of _.
So the map 

    (η_A)^*  :  U(L A),X_ → A,X_

is bijective for every object X of _. The adjunction (U,R) lets
us rewrite the left hand side as L A, R X_,
and the adjunction (L,U) lets
us rewrite this further to A, U(R X)_.
Under these substitutions, the map (η_A)^* becomes the map
induced by the adjunction counit ϵ_X:U(R X)→ X.
This adjunction counit is thus a natural isomorphism, and so R
is also fully faithful.

(ii) The lax monoidal structure of the right adjoint R is a formal
consequence of the strong monoidal structure of the forgetful functor U.
Indeed, for every pair of orthogonal spectra A and B
this strong monoidal structure and the adjunction counits provide a morphism

    U ( ( R A ) ^_ ( R B ) )    U (R A ) _^ U (R B) 
     ϵ_A^ϵ_B   A _^  B

whose adjoint ( R A ) ^_ ( R B ) → R(  A _^  B )
is associative, commutative and unital.

(iii) 
The strong symmetric monoidal structure on U and the adjunction units
provide a morphism

    A ^_ B η_A ^η_B  U( L A) ^_ U ( L B )
      U ( ( L A) ^_  ( L B ) )

whose adjoint λ_A,B:L(  A _^  B ) → ( L A ) ^_ ( L B )
is associative, commutative and unital.
We claim that the morphism  λ_A,B is in fact an isomorphism in _
(and hence it can be turned around).
We can assume that A and B are -cofibrant so that LA=A and LB=B. 
Since  is multiplicative, the pointset level smash product A B
is again -cofibrant by Proposition <ref> (i).
So the value of the left adjoint L on A B is also given by A B.



[Recollements]
Theorem <ref> implies that for all pairs of
nested global families ⊆ the diagram
 of triangulated categories and exact functors

    @C=15mm(;) @<-.3ex>[r]^-i_*   _ @<-.3ex>[r]^-U   @<.4ex>@/^1pc/[l]^-i^!@<-.4ex>@/_1pc/[l]_-i^*   _ @<.4ex>@/^1pc/[l]^-R @<-.4ex>@/_1pc/[l]_-L

is a recollementsubjectrecollement in the sense of <cit.>.
Here (;) denotes the `-global homotopy category with
support outside ', i.e., the full subcategory of _
of spectra all of whose -equivariant homotopy groups vanish.
The functor i_*:(;)→_ is the inclusion, and
i^* (respectively i^!) is a left adjoint (respectively right adjoint) of i_*.



In Theorem <ref>
the left adjoint L:_→_ of the forgetful functor U:_→_
is obtained as the total left derived functor of the identity functor
on orthogonal spectra with respect to change of model structure
from the -global to the  -global model structure.
So one can calculate the value of the left adjoint L on an orthogonal spectrum X 
by choosing any cofibrant replacement in the  -global model structure,
i.e., an -equivalence X_→ X with -cofibrant source.
The global homotopy type (so in particular the -global homotopy type) 
of X_ is then well-defined.
Indeed, since the identity is a left Quillen functor
from the -global to the global model structure
(the special case =),
every acyclic cofibration in the -global model structure
is a global equivalence. Ken Brown's lemma 
(see the proof of <cit.>,
<cit.> or <cit.>) then implies that
every -equivalence between -cofibrant orthogonal spectra
is a global equivalence. 

It seems worth spelling out the extreme case
when =e is the minimal global family of trivial groups and
when = is the maximal global family of all compact Lie groups:
An orthogonal spectrum X is e-cofibrant if for every
m≥ 0 the latching morphism L_m X→ X(^m) is an O(m)-cofibration
and O(m) acts freely on the complement of the image. These are precisely the
orthogonal spectra called `q-cofibrant' in <cit.>.
So every non-equivariant stable equivalence between 
q-cofibrant orthogonal spectra is a global equivalence.



The formal properties of the change-of-family functors 
established in Theorem <ref> facilitate an
easy and rather formal argument to identify the Picard group
of the global stable homotopy category. The result is that there
are no `exotic' invertible objects, i.e., the only smash invertible
objects of  are the suspensions and desuspensions of the global sphere spectrum.
The same is true more generally for the -global stable homotopy
category relative to any multiplicative global family , 
see Theorem <ref> below.

We recall that an object X of a monoidal category (,,I)
is invertible if there is another object Y such that both
X Y and Y X are isomorphic to the unit object I.
If the isomorphism classes of invertible objects form a set, 
then the Picard groupsubjectPicard group
() is this set, with the group structure induced
by the monoidal product.
Every strong monoidal functor between  monoidal categories takes invertible objects
to invertible objects, and thus induces a group homomorphism of Picard groups.

Part of the calculation of () involves a very general argument that we spell
out explicitly. 


Let (,,I) be a monoidal category,
P:→ a strong monoidal functor  
and ϵ:P→_ a monoidal transformation
such that ϵ_I:P I→ I is an isomorphism.
Then the induced endomorphism

    (P) :  () → ()

is the identity.


We let X be any invertible object, and Y an inverse of X.  
Since ϵ is a monoidal transformation, the composite

    (P X) (P Y)     
    P ( X Y)  ϵ_X Y X Y

agrees with the morphism ϵ_Xϵ_Y,
where the first isomorphism is the strong
monoidal structure on P.
Since X Y is isomorphic to the unit object and ϵ_I is
an isomorphism, the morphism
ϵ_X Y is also an isomorphism.
So the composite ϵ_Xϵ_Y is an isomorphism. Since

    ϵ_Xϵ_Y =  (Xϵ_Y)∘ (ϵ_X P Y)
     =  (ϵ_X  Y)∘ (P Xϵ_Y)

we conclude that  ϵ_X P Y has a left inverse 
and ϵ_X  Y has a right inverse. 
Since Y and P Y are invertible objects, the functors
- Y and - P Y are equivalences of categories, so 
already ϵ_X:P X→ X has both a left and a right inverse. 
So ϵ_X is an isomorphism, and hence (P)[X]=[P X]=[X].


Now we have all necessary ingredients to determine the Picard
group of the -global stable homotopy category.



  For every multiplicative global family , 
  the Picard group of the -global stable 
  homotopy category is free abelian of rank 1, generated by the suspension
  of the global sphere spectrum.subjectPicard group!of the -global stable homotopy category


  The forgetful functor 
  
    U =  U^_e : _ → _e =

  to the non-equivariant stable homotopy category and its left adjoint
  
    L :  → _

  both have strong monoidal structures, 
  the latter by Theorem <ref> (iii).
  Moreover, the adjunction counit ϵ:L U→ is a monoidal transformation,
  and the morphism ϵ_:L (U)→ is an isomorphism.
  We apply Proposition <ref> to the composite endofunctor L U
  of _ and conclude that the composite homomorphism
  
    (_)  (U) ()  (L) (_)

  is the identity. In particular, the homomorphism (U) induced by
  the forgetful functor is injective.
  The Picard group of the non-equivariant stable homotopy category 
  is free abelian of rank 1, generated by the suspension 
  of the non-equivariant sphere spectrum. This generator is the image 
  of the suspension of the global sphere spectrum. So the homomorphism
  (U) is surjective, hence an isomorphism. So (_) is
  also free abelian of rank 1, generated by the suspension 
  of the global sphere spectrum.
  



Now we develop criteria that characterize global homotopy
types in the essential image of one of the adjoints to a
forgetful change-of-family functor.
The following  terminology is convenient here.


Let  be a global family. An orthogonal spectrum is
left inducedsubjectleft induced 
from  if it is in the essential image of the left adjoint
L_:_→. Similarly, an orthogonal spectrum is 
right inducedsubjectright induced  
from  if it is in the essential image of the right adjoint R_:_→.   





We start with a criterion, for certain `reflexive' global families,
that characterizes the left induced homotopy types in terms of geometric fixed points.


A global family  is reflexivesubjectreflexive global familysubjectglobal family!reflexive|seereflexive global family
if for every compact Lie group K 
there is a compact Lie group u K, belonging to ,
and a continuous homomorphism p:K→ u K that is initial among
continuous homomorphisms from K to groups in .  


In other words,  is reflexive if and only if
the inclusion into the category of all compact Lie groups has a left adjoint.
As always with adjoints, the universal pair (u K,p) 
is then unique up to unique isomorphism under K. 
Moreover, the universal homomorphism p:K→ u K is necessarily surjective. 
Indeed, the image of p is a closed subgroup of u K,
hence also in the global family . So if the image of p were strictly smaller
than K, then p would not be initial among morphisms into groups from .
Some examples of reflexive global families are
the minimal global family e of trivial groups, 
the global family  of finite groups and
the global family of abelian compact Lie groups.
The maximal family of all compact Lie groups is also reflexive, 
but in this case the following proposition has no content.

A reflexive global family  is in particular 
multiplicative.subjectmultiplicative global family
Indeed, for G,K∈ the projections p_G:G× K→ G
and p_K:G× K→ K factor through continuous homomorphisms
q_G:u(G× K)→ G respectively q_K:u(G× K)→ K.
The composite

    G× K   p  u(G× K) (q_G,q_K) G× K

is then the identity, so the universal homomorphism
p:G× K→ u(G× K) is injective. Since u(G× K)
belongs to , so does G× K.



Let  be a reflexive global family.
Then an orthogonal spectrum X is left induced from  if and only if
for every compact Lie group K the inflation map

    p^*  :  Φ_*^u K( X ) → Φ_*^K ( X )

associated to the universal morphism p:K→ u K is an isomorphism 
between the geometric fixed point homotopy groups for u K and K.


We let  be the full subcategory of  consisting
of the orthogonal spectra X such that for every compact Lie group K
the inflation map p^*: Φ_*^u K( X )→Φ_*^K ( X ) is an isomorphism.
We need to show that  coincides with the class of spectra left induced from .

Geometric fixed point homotopy groups commute with sums and take exact triangles
to long exact sequences. So  is closed under sums and triangles, i.e.,
it is a localizing subcategory of the global homotopy category. 
Now we claim that for every group G in 
the suspension spectrum of the global classifying space B_ G
belongs to . 
Since  p:K→ u K is initial among morphisms into groups from ,
precomposition with p is a bijection between the sets of
conjugacy classes of homomorphisms into G; moreover, the image of a homomorphism
α:u K→ G agrees with the image of α∘ p:K→ G, 
because p is surjective. 
Proposition <ref> (i) 
identifies the fixed points of the orthogonal space B_ G
as a disjoint union, over conjugacy classes of homomorphisms,
of centralizers of images.
So the restriction map along p is a weak equivalence of fixed points spaces

    p^*  :  ((B_ G)(_u K) )^u K =  ((B_ G)(_u K) )^K  ≃ ((B_ G)(_K) )^K  .

Geometric fixed points commute with suspension spectra
(see Example <ref>), 
in the sense of an isomorphism

    Φ^K_* (Σ^∞_+ B_ G)     π_*^e ( Σ^∞_+ ( (B_ G)(_K))^K)  ,

natural for inflation maps.
So together this implies the claim for the suspension spectrum of B_ G.

Now we have shown that  is a localizing subcategory of the global
stable homotopy category that contains the suspension spectra 
of global classifying spaces of all groups in .
The left adjoint L:_→ is fully faithful and 
_ is generated by the suspension spectra of the global classifying spaces
in  (by Theorem <ref>).
So L:_→ is an equivalence onto the full triangulated subcategory generated by
the suspension spectra Σ^∞_+ B_ G for all G∈.
So the image of L is contained in .

Now suppose that conversely X is an orthogonal spectrum in .
The adjunction counit ϵ_X:L( U X )→ X is an -equivalence, 
so it induces isomorphisms of geometric fixed point groups for all groups in . 
By the hypothesis on X and naturality of the inflation maps p^*, 
the morphism ϵ_X induces isomorphisms 
of geometric fixed point homotopy groups for all compact Lie groups.
So ϵ_X is a global equivalence, 
and in particular X is left induced from .




The same proof as in Proposition <ref>
yields the following relative version of the proposition.
We let ⊂ be global families and assume that  
is reflexive relative to , i.e.,
for every compact Lie group K from the family  there is
a compact Lie group u K, belonging to ,
and a continuous homomorphism p:K→ u K that is initial among
homomorphisms to groups in .
Then an orthogonal spectrum X is in the essential image of the relative left adjoint
L:_→_ if and only if
for every compact Lie group K in  the universal morphism p:K→ u K
induces isomorphisms

    p^*  :  Φ_*^u K( X ) → Φ_*^K ( X )

between the geometric fixed point homotopy groups of u K and K.




The minimal global family =e of trivial groups is reflexive, 
and the unique morphism K→ e to any trivial group is universal.
So Proposition <ref> characterizes the global homotopy types
in the essential image of the left adjoint L:=_e→ from the
non-equivariant stable homotopy category to the global stable homotopy category:
an orthogonal spectrum X is left induced from the trivial family
if and only if for every  compact Lie group K the unique homomorphism
p_K:K→ e induces an isomorphism

    p_K^* :  Φ^e_*( X )  → Φ^K_*( X )  .

The geometric fixed point homotopy groups Φ^e_*( X ) with respect
to the trivial group are isomorphic to π_*^e ( X ), the stable homotopy groups of the
underlying non-equivariant spectrum.
So the global homotopy types in the essential image of the left adjoint L:→
are precisely the orthogonal spectra with `constant geometric fixed points'.



Here are some specific examples of left induced global homotopy types.

[Suspension spectra]
The global sphere spectrum  and the suspension spectrum
of every based space are left induced from the trivial global family e.
Indeed, geometric fixed points commute with suspension spectra in the
following sense: 
if A has trivial G-action, then

    π_* ( Σ^∞ A )   Φ^G_*(Σ^∞ A) ,

compare Example <ref>. 
So the suspension spectrum Σ^∞ A has `constant geometric fixed points',
and it is left induced from the trivial family by the criterion
of Example <ref>.



[Global classifying spaces and semifree orthogonal spectra]
If G is a compact Lie group from a global family , 
then the suspension spectrum of the global classifying space B_ G 
is left induced from . To see this, we can refer to the proof
of Proposition <ref>; alternatively, we may show that
Σ^∞_+ B_G is -cofibrant, i.e., has the left lifting property
with respect to morphisms that are both -level equivalences and
-level fibrations. We recall that B_ G = _G,V =  (V,-)/G 
is a semifree orthogonal space, where V is any faithful G-representation.
So morphisms Σ^∞_+ B_G → X of orthogonal spectra
biject with continuous based G-maps S^V → X(V);
since  S^V can be given the structure of a based G-CW-complex,
it has the left lifting property with respect to 
G-weak equivalences that are also G-fibrations, and the claim follows by
adjointness.

The same kind of reasoning shows that the semifree orthogonal spectra F_G,V
introduced in Construction <ref>
are left induced from  whenever G belongs to  and V is
a faithful G-representation.



[-spaces]subjectGamma-space@-space
  We let  denote the category whose objects are the based sets
  n_+={0,1,…,n}, with basepoint 0, and with morphisms all based maps.
  A -space is a functor from  
  to the category of based spaces which is reduced
  (i.e., the value at 0_+ is a one-point space).

  A -space F:→_* can be extended 
  to a continuous functor on the category
  of based spaces by a coend construction
  
    F(K) =  ∫^n_+∈ F(n_+)×  K^n  ;

  here K is a based space and we use that K^n=_*(n_+,K) 
  is contravariantly functorial in n_+.
  We refer to the extended functor
  as the prolongationsubjectprolongation!of a -space of F 
  and denote it by the same letter.
  This abuse of notation is justified by the fact that the value of the prolongation
  at n_+ is canonically homeomorphic to the original value, 
  see Remark <ref>.
  The coend can be calculated by a familiar quotient
  space construction in the ambient category of all topological spaces,
  compare Proposition <ref>:
  F(K) can be obtained from the disjoint union 
  of the spaces F(n_+)×  K^n, for n≥ 0,
  by dividing out the equivalence relation generated by
  
    (F(α)(x); k_1,…,k_n) ∼ (x;  k_α(1),…,k_α(m))

  for all x∈ F(m_+), all (k_1,…,k_n) in K^n,  and all 
  morphisms α:m_+→ n_+ in .
  Here k_α(i) is to be interpreted as the basepoint of K whenever α(i)=0.
  The non-obvious fact, proved in Proposition <ref> (ii),
  is that this quotient space is automatically compactly generated, and hence
  a coend in the category _*.


  We write [x; k_1,…,k_n] for the equivalence class 
  of a tuple (x; k_1,…,k_n)∈ F(n_+)× K^n in F(K).
  The prolongation is continuous and comes with a continuous, based assembly map
  subjectassembly map!of a -space

    α :  K F(L)  → F(K L)  , α(k [x;  l_1,…,l_n]) =  [x; k l_1,…,k l_n] .

  The assembly map is natural in all three variables and associative and unital.

   We can now define an orthogonal spectrum F()
   symbolF() - evaluation of a -space F on spheres
   subjectGamma-space@-space!evaluation on spheres
  by evaluating the -space F on spheres. In other words, 
  the value of F() at an  inner product space V is 
  
    F()(V)  =  F(S^V)

  and the structure map σ_V,W:S^V F()(W)→ F()(V⊕ W)
  is the assembly map for K=S^V and L=S^W,
  followed by the effect of F on the canonical homeomorphism 
  S^V S^W S^V⊕ W.
  The O(V)-action on F()(V) is via the action on S^V and the continuous
  functoriality of F.





 
  Let F be a -space and G a compact Lie group. 
  
  
  * The projection p:G→π_0 G=G̅ to the group of path components 
    induces an isomorphism    of geometric fixed point homotopy groups 
    
    p^*  :  Φ^G̅_*( F() )  → Φ^G_*( F() )

    of the orthogonal spectrum F().
  
  * The orthogonal spectrum F() obtained by evaluation of F on spheres
    is left induced from the global family  of finite groups.subjectglobal family!of finite groups 
  


  (i) 
  If G is connected, then for every based G-space K the map F(K^G)→(F(K))^G
  induced by the fixed point inclusion K^G→ K is a homeomorphism
  by Proposition <ref>, where G acts trivially 
  on the -space F.
  We can calculate G-fixed points by first taking fixed points with respect
  to the normal subgroup G^∘ (the path component of the identity) and then
  fixed points with respect to the quotient G̅=G/G^∘=π_0 G. 
  So for k≥ 0, the G-geometric fixed points 
  of the orthogonal spectrum F() can be
  rewritten as
  
    Φ^G_k ( F() )    =  _V∈ s(_G)  [S^V^G⊕^k,  F(S^V)^G] 
        _V∈ s(_G)  [ (S^V^G^∘⊕^k)^G̅,  F(S^V^G^∘)^G̅] 
        _W∈ s(_G̅)  [ S^W^G̅⊕^k,  F(S^W)^G̅]
         = Φ^G̅_k ( F()) .

  The third step uses that (_G)^G^∘ is a complete universe for
  the finite group G̅ and as V runs through s(_G),
  the G^∘-fixed points V^G^∘ exhaust (_G)^G^∘.
  The composite isomorphism is inverse to the inflation map p^*.
  The argument for k<0 is similar.
  
  (ii)
  The global family  of finite groups is reflexive, and
  for every compact Lie group K the projection K→π_0 K 
  to the finite group of path components is universal with respect to .
  Part (i) verifies the geometric fixed point criterion, 
  so by Proposition <ref> 
  the orthogonal spectrum F() is left induced from the global family of
  finite groups.    








Now we look more closely at right induced global homotopy types.
For a global family  and a compact Lie group G we
denote by ∩ G the family of those closed subgroups of G that belong to , 
and E(∩ G) is a universal G-space for the family ∩ G.  
We also need the equivariant cohomology theory represented by an orthogonal spectrum X.
If A is a G-space, we define the G-cohomology X^k_G(A) as

    X^k_G(A)  =  Σ^∞_+ _G,VA, X[k] ,

the group of degree k maps in  from the suspension spectrum
of the semifree orthogonal space _G,V A to X.subjectequivariant cohomology theory!of an orthogonal spectrum
Here V is an implicitly chosen faithful G-repre­sen­tation.
By the adjunction between the global stable homotopy category
and the G-equivariant stable homotopy category that we discuss in
Theorem <ref> below, the group X^k_G(A)
is isomorphic to the value at A of the G-cohomology theory
represented by the underlying G-spectrum X_G.


An orthogonal spectrum X is right induced from a global family 
if and only if for every compact Lie group G
and every cofibrant G-space A the map

    X^*_G(A) → X^*_G(A× E(∩ G))

induced by the projection A× E(∩ G)→ A is an isomorphism.


For every G-space A the projection from A× E(∩ G) to A
is an (∩ G)-equivalence; moreover, if A is cofibrant, 
then the source is (∩ G)-projective, so the suspension spectrum
of the semifree orthogonal space

    Σ^∞_+ _G,V (A× E(∩ G))

is left induced from the global family . 
This implies that 

    L_(U_( Σ^∞_+ _G,V A))  Σ^∞_+ _G,V (A× E(∩ G))

in the global stable homotopy category. Hence

    X^k_G( A×   E(∩ G))  =  Σ^∞_+ _G,V( A× E(∩ G)),X[k]
        L_(U_(Σ^∞_+ _G,V A)) ,X[k] Σ^∞_+ _G,V A ,R_(U_ (X))[k] .

for every orthogonal spectrum X.
Under this composite isomorphism, the map of the proposition becomes 
the map 

    X_G^k(A) =  Σ^∞_+ _G,V A, X[k] →Σ^∞_+ _G,V A, R_(U_(X))[k]

induced by the adjunction unit X→ R_(U_(X)).

If X is right induced from ,
then this adjunction unit is an isomorphism,
hence so is the map X^*_G(A)→ X^*_G(A× E(∩ G)).
If, on the other hand, this map is an isomorphism for all G-spaces A,
then for A=∗ we deduce that the map

    Σ^∞_+ B_ G, X[k] →Σ^∞_+ B_ G, R_(U_(X))[k]

is an isomorphism. Since the suspension spectrum of B_ G
represents π_0^G (by Theorem <ref> (i)), 
this shows that the adjunction unit  X→ R_(U_(X)) is a global equivalence.
So X is right induced from .





Essentially the same proof also shows the
following relative version of Proposition <ref>.
We let ⊆ be two nested global families.
Then an orthogonal spectrum X is in the essential image
of the relative right adjoint R:_→_ if and only if 
for every group G in  and every cofibrant G-space A the map

    X^*_G(A) → X^*_G(A× E(∩ G))

induced by the projection A× E(∩ G)→ A is an isomorphism.





We let X be a global Ω-spectrum with the property that for every
inner product space V, 
the O(V)-space X(V) is cofree,subjectcofree equivariant space i.e.,
for some (hence any) universal free O(V)-space E the map

    const :   X(V) → (E, X(V) )

that sends a point to the corresponding constant map is an O(V)-weak equivalence.
We claim that then the orthogonal spectrum X 
is right induced from the trivial global family e.
We use the criterion of Proposition <ref>
and show that for every compact Lie group G,
every cofibrant G-space A and every integer k the map X^k_G(Π)
induced by the projection Π:A× E G→ A is an isomorphism.

We start with the case k=0.
We let V be any faithful G-representation.
The projection Π is a weak equivalence
of underlying non-equivariant spaces, and source and target are cofibrant
as G-spaces. So the G-map

    S^VΠ_+ :  S^V (A× E G)_+ →  S^V A_+

is also a weak equivalence
of underlying non-equivariant spaces, and source and target are G-cofibrant
in the based sense. 
Since X(V) is cofree as an O(V)-space, it is also cofree as a G-space,
where G acts via the representation homomorphism G→ O(V).
So the induced map

    [ S^VΠ_+ , X(V) ]^G  :[ S^V A_+ , X(V) ]^G  → [ S^V(A× E G)_+, X(V)]^G

is bijective.
Since X is a global Ω-spectrum
and the orthogonal suspension spectrum Σ^∞_+_G,VA is flat, 
the localization map

    (Σ^∞_+ _G,VA,  X) / homotopy → Σ^∞_+ _G,VA,  X =   X^0_G(A)

is bijective.
By the freeness property, the left hand side bijects with the set [S^V A_+, X(V)]^G.
So by the previous paragraph the map
X_G^0(Π):X^0_G(A)→ X^0_G(A× E G)
is bijective.

For k>0 we apply the same argument to the
global Ω-spectrum ^k X (which also has cofree levels)
and exploit the natural isomorphism

    (^k X)^0_G(A) =  Σ^∞_+ _G,VA, ^k X Σ^∞_+ _G,VA,  X[k] =   X^k_G(A) .

 For k<0 we apply the same argument to the
global Ω-spectrum Ω^-k X (which also has cofree levels)
and exploit the natural isomorphism

    (Ω^-k X)^0_G(A) =  Σ^∞_+ _G,VA, Ω^-k X Σ^∞_+ _G,VA,  X[k] =   X^k_G(A) .



As we explain in more detail in Example <ref>
below, the global K-theory spectrum  
is right induced from the global family
of finite cyclic groups. This fact is a reformulation of the generalization, 
due to Adams, Haeberly, Jackowski 
and May <cit.>,
of the Atiyah-Segal completion theorem.


[Global Borel theories]
We let E be a non-equivariant generalized cohomology theory.
Then we obtain a global functor E  by setting

    E(G)  =  E^0(B G)  ,
 
the 0-th E-cohomology of the classifying space of the group G.
The contravariant functoriality in group homomorphisms α:K→ G
comes from the covariant functoriality of the classifying space construction.
The transfer map for a subgroup inclusion H⊂ G 
is the Becker-Gottlieb transfer <cit.>

    :  Σ^∞_+ B G  → Σ^∞_+ B H

associated to the fiber bundle E G/H→ E G/G =B G
with fiber G/H, using that E G/H is a classifying space for H.
Strictly speaking, in <cit.>subjectBecker-Gottlieb transfer 
Becker and Gottlieb only define a stable transfer map for 
locally trivial fiber bundles with smooth compact manifold fiber
whenever the base is a finite CW-complex. To get the transfer above
one approximates E G (and hence B G) by its finite-dimensional skeleta.
The verification of the double coset formula for this global functor
is due to Feshbach <cit.>.

More generally, we can consider the Borel equivariant cohomology
theory represented by E. For a compact Lie group G and a cofibrant G-space A,
its value is

    E^*(E G×_G A)  ,

the E-cohomology of the Borel construction 
(also known as homotopy orbit construction).
Here E G is a universal free G-space, which is unique up to equivariant
homotopy equivalence.
We claim that these Borel cohomology theories associated to E
are represented by a specific global homotopy type, namely the result
of applying the right adjoint 

    R  :   →

to the forget functor U:→ 
to any non-equivariant spectrum that represents E.
To verify this claim we choose a faithful G-representation V
and recall from Proposition <ref> (i)
that the G-space (V,^∞) is a universal free G-space.
So for every G-space A the underlying non-equivariant homotopy type 
of the closed orthogonal space _G,V A is 

    (_G,VA)(^∞)  =  (V,^∞)×_G A  =  E G×_G A .

The adjunction between U and R thus provides an isomorphism

    (R E)^0_G(A) =  Σ^∞_+ _G,V A, R E     ( U(Σ^∞_+ _G,V A), E ) 
       =  ( Σ^∞_+ (E G×_G A), E )  =  E^0(E G×_G A) .

When A is a one-point G-space, then E G×_G ∗= B G,
and this bijection gives rise to a composite
isomorphism

    π_0^G(R E)  Σ^∞_+ _G,V , R E   E^0(B G) ,

where the first one is inverse to evaluation at the stable tautological class
e_G=e_G,V∈π_0^G(Σ^∞_+ _G,V).
We claim that the isomorphisms (<ref>) 
are compatible with restriction maps
arising from continuous group homomorphisms α:K→ G.
For this purpose we also choose a faithful K-representation W.
This data gives rise to a composite morphism of global classifying spaces

    B_α :  B_ K  =  _K,α^*(V)⊕ W ρ_α^*(V),W/K _K,α^*(V)  proj  _G,V =  B_G .

The first morphism restricts a linear isometric embedding from α^*(V)⊕ W
to α^*(V), and the second morphism is the quotient map from
K-orbits to G-orbits.
On the underlying non-equivariant homotopy types
(i.e., after evaluating at ^∞), the morphism B_α
classifies the homomorphism α. Moreover, the morphism
has the `correct' behavior on the unstable tautological classes. i.e.,

    (B_α)_*(u_K)  =  α^*(u_G)    in π_0^K(B_ G) ,

by direct verification from the definitions.
The analogous relation for the stable tautological classes

    (Σ^∞_+ B_α)_*(e_K)     =  
     (Σ^∞_+ B_α)_*(σ^K(u_K))  =  σ^K( (B_α)_*(u_K)) 
       =  σ^K( α^*(u_G))  
    = α^*(σ^G(u_G))  = α^*(e_G)

in the stable group π_0^K(Σ^∞_+ B_ G)
then follows by naturality 
of the suspension maps σ^K:π_0^K(Y)→π_0^K(Σ^∞_+ Y)
and its compatibility with restriction. This proves the compatibility 
of the isomorphisms (<ref>) with restriction maps.
Compatibility with transfers is essentially built in,
as both the transfer in Construction <ref> 
and the Becker-Gottlieb transfer in <cit.>
are defined as the Thom-Pontryagin construction based on smooth equivariant
embedding of G/H into a G-representation;
we omit the formal proof. In any case, the group isomorphisms (<ref>) 
together form an isomorphism of global functors 
between _0(R E) and E.
This proves the claim that the `global Borel theories' are 
precisely the ones right induced from non-equivariant stable homotopy theory.



We introduce a specific pointset level lift

    b  :   →

of the right adjoint R:→ to the category of orthogonal spectra.
Given an orthogonal spectrum E  
we define a new orthogonal spectrum b Esymbolb E - global Borel theory 
as follows.
For an inner product space V we setsubjectglobal Borel theory|( 

    (b E)(V) =  ((V,^∞),E(V))  ,

the space of all continuous maps from (V,^∞) to E(V).
The orthogonal group O(V) acts on this mapping space by conjugation, through
its actions on (V,^∞) and on E(V).
We define structure maps σ_V,W:S^V (b E)(W)→ (b E)(V⊕ W)
as the composite

    S^V((W,^∞),E(W))   assembly       ((W,^∞),S^V E(W)) 
    (_W, σ_V,W^E)    ((V⊕ W,^∞),E(V⊕ W))

where _W:(V⊕ W,^∞)→(W,^∞)
restricts an isometric embedding from V⊕ W to W.
In the functorial description of orthogonal spectra, the structure maps
are given by

    (V,W)((V,^∞),E(V))    → ((W,^∞),E(W)) 
      
    (w,φ)   f    ⟼ { ψ↦ X(w,φ)(f(ψ∘φ)) } .



The endofunctor b on the category of orthogonal spectra
comes with a natural transformation

    i_E  :  E  → b E

whose value at an inner product space V sends a point
x∈ E(V) to the constant map (V,^∞)→ E(V) with value x.
Since (V,^∞) is contractible, the morphism
i_E:E→ b E is a non-equivariant level equivalence,
hence a non-equivariant stable equivalence.


The next result shows that the global Borel construction b 
takes Ω-spectra to global Ω-spectra,
and that the functor b realizes, in a certain precise way, 
the right adjoint to the forgetful functor
from the global to the non-equivariant stable homotopy category.
Since the morphism i_E:E→ b E is a non-equivariant stable
equivalence, it becomes invertible in the non-equivariant stable
homotopy category .
Part (ii) of the following proposition shows 
that the morphism i_E^-1:b E→ E is the counit of the adjunction (U,R).



Let E be an orthogonal Ω-spectrum.


  * The orthogonal spectrum
b E is a global Ω-spectrum and 
right induced from the trivial global family.

  * For every orthogonal spectrum A both of the two group homomorphisms

    A,b E  U   (A,b E)  (i_E)^-1_* (A, E) .

are bijective. 


 (i) We let G be a compact Lie group and V and W two
G-representations such that W is faithful. Since E is an 
Ω-spectrum, the adjoint structure map

    σ̃^E_V,W :  E(W) → Ω^V E(V⊕ W)

is a non-equivariant weak equivalence. 
The G-space (W,^∞) 
is cofibrant by Proposition <ref> (ii). 
Because W is a faithful G-representation, the induced G-action on
(W,^∞) is free. So the induced map

    ((W,^∞),σ̃^E_V,W) :  
    (b E)(W)  =    ((W,^∞),E(W))
       → ((W,^∞),Ω^V E(V⊕ W))

is a G-weak equivalence. Moreover, the restriction map
_W:(V⊕ W,^∞)→(W,^∞)
is a G-homotopy equivalence (by Proposition <ref> (ii)), 
hence it induces another G-homotopy equivalence

    (_W,Ω^V E(V⊕ W)) :     ((W,^∞),Ω^V E(V⊕ W)) 
       → ((V⊕ W,^∞),Ω^V E(V⊕ W))

on mapping spaces.
The target of this last map is G-homeomorphic to

    _*(S^V,((V⊕ W,^∞),E(V⊕ W)))  = Ω^V( (b E)(V⊕ W) ) ;

under this homeomorphism, the composite of the two G-weak equivalences
becomes the adjoint structure map

    σ̃^b E_V,W :  (b E)(W) → Ω^V( (b E)(V⊕ W) ) .

So we have shown that σ̃^b E_V,W is a G-weak equivalence,
and that means that b E is a global Ω-spectrum.

The O(V)-space (V,^∞) is a universal free O(V)-space by
Proposition <ref> (i).
So the O(V)-space (b E)(V)=((V,^∞),E(V))
is cofree. 
Since b E is also a global Ω-spectrum,
the criterion of Example <ref>
shows that it is right induced from the trivial global family.

Part (ii) is a formal consequence of (i): since b E is right induced
from the trivial global family, the forgetful functor induces a bijection
U:A,b E(A,b E). Since the morphism i_E:E→ b E 
becomes an isomorphism in , it induces another bijection on (A,-).



We endow the functor b with a lax symmetric monoidal transformation

    μ_E,F :  b E b F  → b(E F) .

To construct μ_E,F we start from the (O(V)× O(W))-equivariant maps

    ((V,^∞), E(V))    ((W,^∞), F(W)) 
              ((V,^∞)×(W,^∞), E(V) F(W)) 
    (_V,W,i_V,W)    ((V⊕ W,^∞), (E F)(V⊕ W))

that constitute a bimorphism from (b E,b F) to b(E F).
Here

    _V,W : (V⊕ W,^∞) → (V,^∞)×(W,^∞)

maps an embedding of V⊕ W to its restrictions to V and W.
The morphism μ_E,F is associated to this bimorphism
via the universal property of the smash product (<ref>).



Various `completion' mapssubjectcompletion map 
(also called `bundling maps') fit in here as follows. 
For this we suppose that E is a commutative orthogonal 
ring spectrum and a positive Ω-spectrum (in the non-equivariant sense).
Then the morphism i_E:E→ b E is a kind of `global completion map'. 
For every compact Lie group G it induces a ring homomorphism
of G-equivariant homotopy groups

    π_0^G(E)  → π_0^G( b E )   E^0( B G )  .

When  E= is the sphere spectrum and G is finite, 
Carlsson's theorem <cit.> 
(proving the Segal conjecture) shows that the map

    (G)π_0^G()  → π^0(B G)

is completion of the Burnside ring 
at the augmentation ideal.subjectaugmentation ideal!of the Burnside ring
The sphere spectrum is the suspension spectrum of a 
global classifying space of the trivial group; 
more generally, for the global classifying space B_ K 
of a finite group K the `forgetful' map

    (K,G)π_0^G(Σ^∞_+ B_K)  → ( Σ^∞_+ B G, Σ^∞_+ B K )

is again completion at the augmentation ideal of the Burnside ring A(G),
see <cit.>.

Since the global Borel theory functor b:→
is lax symmetric monoidal,
it takes orthogonal ring spectra to orthogonal ring spectra, 
in a way preserving commutativity and module structures.
Since the transformation i_E is monoidal, it becomes a homomorphism 
of orthogonal ring spectra when E is an orthogonal ring spectrum.
We let →^f be a `positively fibrant replacement',
i.e., a morphism of commutative orthogonal ring spectra
that is a non-equivariant stable equivalence and whose target
is a positive Ω-spectrum;
such a replacement exists and is homotopically unique by the
positive model structure for commutative orthogonal ring spectra
of <cit.>.
The spectrum =b(^f)symbol@ - completed sphere spectrum thus comes with a commutative ring spectrum structure,
and we call it the completed sphere spectrum.subjectcompleted sphere spectrumsubjectsphere spectrum!completed|seecompleted sphere spectrum
Moreover, for every ^f-module spectrum E the map

    b E =  b(^f)  b E  μ_^f,E 
    b(^f E)  b(act) b E

makes the orthogonal spectrum b E into a module spectrum 
over the completed sphere spectrum. 
Since  is non-equivariantly stably equivalent to ,
this shows that for every group G the equivariant homotopy group

    π_k^G(b E)    E^-k(B G)

is naturally a module over the commutative ring π_0^G( ). 

For the global K-theory spectrum 
(compare Construction <ref> below)
and any compact Lie group G, the bundling map

    (G)π_0^G (  )  → K U^0(B G)
subjectaugmentation ideal!of the unitary representation ring
takes a virtual G-representation to the class of the associated virtual vector bundle
over B G. The Atiyah-Segal completion theorem <cit.>
shows that this map is completion at the augmentation ideal of the representation ring.
For the Eilenberg-Mac Lane spectrum 
(see Construction <ref> below), 
the global functor G↦ H^0(B G;) is constant with value ; the map 

    π_0^G()  → H^0(B G;)

is surjective and an isomorphism modulo torsion 
for all compact Lie groups whose identity path component 
is commutative (compare Example <ref>).

subjectglobal Borel theory|) 



Now we fix a compact Lie group G and relate the global stable homotopy category 
to the G-equivariant stable homotopy category (based on a complete universe). 
We denote by G-symbol  G- - G-equivariant stable homotopy category 
the G-equivariant stable homotopy category,subjectstable homotopy category!G-equivariant
i.e., any localization of the category G of orthogonal G-spectra
at the class of _*-isomorphisms.
Various stable model structures have been established with 
 _*-isomorphisms as weak equivalences, for example by
Mandell-May <cit.>, Stolz <cit.>
and Hill-Hopkins-Ravenel <cit.>.
In particular, as for every stable model category,
the homotopy category  G-
comes with a preferred structure of a triangulated category.

A functor

    (-)_G  :   → G ,    X  ⟼ X_G

from orthogonal spectra to orthogonal G-spectra
is given by endowing an orthogonal spectrum with the trivial G-action.
Since the trivial action functor takes global equivalences 
to _*-isomorphisms,
the universal property of localizations provides 
a `forgetful' functor on the homotopy categories

    U = U_G :   → G- .

Moreover, U is canonically an exact functor of triangulated categories. 
We will show now that the forgetful functor has both a left and a right adjoint.

The `equivariant' smash product of orthogonal G-spectra
is simply the smash product of the underlying non-equivariant
orthogonal spectra with diagonal G-action.
So the trivial action functor (-)_G:→ G is
strong symmetric monoidal.
The smash product of orthogonal spectra and of orthogonal G-spectra 
can be derived to symmetric monoidal products on  and on G- 
(see Corollary <ref>). The forgetful functor
is strongly monoidal with respect to these derived smash products.
Indeed, the derived smash product in  can be calculated by
flat approximation up to global equivalence;
a flat orthogonal spectrum, endowed with trivial G-action,
is G-flat, and hence can be used to calculate the derived smash product
 G-, by the flatness theorem (Theorem <ref>). 


When G=e is a trivial group, the next theorem reduces to the change of family
functor of Theorem <ref>, with = and =e.



  For every compact Lie group G the forgetful functor
  
    U  :   → G-

  preserves sums and products, and it has a left adjoint and a right adjoint.
  The left adjoint has a preferred lax symmetric comonoidal structure.
  The right adjoint has a preferred lax symmetric monoidal structure.


  Sums in  and G- are represented 
  in both cases by the pointset level wedge.
  For  we state this explicitly in Proposition <ref> (i);
  for G- we can run the argument based on the stable model
  structure for orthogonal G-spectra established in <cit.>.
  On the pointset level, the forgetful functor preserves wedges,
  so the derived forgetful functor preserves sums.
  As spelled out in Corollary <ref> (iv),
  the existence of the right adjoint is a formal consequence
  of the fact that  is compactly generated 
  and that the functor U preserves sums.

  The forgetful functor also preserves infinite products,
  but the argument here is slightly more subtle because products in 
  are not generally represented by the pointset level product,
  and because equivariant homotopy groups do not in general commute
  with infinite pointset level products,
  compare Remark <ref>.
  We let {X_i}_i∈ I be a family of orthogonal spectra.
  By replacing each factor by a globally equivalent spectrum, if necessary,
  we can assume without loss of generality that each X_i is a global Ω-spectrum.
  Since global Ω-spectra are the fibrant objects
  in a model structure underlying , 
  the pointset level product ∏_i∈ I X_i then represents
  the product in .
  
  Even though X_i is a global Ω-spectrum,
  the underlying orthogonal G-spec­trum (X_i)_G need not
  be a G-Ω-spectrum. However, as we spell out in the proof of
  Proposition <ref> (ii), the natural map
  
    π_k^G(∏_i∈ I  X_i)  → ∏_i∈ I π_k^G(X_i)

  is an isomorphism for all integers k.
  Again we can run the argument of Proposition <ref> (ii)
  in the stable model structure 
  for orthogonal G-spectra <cit.>, and conclude that
  in this situation, the pointset level product is also a product in G-.
  So the derived forgetful functor preserves products.
  The existence of the left adjoint is then again a formal consequence
  of the fact that  is compactly generated,  
  compare Corollary <ref> (v).

  The same formal argument as in part (iii) of Theorem <ref>
  shows how to turn the strong monoidal structure of the forgetful functor U
  into a lax comonoidal structure 
  L(  A  ^  B ) → ( L A ) ^ ( L B ) of the left adjoint.
  In contrast to Theorem <ref> (iii), however,
  this morphism is usually not an isomorphism, so we cannot turn it
  around into a monoidal structure on L.
  The same formal argument as in Theorem <ref> (ii)
  constructs the lax monoidal structure on R from the strong monoidal
  structure of the forget functor U.


Theorem <ref> looks similar to the change-of-family
Theorem <ref>, but there is one important difference:
if the group G is non-trivial, then neither of the two adjoints
to the forgetful functor  U:→ G- is fully faithful.


  We mention an alternative way to construct the two adjoints
  to the forgetful functor U : → G-, by exhibiting the
  pointset forgetful functor (-)_G:→ G
  as a left respectively a right Quillen functor for suitable model structures. 
  We sketch this for the left adjoint, where we can use 
  the stable model structure of orthogonal G-spectra
  established by Mandell and May in <cit.>.
  However, we cannot argue directly with the functor (-)_G:→ G,
  since it is not a right Quillen functor.
  Indeed, if it were a right Quillen functor, then it would  preserve
  fibrant objects. However, a global Ω-spectrum is typically
  not a G-Ω-spectrum when given the trivial G-action.
  
  What saves us is that a global Ω-spectrum is `eventually'
  a G-Ω-spectrum, i.e., starting at faithful representations.
  This lets us modify (-)_G into a right Quillen functor as follows. 
  We choose a faithful G-representation V and let
  
    Ω^V ^V  :   → G

  denote the functor that takes an orthogonal spectrum X to the
  orthogonal G-spectrum with U-th level
  
    ( Ω^V ^V X )(U)  =  _*(S^V, X(U⊕ V)) .

  We emphasize that the G-action on  Ω^V ^V X is non-trivial,
  despite the fact that we started with an orthogonal spectrum without a G-action.
  A natural morphism of orthogonal G-spectra 
  λ̃^V_X:X→Ω^V ^V X is given by the adjoint
  of the morphism λ^V_X:X S^V→^V X
  defined in (<ref>);
  the morphism λ̃^V_X 
  is a _*-isomorphism by Proposition <ref> (ii).
  In particular, the functor Ω^V ^V also takes
  global equivalences of orthogonal spectra to _*-isomorphisms
  of orthogonal G-spectra, and the derived functor of Ω^V ^V 
  is naturally isomorphic to the forgetful functor U:→ G-.
  The argument can then be completed by showing that  
  the functor Ω^V ^V is a right Quillen functor
  from the global model structure on orthogonal spectra
  to the stable model structure on orthogonal G-spectra
  from <cit.>.
  Then by general model category theory, the derived functor
  of Ω^V ^V, and hence also the forgetful functor U,
  has a left adjoint.
  
  The existence of the right adjoint to U can also be established
  by model category reasoning. For this one can use the 𝕊-model
  structure on orthogonal G-spectra 
  constructed by Stolz <cit.>.
  We leave it to the interested reader to verify that the cofibrant objects
  in the 𝕊-model structure are precisely 
  the G-flat orthogonal G-spectra in the sense of Definition <ref>.
  This shows that the forgetful functor
  (-)_G:→ G is a left Quillen functor from the
  global model structure on orthogonal spectra to the stable 𝕊-model
  structure on orthogonal G-spectra.


The left adjoint L:G-→ to the forgetful functor
is an exact functor of
triangulated categories that preserves infinite sums.
The G-equivariant stable homotopy category is compactly generated
by the unreduced suspension spectra of all the coset spaces G/H,
for all closed subgroups H of G. So L is essentially determined by
its values on these generators.
The sequence of natural bijections

    (L(Σ^∞_+ G/H), X)  G-(Σ^∞_+ G/H,  U X)   π_0^H (X)   (Σ^∞_+ B_H , X)

shows that the left adjoint L takes the unreduced suspension spectrum
of the coset space G/H to the suspension spectrum  of the 
global classifying space of H.
In the special case H=G the spectrum Σ^∞_+ G/H
is the equivariant sphere spectrum _G, and we obtain that

    L(_G)  Σ^∞_+ B_G  .

Now

    G-(_G,_G)   π_0^G ()   (G)  =  (e,G)

is the Burnside ring, whereas

    (L(_G),L(_G))   π_0^G (Σ^∞_+ B_ G)  
       (G,G)

is the double Burnside ring. 
The map L:G-(_G,_G) →(L(_G),L(_G)) 
corresponds to the ring homomorphism

    (G) =  (e,G)  → (G,G)  ,   _H^G∘ p_H^*  ⟼ _H^G∘^G_H

from the Burnside ring to the double Burnside ring; 
this homomorphism is never surjective unless G is trivial, 
so the left adjoint is not full.





 The discussion in this section could be done relative
to a global family , as long as  contains the compact Lie group
G under consideration (and hence also all its subgroups).
Indeed, if  contains G, then every -equivalence
of orthogonal spectra is a _*-isomorphism of 
underlying orthogonal G-spectra.
Hence the trivial action functor descends to
a `forgetful' functor on the homotopy categories

    U_G^ :  _ → G-

by the universal property of localizations. The same arguments as
in Theorem <ref> show the existence of both
adjoints to this forgetful functor, with the same kind of monoidal properties. 

Theorem <ref> discusses the maximal case
of the global family = of all compact Lie groups.
The minimal case is the global family G generated by G,
i.e., the class of compact Lie groups that are isomorphic to
a quotient of a closed subgroup of G. All the forgetful functors U^_G
then factor as composites 

    _ U_G^ _G U_G^G  G-

of a change-of-family functor and a family-to-group functor.
The various adjoints then compose accordingly.




Whenever G is non-trivial, then the global homotopy category
_G associated to the global family generated by G is different
from the G-equivariant stable homotopy category G-.
In other words, if G is non-trivial, then the forgetful family-to-group functor 
U_G^G: _G→ G- is not an equivalence,
and neither of its adjoints is fully faithful.
subjectglobal stable homotopy category|)





subjectglobal stable homotopy category!for finite groups|(
subjectglobal family!of finite groups|( 
In the rest of this section we turn to the global family  of finite groups
and describe the associated global stable homotopy category _ rationally. 
By our previous results, _ is a compactly generated triangulated category with
a symmetric monoidal derived smash product.
We call an object X of the category _ rationalsubjectorthogonal spectrum!rational
if the equivariant homotopy groups π_k^G(X) are uniquely divisible 
(i.e., -vector spaces) for all finite groups G.
In this section we will give an algebraic model of the 
rational -global stable homotopy category,
i.e., the full subcategory ^_ of rational spectra in _.
Theorem <ref> below shows that the homotopy types in ^_
are determined by a chain complex of global functors,
up to quasi-isomorphism.
More precisely, we construct an equivalence of triangulated categories
from  ^_ to the unbounded derived category of rational global functors
on finite groups.


We let G and K be compact Lie groups.
We recall from Proposition <ref> 
that the evaluation map

    (G,K)  → π_0^K(Σ^∞_+ B_ G)  ,   τ⟼τ(e_G)

is an isomorphism, where e_G∈π_0^G(Σ^∞_+ B_ G)
is the stable tautological class.
More precisely, the definition of the global classifying space B_ G
involves an implicit choice of faithful G-representation V that is omitted 
from the notation, and e_G is the class denote e_G,V 
in (<ref>).
The unreduced suspension spectrum of every orthogonal spaces is globally connective
(see Proposition <ref>),
so the group π_k^K(Σ^∞_+ B_ G) is trivial for k<0.

subjectglobal classifying space!of a finite group   
Let G and K be finite groups. Then for every k>0, 
the equivariant homotopy group π_k^K(Σ^∞_+ B_ G) is torsion.

 
We show first that the geometric fixed point group 
Φ_k^K(Σ^∞_+ B_ G) is torsion for k>0.subjectgeometric fixed points
This part of the argument needs G to be finite, but K could be any compact Lie group.
Geometric fixed points commute with suspension spectra, i.e., 
the groups Φ_*^K(Σ^∞_+ B_ G)
are isomorphic to the non-equivariant stable homotopy groups
of the fixed point space ((B_G)(_K))^K.
Proposition <ref> (i) 
identifies these fixed points as

    (( B_G)(_K))^K  ≃ ∐_[α]∈Rep(K,G)   B C(α) ,

where the disjoint union is indexed by conjugacy
classes of homomorphisms from K to G, 
and C(α) is the centralizer of the image of α:K→ G.
Since G is finite, so are all the centralizers C(α),
hence the classifying space B C(α) has no rational homology,
hence no rational stable homotopy, in positive dimensions.
So we conclude that the rationalized stable homotopy groups 
of the space ((B_G)(_K))^K vanish in positive dimensions. 

If K is also finite, then the k-th rationalized equivariant 
stable homotopy group of any orthogonal K-spectrum can be recovered from the 
k-th rationalized geometric fixed point homotopy groups
for all subgroups L of K, as described in 
Corollary <ref>.
So when both G and K are finite, then 
also the equivariant homotopy group π_k^K(Σ^∞_+ B_ G) is torsion
for all k>0.


The conclusion of Proposition <ref> 
is no longer true if we drop the finiteness hypothesis on one
of the two groups G or K.
For example, for G=e we have Σ^∞_+ B_ G=, 
and the dimension shifting transfer _e^U(1)(1) is an element 
of infinite order in the group π_1^U(1)().
On the other hand, for K=e the group π_k^K(Σ^∞_+ B_ G)
is the non-equivariant stable homotopy group of the 
ordinary classifying space B G. For G=U(1) the group
π_k^e(Σ^∞_+ B U(1)) contains a free summand of rank 1
whenever k≥ 0 is even.



Now we can establish the algebraic model for the rational -global
homotopy category.
We let  be a pre-additive category, such as the 
-Burnside category _.
We denote by  the category of additive functors from
 to the category of -vector spaces.
This is an abelian category, and the represented functors
(a,-), for all objects a of , form a set
of finitely presented projective generators of .
The category of -graded chain complexes in the abelian category 
then admits the projective model structure with the quasi-isomorphisms
as weak equivalences. The fibrations in the projective model structure
are those chain morphisms that are surjective in every chain complex degree
and at every object of .
This projective model structure for complexes of -modules is a
special case of <cit.>.
Indeed, the projective (in the usual sense) -modules together
with the epimorphisms form a projective class 
(in the sense of <cit.>),
and this class is determined
(in the sense of <cit.>)
by the set of represented functors.

We also need the rational version of the -global model structure,
for a global family .
We call a morphism f:X→ Y of orthogonal spectra a
rational -equivalencesubjectF-equivalence@-equivalence!rational
if the map

    π_k(f) :  π_k^G(X) → π_k^G(Y)

is an isomorphism for all integers k and all groups G in the family .



   
  Let  be a global family.
  
  
  * The rational -equivalences and the -cofibrations 
    are part of a model structure on the category of orthogonal spectra, 
    the rational -global model structure.subjectF-global model structure@-global model structure!rational
  
  * The fibrant objects in the rational -global model structure 
    are the -Ω-spectra X such that for all 
    G∈ the equivariant homotopy groups π_*^G(X) are uniquely divisible.
  
  * The rational -global model structure is cofibrantly generated, 
    proper and topological. 
  


Theorem <ref> is obtained
by Bousfield localization of the -global model structure on
orthogonal spectra, and one can use a similar proof as for the rational stable
model structure on sequential spectra in <cit.>.
We omit the details. 

 
There is a chain of Quillen equivalences between
the category of orthogonal spectra with the 
rational -global model structure and the category of chain complexes of
rational global functors on finite groups. In particular, this induces
an equivalence of triangulated categories

    ^_    ( ^_)  .

The equivalence can be chosen so that the homotopy group global functor
on the left hand side corresponds to the homology global functor 
on the right hand side.


We prove this as a special case of the `generalized tilting theorem'
of Brooke Shipley and the author. 
Indeed, by Theorem <ref> 
the suspension spectra of the global classifying spaces B_G 
are compact generators of the global homotopy category
_ as G varies through all finite groups.
So the rationalizations  (Σ^∞_+ B_G)_ 
are compact generators of the rational global homotopy category _^. 
If k is any integer, then 
the morphism vector spaces between two such objects are given by

    (Σ^∞_+ B_ K)_[k], (Σ^∞_+ B_ G)_    π_k^K((Σ^∞_+ B_ G)_)   π_k^K( Σ^∞_+ B_ G ) 
        (G,K)       for k=0, and
         0       for k0.

The vanishing for k>0 is Proposition <ref>;
the vanishing for k<0 is Proposition <ref>.


The rational -global model structure on orthogonal spectra is 
topological (hence simplicial), cofibrantly generated, proper and stable; 
so we can apply the Tilting Theorem <cit.>. 
This theorem yields a chain
of Quillen equivalences between orthogonal spectra 
in the rational -global model structure and the category of chain complexes of
_-modules, i.e., 
additive functors from the rationalized Burnside category _
to abelian groups. 
This functor category is equivalent to the category of additive functors
from _ to -vector spaces, and this proves the theorem.


 There is an important homological difference between
global functors on finite groups and Mackey functors for one fixed finite group.
Indeed, for a finite group G, 
the category of rational G-Mackey functors is equivalent to a product, 
indexed over conjugacy classes (H) of subgroups of G, of the
module categories over the rational group rings [W_G H]
of the Weyl groups, see Theorem <ref> (ii). 
In particular, the abelian  category of rational G-Mackey functors
is semisimple, every object is projective and injective
and the derived category is equivalent, by taking homology,
to the category of graded rational G-Mackey functors.

There is no analog of this for rational -global functors.
For example, the rationalized augmentation

    = (e,-)  →

from the rationalized Burnside ring global functor to the constant global functor
for the group  does not split on finite groups.
The new phenomenon is that any splitting would have to be natural for inflation maps.
Let us be even more specific.
In the constant global functor  we have

    _e^C_2(1) = 2· p^*(1)
          in    (C_2)= ,

where p:C_2→ e is the unique group homomorphism.
So for every morphism of global functors φ:→ N
the image φ(e)(1) of the unit element under the map
φ(e):→ N(e) must satisfy

    _e^C_2(φ(e)(1))  =  2· p^*(φ(e)(1)) .

But in the Burnside ring (e), and also in its rationalization,
0 is the only element in the kernel of _e^C_2-2· p^*; 
so every morphism of global functors from  to  is zero.




While the abelian category ^_ is not semisimple,
we can still `divide out transfers' and thereby replace ^_ by an equivalent, 
but simpler category.
We let symbol   - category of finite groups and conjugacy classes
of epimorphisms denote the category of finite groups and conjugacy classes
of surjective group homomorphisms. We write

    =  (^,)

for the category of `right -modules', i.e., contravariant functors
from  to the category of abelian groups.
To a -global functor M:_→ we can associate a right -module
τ M:^→, the reduced functor as follows.
On objects we set

    (τ M)(G)  =  τ_G M =  M(G)/ t_G M  ,

the quotient of the group M(G) by the subgroup t_G M
generated by the images of all transfer maps _H^G:M(H)→ M(G) 
for all proper subgroups H of G.
If α:K→ G is a surjective group homomorphism and
H≤ G a proper subgroup, then L=α^-1(H) is a proper
subgroup of K and the relation

    α^*∘_H^G  =  _L^K ∘ (α|_L)^* :  M(H)→ M(K)

shows that the inflation
map α^*:M(G)→ M(K) passes to a homomorphism
α^*:(τ M)(G)→ (τ M)(K) of quotient groups.
We will now argue that the reduction functor

    τ :  _ →

is rationally an equivalence of categories, 
compare Theorem <ref>.
By construction, the projection maps M(G)→τ M(G) 
form a natural transformation from the restriction of
the global functor M to the category ^ to τ M.


 We let A be an abelian group and A
the constant global functor with value A,
compare Example <ref> (iii).subjectglobal functor!constant 
Then (τA)(G)= A/c A,
where c is the greatest common divisor of the indices of all proper subgroups
of G. If G is not a p-group for any prime p, then this
greatest common divisor is 1. If G is a non-trivial p-group, then
G has a proper subgroup of index p. So we have

    (τA)(G)  =    A    if G=e,
    
      A/p A    if G is a non-trivial p-group, and
      0    else.

The inflation maps in τA are quotient maps.


 The Burnside ring (K)=(e,K)subjectBurnside ring global functor 
of a finite group K is
freely generated, as an abelian group, by the transfers _L^K(1)
where L runs through representatives of the conjugacy classes
of subgroups of K. So (τ)(K) is free abelian of rank 1,
generated by the class of the multiplicative unit 1. All restriction maps
preserve the unit, so the reduced functor τ of the global Burnside
ring functor is isomorphic to the constant functor ^→
with value .

We denote by _G=(G,-) the global functor represented by a finite group G.
We will now present
τ(_G) as an explicit quotient of a sum of representable -modules.
For every closed subgroup H of G the restriction map _H^G
is a morphism in (G,H). If H is finite, the 
Yoneda lemma provides a unique morphism

    _H  →τ(_G)

of -functors from the representable functor
_H=[(-,H)] to τ(_G) that sends the identity of H
to the class of _H^G in τ (_G)(H).
For every element g∈ G the conjugation isomorphism
c_g:H→ H^g given by c_g(γ)=g^-1γ g induces
an isomorphism

    g_⋆∘ -  :  _G(H^g)  → _G(H)

by postcomposition. We have 

    g_⋆∘ [_H^g^G]  =  [g_⋆∘_H^g^G]  =  
    [_H^G∘ g_⋆]  =  [_H^G]

in (τ_G)(H); so the Yoneda lemma shows that the triangle of -functors

    @R=6mm@C=6mm_H [dr] [rr]^-c_g∘ -      _H^g[dl]
       τ(_G)

commutes. The direct sum of the transformations _H→τ(_G)
thus factors over a natural transformation

    (⊕_H≤ G _H)/G  → τ (_G)  .

The source of this morphism can be rewritten if we choose representatives
of the conjugacy classes of subgroups in G:

    ⊕_(H)  ( _H / W_G H ) → τ (_G)  .

Now the sum is indexed by conjugacy classes.





For every finite group G, the morphism (<ref>)
is an isomorphism of -modules.


By Theorem <ref> 
the abelian group _G(K)=(G,K) is
freely generated by the elements _L^K∘α^*
where (L,α) runs through representatives of the (K× G)-conjugacy classes
of pairs consisting of a subgroup L of K
and a homomorphism α:L→ G. 
So (τ_G)(K) is a free abelian group with basis the
classes of α^* for all conjugacy classes of
homomorphisms α:K→ G. 

On the other hand, the group 
( _H / W_G H )(K) is free abelian with basis
given by W_G H-orbits of conjugacy classes of epimorphisms α:K→ H.
The map (<ref>) sends the basis
element represented by α to the basis element represented
by the composite of α with the inclusion H→ G.
So the homomorphism (<ref>) 
takes a basis of the source to a basis of the target, and is thus an isomorphism.


We recalled in Proposition <ref>
above how the value of a G-Mackey functor M,
for a finite group G, can rationally be recovered
from the groups (τ M)(H) for all subgroups H of G:
the map

    ψ^M_G  :  M(G) → ( ∏_H≤ G  (τ M)(H))^G

whose H-component is the composite

    M(G)  ^G_H  M(H) proj  (τ M)(H)

becomes an isomorphism after tensoring with .
When applied to a -global functor M, we see that M can rationally be
recovered from the -module τ M. This is the key input to the following
equivalence of categories.
I suspect that the following result is well known,
but I have been unable to find an explicit reference.

 
 
The restriction of the functor τ to the full subcategory
of rational -global functors is  an equivalence of categories

    τ :  _^ → _ =  (^ op,)

onto the category of rational -modules.


Since global functors are an enriched functor category and the functor

    τ :  _ →

commutes with colimits, τ has a right adjoint

    ρ :   → _ .

If ρ X is an -module, the value of the -global functor ρ X
at a finite group G is necessarily given by

    (ρ X)(G) =  (τ(_G), X) ,

the group of -module homomorphisms from τ(_G) to X.
The global functoriality in G is via τ, i.e., as the composite

    (G,K)  (ρ X)(G)   → (_K,_G)  (ρ X)(G)
       τ (τ(_K),τ(_G))  (ρ X)(G) ∘  (ρ X)(K) .

We rewrite the definition of (ρ X)(G)
using the description of the -module τ(_G)
given in (<ref>).
Indeed, by Proposition <ref>,
precomposition with (<ref>) induces an isomorphism

    (ρ X)(G)  (  (⊕_H≤ G _H)/G, X)   ( ∏_H≤ G X(H))^G .

So for every global functor M,
this description shows that 
ρ(τ M)(G) is isomorphic to the target of the morphism ψ^M_G.
A closer analysis reveals that for X=τ M, 
the above isomorphism identifies ψ^M_G
with the value of the adjunction unit η:M→ρ(τ M) at G.
So Proposition <ref>
shows that for every rational -global functor M
the adjunction unit η_M:M→ρ(τ M) is an isomorphism.
This implies that the restriction of the left adjoint τ 
to the category of rational -global functors is fully faithful.

Now we consider a rational -module X. Then ρ X
is a rational -global functor, so η_ρ X:ρ X→ρ(τ(ρ X))
is an isomorphism by the previous paragraph.
Since η_ρ X is right inverse to ρ(ϵ_X),
the morphism ρ(ϵ_X):ρ(τ(ρ X))→ρ X
is an isomorphism of -global functors.
By Proposition <ref> the represented -module
_G is a direct summand of τ(_G).
So the group

    (_G,X)   X(G)

is a direct summand of the group

    (τ(_G),X)   (_G,ρ X)   (ρ X)(G)  ,

and this splitting is natural for morphisms of -modules in X.
In particular, the morphism ϵ_X(G):(τ(ρ X))(G)→ X(G)
is a direct summand of the morphism

    ρ(ϵ_X)(G) : (ρ(τ(ρ X)))(G) → (ρ X)(G)  .

The latter is an isomorphism by the previous paragraph,
so the morphism ϵ_X(G) is also an isomorphism. 
This shows that for every rational -module X
the adjunction counit ϵ_X:τ(ρ X)→ X is an isomorphism.

Altogether we have now seen that when restricted to rational objects on both sides,
the unit and counit of the adjunction (τ,ρ) are isomorphisms. 
This proves the theorem.




The rational equivalence τ of abelian categories prolongs
to an equivalence of derived categories by applying τ dimensionwise
to chain complexes. The combination with the equivalence of triangulated
categories of Theorem <ref> is then a chain of
two exact equivalences of triangulated categories 

    ^_ [  ]  (^_)
     [](τ) (_) .

The next proposition shows that this composite equivalence is
an algebraic model for the geometric fixed point homotopy groups.

For every orthogonal spectrum X and every compact Lie group G,
the geometric fixed point map Φ:π^G_0(X)→Φ_0^G(X) 
annihilates all transfers from proper subgroups 
by Proposition <ref>. So the geometric fixed
point map factors over a homomorphism

    Φ̅ :  τ (_k(X))(G)  → Φ_k^G(X)

that we called the reduced geometric fixed point map above.subjectgeometric fixed point homomorphism!reduced
The geometric fixed point maps are compatible with inflations
(Proposition <ref> (iii)),
so as G varies among finite groups, the reduced geometric fixed
point maps form a morphism of -modules.
When we apply Proposition <ref> 
to the underlying orthogonal G-spectrum of an orthogonal spectrum,
it specializes to the following:

 
For every orthogonal spectrum X, every finite group G and every integer k
the mapsubjectgeometric fixed points!rationally 

    Φ̅ :  τ( _k(X) ) (G)  → Φ^G_k(X)

becomes an isomorphism after tensoring with .
So for varying finite groups G, these maps
form a rational isomorphism of -functors
τ(_k(X))Φ_k(X).
 


As a corollary we obtain that the combined equivalence κ of triangulated 
categories (<ref>)
from the rational finite global homotopy category ^_
to the derived category of the abelian category _
comes with a natural isomorphism

    Φ_*^G( X)    H_*(κ(X))    ,

for every object X of ^_,
between the geometric fixed point homotopy groups and
the homology -modules of κ(X).




subjectglobal stable homotopy category!for finite groups|)
subjectglobal family!of finite groups|) 






CHAPTER: ULTRA-COMMUTATIVE RING SPECTRA




This chapter is devoted to ultra-commutative ring spectra,
our model for extremely highly structured, multiplicative global stable
homotopy types.
On the point set level, these objects are simply 
commutative orthogonal ring spectra;
we use the term `ultra-commutative' to emphasize 
that we care about their homotopy theory with respect to
multiplicative morphisms that are global equivalences.subjectring spectrum!ultra-commutative|seeultra-commutative ring spectrumsubjectultra-commutative ring spectrum|(
We refer to the introduction of Chapter <ref> 
for further justification of the adjective `ultra-commutative'.
In short, the slogan “E_∞=commutative” is not true globally, 
and a strictly commutative multiplication encodes 
a large amount of additional structure that deserves a special name.

Section <ref>
introduces the formal setup for power operations on ultra-commu­ta­tive ring spectra.
We define global power functors as global Green functors 
equipped with additional power operations,
satisfying a list of axioms reminiscent of the properties
of the power maps x↦ x^m in a commutative ring.
We show in Theorem <ref> 
that the global functor _0(R) of an ultra-commutative ring spectrum R
supports such power operations, and is an example of a global power functor.

Section <ref> 
is primarily of algebraic nature, and gives both a monadic and
a comonadic description of the category of global power functors.
We introduce the comonad of `exponential sequences'
on the category of global Green functors, and show that its
coalgebras are equivalent to global power functors.
A formal consequence is that the category of global power functors
has all limits and colimits, and that they are created in the category of
global Green functors.
We discuss localization of global Green functors and global power functors
at a multiplicative subset  of the underlying ring, including
rationalization of global power functors.

In Section <ref> we discuss various examples of 
global power functors, such as the Burnside ring global power functor, 
the global functor represented by an abelian compact Lie group,
free global power functors, constant global power functors,
and the complex representation ring global functor.
In Section <ref> we set up the global model structure 
on the category of ultra-commutative ring spectra, compare 
Theorem <ref>.
In Theorem <ref>  we calculate the algebra
of natural operations on the 0-th homotopy groups of ultra-commutative ring spectra:
we show that these operations are freely generated by restrictions,
transfers and power operations. 
Theorem <ref> shows that every global
power functor can be realized by an ultra-commutative ring spectrum.






§ POWER OPERATIONS


subjectglobal power functor|(

In this section we introduce the formal setup for encoding
the power operations on ultra-commutative ring spectra.
In Definition <ref> we define global power functors,
which are global Green functors equipped with additional power operations,
satisfying a list of axioms reminiscent of the properties
of the power maps x↦ x^m in a commutative ring.
Theorem <ref> shows 
that the global functor _0(R) of an ultra-commutative ring spectrum R
supports power operations, and is an example of a global power functor.
As we shall see in Theorem <ref> below,
all the natural operations on _0(R) are generated by restrictions,
transfers and power operations,
so we are not missing any additional structure.
Moreover, every global power functor is realized by an
ultra-commutative Eilenberg-Mac Lane ring spectrum,
see Theorem <ref> below.
For a different perspective of global power functors
(restricted to finite groups), 
including the relationship to the concepts of λ-rings, 
τ-rings and β-rings,
we refer the reader to Ganter's paper <cit.>.



An ultra-commutative ring spectrum is a commutative orthogonal ring spectrum.
We write ucom for the category of ultra-commutative ring spectra.symbol  ucom - category of ultra-commutative ring spectra

subjectring spectrum!ultra-commutative|seeultra-commutative ring spectrum
subjectultra-commutative ring spectrum

In Section <ref> we introduced 
power operations and transfers on the equivariant homotopy sets of
ultra-commutative monoids.
For every ultra-commutative ring spectrum R, 
the orthogonal space Ω^∙ R 
inherits a commutative multiplication, making it an ultra-commutative monoid
(compare Example <ref>).
Moreover, π_0^G(Ω^∙ R)=π_0^G(R),
so this endows the 0-th stable equivariant homotopy groups π_0^G(R) 
with multiplicative power operationssubjectpower operation!in an ultra-commutative ring spectrum
and transfers, natural for homomorphisms of ultra-commutative ring spectra.
Since these operations come from the multiplicative 
(as opposed to the `additive' structure) of the ring spectrum, 
we now switch to a multiplicative notation and write

    P^m  :  π_0^G ( R ) → π^Σ_m≀ G_0 ( R )

(instead of [m]) for the multiplicative power operations, and we write N_H^G
(instead of _H^G) for multiplicative transfers.
Multiplicative transfers are often referred to as norm maps,
and that is also the terminology we will use. 
Since we will work with power operations a lot, 
we take the time to expand the definition: the operation P^m takes the class
represented by a based G-map f:S^V→ R(V), for some G-representation V,
to the class of the (Σ_m≀ G)-map

    S^V^m = (S^V)^ m  f^ m R(V)^ m  μ_V,…,V  R(V^m)  ,

where μ_V,…,V
is the iterated, (Σ_m≀ G)-equivariant multiplication map of R.


A global Green functorsubjectglobal Green functor
subjectGreen functor!global|seeglobal Green functor
is a commutative monoid in the category of global functors
under the monoidal structure given by the box product 
of Construction <ref>.subjectbox product!of global functors 
We write  for the category of global Green functors.symbol   - category of global Green functors 




As we explain after Definition <ref>,
the commutative multiplication on a global Green functor R
can be made more explicit in two equivalent ways:


  * as a commutative ring structure on the group R(G)
for every compact Lie group, subject to the requirement
that all restrictions maps are ring homomorphisms and the transfer maps
satisfy reciprocity;

  * as a unit element 1∈ R(e) and biadditive, commutative, associative and unital
external pairings × :R(G)× R(K)→ R(G× K) that are
morphisms of global functors in each variable separately.



We clarify next how the power operations 
of ultra-commutative ring spectra interact with the other structure
on equivariant stable homotopy groups, such as the addition,
restriction and transfer maps.
For m≥ 2 the power operation P^m is not additive, but it satisfies
various properties reminiscent of the map x↦ x^m in a commutative ring.
We formalize these properties into the concept of a global power functor.
Conditions (i) through (vi) in the following definition 
express the fact that a global power functor has an underlying `multiplicative'
global power monoid, 
in the sense of Definition <ref>,
if we forget the additive structure.
The definition makes use of certain embeddings between products
and wreath products:

    Φ_i,j :  (Σ_i≀ G) × (Σ_j≀ G)   →  Σ_i+j≀ G 
    
     ((σ;  g_1,…,g_i),  (σ';  g_i+1,…,g_i+j))     ⟼(σ+σ';  g_1,…,g_i+j)

and

    Ψ_k,m :  Σ_k≀ (Σ_m≀ G)        →    Σ_k m≀ G 
    
     (σ;  (τ_1;  h^1),…,(τ_k;  h^k))     ⟼(σ♮(τ_1,…,τ_k);  h^1+…+h^k) .

These monomorphisms were defined
in Construction <ref>.


 subjectglobal power functor
A global power functor is a global Green functor R 
equipped with maps

    P^m  :  R(G)  → R(Σ_m≀ G)

for all compact Lie groups G and m≥ 1,
called power operations, that satisfy the following relations.


  * (Unit)  P^m(1)=1 for the unit 1∈ R(e).

  * (Identity) P^1= under the identification GΣ_1≀ G
sending g to (1;g).

  * (Naturality) 
For every continuous homomorphism α:K→ G between compact Lie groups
and all m≥ 1 the relation

    P^m∘α^*  =  (Σ_m≀α)^*∘ P^m

holds as maps R(G)→ R(Σ_m≀ K).

  * (Multiplicativity) 
For all compact Lie groups G, 
all m≥ 1 and all classes x,y∈ R(G) the relation

    P^m(x·  y)  =  P^m(x)· P^m(y)

holds in the group R(Σ_m≀ G). 

  * (Restriction)
For all compact Lie groups G, all m >k>0 and all x∈ R(G)
the relation

    Φ^*_k,m-k( P^m(x) ) =  P^k(x)× P^m-k(x)

holds in R((Σ_k≀ G)×(Σ_m-k≀ G)).

  * (Transitivity)
For all compact Lie groups G, all k,m≥ 1 and all x∈ R(G) the relation

    Ψ^*_k,m( P^k m(x) ) =  P^k( P^m (x) )

holds in R(Σ_k≀(Σ_m ≀ G)).

  * (Additivity) 
For all compact Lie groups G, all m≥ 1, and all x,y∈ R(G)
the relation

    P^m(x+y)  =  ∑_k=0^m _k,m-k (P^k(x)× P^m-k(y))

holds in R(Σ_m≀ G), where _k,m-k is the transfer associated
to the embedding Φ_k,m-k:(Σ_k≀ G) × (Σ_m-k≀ G)→Σ_m≀ G defined in (<ref>). Here P^0(x) is the multiplicative unit 1.

  * (Transfer) For every closed subgroup H of 
a compact Lie group G  and every m≥ 1 the relation

    P^m∘_H^G  =  _Σ_m≀ H^Σ_m≀ G∘ P^m

holds as maps R(H)→ R(Σ_m≀ G).

A morphism of global power functors is a morphism of
global Green functors that also commutes with the power operations.subjectmorphism!of global power functors
We write  for the category of global power functors.symbol   - category of global power functors 



In a global power functor the relation P^m(0)=0
also holds for every m≥ 1 and all G.
Indeed, the additivity and unit relations give

    ∑_k=1^m-1 _k,m-k (P^k(0)× 1 ) =  P^m(0+ 1) - P^m(0) - P^m(1)
      =  -P^m(0) .

Starting from P^1(0)=0 this shows inductively that P^m(0)=0.  



[Global power functors versus global Tambara functors]

A global power functor gives rise to two underlying global power monoids, 
the additive and the multiplicative one.
As we explained in Construction <ref>, 
applied to the multiplicative global power monoid,
the power operations P^m lead to `multiplicative transfers',
N_H^G:R(H)→ R(G) that are called norm maps,
for every subgroup H of finite index in G.
For the convenience of the reader, we recall the construction of the norm maps.
We suppose that [G:H]=m,
and we choose an H-basis of G, i.e., an ordered m-tuple 
g̅=(g_1,…,g_m) of elements in disjoint H-orbits such that

    G  =  ⋃_i=1^m  g_i H  .

The wreath product Σ_m≀ H acts freely and transitively 
from the right on the set of all such H-bases of G, by the formula

    (g_1,…,g_m)· (σ; h_1,…,h_m) =  
    (g_σ(1) h_1,…,g_σ(m)h_m) .

We obtain a continuous homomorphism Ψ_g̅:G→Σ_m≀ H by requiring that

    γ·g̅ =  g̅·Ψ_g̅(γ)   .

The norm N_H^G:R(H)→ R(G)subjectnorm mapsymbolN_H^G - norm map is then the composite

    R(H)   P^m  R(Σ_m≀ H)   Ψ_g̅^*  R(G)  .

Any other H-basis is of the form
g̅ω for a unique ω∈Σ_m≀ H. 
We have Ψ_g̅ω=c_ω∘Ψ_g̅ as maps G→Σ_m≀ H,
where c_ω(γ)=ω^-1γω. Since inner automorphisms
induce the identity in any Rep-functor, we have

    Ψ_g̅^* = Ψ_g̅ω^*  :  R(Σ_m≀ H)  → R(G) .

So the norm N_H^G does not depend on the choice of basis g̅. 

The norms maps satisfy a number of important relations,
by Proposition <ref> applied to the multiplicative monoid
of the global power functor R. There relations – turned into multiplicative notation –
are as follows.


  * (Transitivity) 
We have N_G^G=_R(G) and for nested subgroups
H⊆ G⊆ F of finite index the relation

    N_G^F ∘ N_H^G  =   N_H^F

holds as maps  R(H)→ R(F).

  * (Multiplicative double coset formula)
For every subgroup K of G (not necessarily of finite index)
the relationsubjectdouble coset formula!for norm map

    ^G_K∘ N_H^G   =  ∏_[g]∈ K\ G/H
    N^K_K∩^g H∘ g_⋆∘^H_K^g∩ H

holds as maps R(H)→ R(K).
Here [g] runs over a set of representatives of the finite
set of K-H-double cosets.

  * (Inflation)
For every continuous epimorphism  α:K→ G of compact Lie groups
the relation

    α^*∘ N_H^G  =  N_L^K∘ (α|_L)^*

holds as maps from R(H)→ R(K), where L=α^-1(H).

  * For every m≥ 1 the m-th power operation can be recovered as

    P^m  =  N_K^Σ_m≀ G∘ q^*  ,

where K is the subgroup of Σ_m≀ G
consisting of all (σ; g_1,…,g_m) such that σ(m)=m
and q:K→ G is defined by q(σ; g_1,…,g_m)=g_m.

In particular, the power operations define the norm maps,
but they can also be reconstructed from the norm maps.
So the information in a global power functor could be packaged
in an equivalent but different way using norm maps instead of
power operations. The algebraic structure that arises then
is the global analog of a TNR-functorsubjectTNR-functor 
in the sense of Tambara <cit.>, nowadays also called
a Tambara functor;subjectTambara functor|seeTNR-functor
here the acronym stands form `Transfer, Norm and Restriction'.
This observation can be stated as an equivalence of categories 
between global power functors and a certain category of `global TNR functors'; 
we shall not pursue this further.
Our reason for favoring power operations over norm maps
is that they satisfy explicit and intuitive formulas 
with respect to the rest of the structure 
(restriction, transfer, sum, product,…).
The norm maps also satisfy universal formulas when applied to sums
and transfers, but I find them harder to describe and to remember.

For a fixed finite group G, Brun <cit.> 
has constructed norm maps on the 0-th equivariant homotopy group Mackey functor 
of every commutative orthogonal G-ring spectrum, and he showed
that this structure is a TNR-functor. 
So when restricted to finite groups, the global power functor structure
on _0(R) for an ultra-commutative ring spectrum,
obtained in the following Theorem <ref>, 
could also be deduced by using Brun's TNR-structure 
for the underlying orthogonal G-ring spectrum R_G, and then turning the norm maps
into power operations as (<ref>).
However, Brun's construction is rather indirect and this would hide
the simple and explicit nature of the power operations.



In order to show that the 0-th equivariant homotopy group functor
of an ultra-commutative ring spectrum satisfies the transfer axiom (viii)
of a global power functor, we study the interplay between power operations,
the Wirth­müller isomorphism and the degree shifting transfer.
To state the results we first have to generalize power operations
from equivariant homotopy groups to equivariant homology theories.

We let R be an orthogonal spectrum, G a compact Lie group and A
a based G-space. We define the G-equivariant R-homology group of A 
as the groupsubjectequivariant homology group!of an orthogonal spectrum

    R_0^G(A)  =  π_0^G(R A) =  _V∈ s(_G)  [S^V, R(V) A ]^G .

Every continuous group homomorphism α:K→ G induces a 
restriction homomorphism

    α^*  :  R_0^G(A)  → R_0^K(α^*(A))

that generalizes the restriction homomorphism α^*:π_0^G(R)→π_0^K(R).
Again α^* is defined by applying restriction of scalars to any representative
of a given equivariant homology class.

Now we let R be an orthogonal ring spectrum (not necessarily commutative).
Then the equivariant homology theories represented by R inherit 
multiplications in the form of bilinear maps

    × :  R_0^G(A) × R_0^G(B) → R_0^G(A B)  .

We define this pairing simply as the composite

    R_0^G(A) × R_0^G(B) =  π_0^G(R A) ×π_0^G   (R B)  ×  π_0^G((R A)  (R B))
        μ_*   π_0^G(R A  B) =  R_0^G(A B)  ,

where the first map is the homotopy group pairing of 
Construction <ref>
and μ:(R A)  (R B)→ R A  B
stems from the multiplication of R.
For A=B=S^0 this construction reduces to the pairings 
of equivariant homotopy groups (<ref>).

Now we suppose that the ring spectrum R is ultra-commutative.
Given a based G-space A, we write A^(m)=A^ m for its m-fold
smash power, which is naturally a based (Σ_m≀ G)-space.
Then we define the m-th power operation

    P^m  :  R_0^G(A) → R_0^Σ_m≀ G(A^(m))

by the obvious generalization of (<ref>): 
the operation P^m takes the class
represented by a based G-map f:S^V→ R(V) A, for some G-representation V,
to the class of the (Σ_m≀ G)-map

    S^V^m = (S^V)^(m)  f^(m)    (R(V) A)^(m)
     shuffle     R(V)^(m) A^(m) μ_V,…,V A^(m) R(V^m) A^(m) ,

where μ_V,…,V is the iterated multiplication map of R.
subjectpower operation!in equivariant homology groups
We omit the straightforward verification that the power operations 
in equivariant R-homology are compatible with restriction maps:
for every continuous homomorphism α:K→ G between compact Lie groups
and every based G-space A, the relation

    P^m∘α^*  =  (Σ_m≀α)^*∘ P^m

holds as maps from R_0^G(A) to R_0^Σ_m≀ K( α^*(A)^(m)),
exploiting that (Σ_m≀α)^*(A^(m))=α^*(A)^(m)
as (Σ_m≀ K)-spaces.

The following proposition makes precise how the power operations 
in equivariant R-homology interact with 
the Wirthmüller isomorphism of Theorem <ref>.
To give the precise statement we have to introduce additional notation.
We let H be a closed subgroup of a compact Lie group G.
As before we let L = T_e H( G/H) denote the tangent H-representation, 
the tangent space of G/H at the distinguished coset e H. 
We write

    γ :  (G/H)^m   (Σ_m≀ G) / (Σ_m≀ H)  , (g_1 H,…, g_m H) ⟼ (1; g_1,…, g_m)·(Σ_m≀ H)

for the distinguished (Σ_m≀ G)-equivariant diffeomorphism.
The differential of γ at (e H,…,e H) 
is a (Σ_m≀ H)-equivariant linear isometry

    (d γ)_(e H,…,e H) :  L^m   
    T_e(Σ_m≀ H)( (Σ_m≀ G) / (Σ_m≀ H) )  .

In the next proposition and its corollaries, we will use this
equivariant isometry to identify L^m with the tangent representation
of Σ_m≀ H inside Σ_m≀ G.



 
Let R be an ultra-commutative ring spectrum and H 
a closed subgroup of a compact Lie group G. Then the following diagram commutes

    @C=15mm@R=7mm 
    R_0^G(G/H_+) [r]^-^G_H_-[d]_P^m   
    R_0^H(S^L) [d]^P^m
    
    R_0^Σ_m≀ G(( G / H)^m_+)  [d]^_γ_*     
    R_0^Σ_m≀ H( ( S^L)^(m)) [d]^
    
    R_0^Σ_m≀ G(  (Σ_m≀ G) /  (Σ_m≀ H)_+ ) 
    [r]_-^Σ_m≀ G_Σ_m≀ H^-    R_0^Σ_m≀ H(S^L^m)

where the horizontal maps are the respective Wirthmüller isomorphisms.subjectWirthmüller isomorphism


We choose a slice as in the definition of the Wirthmüller map in
Construction <ref>, 
i.e., a smooth embedding s:D(L)→ G that satisfies

    s(0)=1 ,    s(h· l)=h· s(l)· h^-1     and     s(-l)=s(l)^-1

for all (l,h)∈ D(L)× H, and such that the differential at 0 of the composite

    D(L)   s  G  proj G/H

is the identity.
The collapse map

    λ_H^G =  l_H^G/H  :  G/H_+  → S^L

is then given by the formula

    λ_H^G(g H)  =  
      l / (1-|l|)       if g=s(l)· h with (l,h)∈ D(L)× H, and 
      ∗    if g is not of this form.

We define a slice for the pair (Σ_m≀ G,Σ_m≀ H)
from the slice s for the pair (G,H), namely as the smooth embedding

    s̅ :  D(L^m) → Σ_m≀ G  ,   s̅(l_1,…, l_m) =  (1;  s(l_1),…, s(l_m)) .

Clearly, s̅(0,…,0) is the multiplicative unit,

    s̅(-l_1,…,-l_m)    =  (1;  s(- l_1),…, s(-l_m))
       =  (1;  s(l_1)^-1,…, s(l_m)^-1) =  s̅(-l_1,…,-l_m)^-1 ,

and

    s̅((σ^-1; h_1,…,   h_m)·(l_1,…, l_m)) 
    = s̅(h_σ(1)l_σ(1),…,h_σ(m) l_σ(m))
       = (1; s(h_σ(1)l_σ(1)),…,s(h_σ(m) l_σ(m)))
       = (1; h_σ(1)s(l_σ(1))h_σ(1)^-1,…,
    h_σ(m)s(l_σ(m))h_σ(m)^-1)
       =  (σ^-1; h_1,…,h_m)· (1; s(l_1),…, s(l_m))· 
    (σ; h_σ(1)^-1,…,h_σ(m)^-1)
       =  (σ^-1; h_1,…,h_m)·s̅(l_1,…, l_m)· 
    (σ^-1; h_1,…,h_m)^-1 ,

for all (l_1,…,l_m)∈ D(L^m) and all
(σ^-1; h_1,…,h_m)∈Σ_m≀ H.
Finally, the differential of the composite

    D(L^m)  s̅  Σ_m≀ G  proj
    (Σ_m≀ G) / (Σ_m≀ H)  γ^-1 (G/H)^m

is the identity, so we have indeed defined a slice.
We let

    λ_Σ_m≀ H^Σ_m≀ G : (Σ_m≀ G) /  (Σ_m≀ H)_+ → S^L^m

denote the collapse map based on the slice s̅.
The composite

    (G / H )^m_+   γ_+ (Σ_m≀ G) /  (Σ_m≀ H)_+ λ_Σ_m≀ H^Σ_m≀ G S^L^m

sends a point ( s(l_1)H,…, s(l_m)H) with (l_1,…,l_m)∈ D(L^m) to the point

    ( 1-√(|l_1|^2+…+|l_m|^2))^-1·(l_1,…,l_m)   ;

all other points of (G/H)^m are sent to the basepoint at infinity.
A scaling homotopy thus witnesses that the following
diagram commutes up to (Σ_m≀ H)-equivariant based homotopy:

    @C=15mm 
    (G / H )^m_+ [r]^-(λ_H^G)^(m)[d]^_γ_+   
    (S^L )^(m)[d]^
    
    (Σ_m≀ G) /  (Σ_m≀ H)_+
    [r]_-λ_Σ_m≀ H^Σ_m≀ G    S^L^m

Now we contemplate the diagram:

    @C=8mm@R=7mm 
    R_0^G( G / H_+) [r]^-^G_H[d]_P^m   
    R_0^H( G / H_+) [r]^-(λ_H^G)_*[d]_P^m   
    R_0^H(S^L) [d]^P^m
    
    R_0^Σ_m≀ G(( G / H ) ^m_+) 
    [r]^-^Σ_m≀ G_Σ_m≀ H[d]^_ (γ_+)_*     
    R_0^Σ_m≀ H(( G / H)^m_+) 
    [r]^- ((λ_H^G)^m)_*[d]_(γ_+)_*^    
    R_0^Σ_m≀ H(( S^L)^(m)) [d]^
    
    R_0^Σ_m≀ G( (Σ_m≀ G) / (Σ_m≀ H) _+ ) 
    [r]_-^Σ_m≀ G_Σ_m≀ H   
    R_0^Σ_m≀ H((Σ_m≀ G) / (Σ_m≀ H) _+) 
    [r]_-(λ_Σ_m≀ H^Σ_m≀ G)_*    R_0^Σ_m≀ H( S^L^m)

The two squares on the left commute by compatibility of power operations
with restriction and by naturality of restriction.
The upper right square commutes by naturality of power operations.
The lower right square commutes by the previous paragraph.
Since the upper and lower horizontal composites are 
the respective Wirthmüller maps, this proves the proposition.


A direct consequence of the previous proposition is that power
operations are compatible with dimension shifting 
and degree zero transfer maps.


Let R be an ultra-commutative ring spectrum and H 
a closed subgroup of a compact Lie group G. Then the following two diagrams commute:

    @C=10mm@R=6mm 
    R_0^H(S^L) [d]_P^m[r]^-_H^G   π_0^G(R)  [dd]^P^m   π_0^H(R)   [dd]_P^m[r]^-_H^G   π_0^G(R) [dd]^P^m
    
    R_0^Σ_m≀ H( (S^L)^(m))   [d]_   
    
    R_0^Σ_m≀ H(S^L^m)  [r]_-_Σ_m≀ H^Σ_m≀ G   π_0^Σ_m≀ G(R)    π_0^Σ_m≀ H(R)  [r]_-_Σ_m≀ H^Σ_m≀ G   π_0^Σ_m≀ G(R)



By Theorem <ref>, the external transfer

    G⋉_H-  :  R_0^H(S^L) =  π_0^H(R S^L) 
     → π_0^G(R G/H_+)  =  R_0^G(G/H_+)

is inverse to the Wirthmüller isomorphism, 
up to the effect of the antipodal map of S^L.
The identification (S^L)^(m) S^L^m takes (S^-_L)^(m)
to S^-_L^m, so the following diagram commutes by naturality of power operations:

    @C=12mm@R=7mm 
    R_0^H(S^L) [r]^-ε_L_-[d]_P^m   
    R_0^H( S^L) [d]^P^m
    
    R_0^Σ_m≀ H( ( S^L)^(m))  [d]_     
    R_0^Σ_m≀ H(( S^L)^(m)) [d]^
    
    R_0^Σ_m≀ H( S^L^m) 
    [r]_-ε_L^m^-    R_0^Σ_m≀ H(S^L^m)

Stacking this diagram next to the commutative diagram of Wirthmüller
isomorphisms given in Proposition <ref>,
and reading the composite diagram backwards yields the commutativity 
of the left part of the following diagram:

    @C=12mm@R=7mm 
    R_0^H(S^L) [d]_P^m[r]_-G⋉_H-@/^1pc/[rr]^(.3)_H^G    R_0^G( (G/H)_+)  [d]^P^m[r]_-(p_+)_*   π_0^G(R)   [d]^P^m
    
    R_0^Σ_m≀ H( (S^L)^(m))   [d]_     
    R_0^Σ_m≀ G((G/H)^m_+)[d]^γ_*_[r]_-(p^m_+)_*   π_0^Σ_m≀ G(R) @=[d] 
    
    R_0^Σ_m≀ H(S^L^m)  [r]^-Σ_m≀ G⋉_Σ_m≀ H-@/_1pc/[rr]_(.3)_Σ_m≀ H^Σ_m≀ G   
    R_0^Σ_m≀ G(  (Σ_m≀ G) / (Σ_m≀ H)_+ ) [r]^-(p_+)_*   π_0^Σ_m≀ G(R)

The dimension shifting transfer _H^G : R_0^H(S^L) →π_0^G(R)
is the composite of the external transfer 
and the effect of the unique G-map p:G/H→∗.
So the first claim follows from the commutativity of the right part of the
above diagram.
The degree zero transfer is obtained from the dimension shifting transfer
by precomposing with the effect of the map S^0→ S^L,
the inclusion of the origin into the tangent representation.
If we raise the inclusion of the origin of S^L to the m-th power,
the canonical homeomorphism (S^L)^(m)→ S^L^m identifies it with
the inclusion of the origin of S^L^m.
So the power operations are also compatible with degree zero transfers.





Much of the next result is contained, at least implicitly,
in Greenlees' and May's construction 
of norm maps <cit.>,
simply because an ultra-commutative ring spectrum is an example of
a `𝒢ℐ_*-FSP' 
in the sense of <cit.>.


 
Let R be an ultra-commutative ring spectrum.
The power operations (<ref>) make the
global functor _0(R) into a global power functor.

 The properties (i) through (vi) only involve the
multiplication, power operations and restriction maps,
so they are special cases of Proposition <ref> 
for the multiplicative ultra-commutative monoid Ω^∙ R.
The transfer relation (viii) is proved in Corollary <ref>.
The most involved argument remaining is required for 
the additivity formula (vii), identifying the behavior of power operations on sums.

We first show an external version of the additivity relation.
We consider two equivariant homology classes x, y∈π_0^G(R).
Since equivariant homotopy groups take wedges to direct sums there is
a unique class

    x⊕ y ∈ π_0^G(R{1,2}_+)

 such that

    p^1_*(x⊕ y)  =  x      and    
    p^2_*(x⊕ y)  =  y  ,

where p^1,p^2:{1,2}_+→ S^0 are the projections determined by (p^i)^-1(0)={i}.
We write Σ_k,m-k≀ G for the image of the homomorphism
Φ_k,m-k:(Σ_k≀ G)×(Σ_m-k≀ G)→Σ_m≀ G
and let

    ψ^k :  (Σ_m≀ G) / (Σ_k,m-k≀ G) → {1,2}^m

be the embedding of (Σ_m≀ G)-sets that sends the preferred coset
e(Σ_k,m-k≀ G) to the point

    (1,…,1_k,2,…,2_m-k) ∈ {1,2}^m .

Here Σ_m≀ G acts on {1,2}^m through the projection to Σ_m,
by permutation of coordinates.
The orbits of the (Σ_m≀ G)-set {1,2}^m are precisely 
the images of the maps ψ^0,…,ψ^m.
We will show the relation

    P^m(x⊕ y) =  ∑_k=0^m   (ψ^k_+)_*( 
    (Σ_m≀ G)⋉_Σ_k,m-k≀ G 
     (P^k(x)× P^m-k(y)))

in the group π_0^Σ_m≀ G(R{1,2}^m_+).
Since π_0^Σ_m≀ G(R -) is additive on wedges, it suffices to show the relation
after projection to each  (Σ_m≀ G)-orbit of {1,2}^m.
So we let 

    ψ̅^k :  {1,2}^m_+ → 
    (Σ_m≀ G) / (Σ_k,m-k≀ G)_+

be the right inverse to ψ^k_+ that sends the other orbits to the
basepoint, i.e., ψ̅^k∘ψ^j_+ is constant for j k.
The relation (<ref>) thus follows if we can show

    ψ̅^k_*( P^m(x⊕ y))  =  
    (Σ_m≀ G)⋉_Σ_k,m-k≀ G (P^k(x)× P^m-k(y))

in the group 

    π_0^Σ_m≀ G( R (Σ_m≀ G)/(Σ_k,m-k≀ G)_+)

for all 0≤ k≤ m. 
We apply the Wirthmüller isomorphism, i.e., the composite

    π_0^Σ_m≀ G(R (Σ_m≀ G)/(Σ_k,m-k≀ G)_+)  
        ^Σ_m≀ G_Σ_k,m-k≀ G
    π_0^Σ_k,m-k≀ G(R   (Σ_m≀ G)/(Σ_k,m-k≀ G)_+)  (l_k)_* π_0^Σ_k,m-k≀ G(R)  .

Here l_k:(Σ_m≀ G)/(Σ_k,m-k≀ G)_+ → S^0
is the projection to the preferred coset.
We obtain

    (l_k)_*(^Σ_m≀ G_Σ_k,m-k≀ G   (ψ̅^k_*( P^m(x⊕ y))))  =  
    (l_k∘ψ̅^k)_*(^Σ_m≀ G_Σ_k,m-k≀ G ( P^m(x⊕ y))) 
       =   ( (p^1)^(k) (p^2)^(m-k))_*( P^k(x⊕ y)× P^m-k(x⊕ y))
       =   P^k( p^1_*(x⊕ y) )× P^m-k(p^2_*(x⊕ y) )=   P^k(x)× P^m-k(y) .

Since the Wirthmüller isomorphism is inverse to the external transfer
(compare Theorem <ref>),
this proves (<ref>),
and hence (<ref>). 

Now we obtain the additivity relation by naturality for the fold map
∇:{1,2}_+→ S^0 with ∇(1)=∇(2)=0.
Then

    P^m(x+y)    =  P^m(∇_*(x⊕ y))  =  
    (∇^(m))_*(P^m(x⊕ y)) 
    
    _(<ref>)   =  ∑_k=0^m   (∇^(m)∘ψ^k_+)_*( (Σ_m≀ G)⋉_Σ_k,m-k≀ G
     (P^k(x)× P^m-k(y)))   
       =  ∑_k=0^m  _Σ_k,m-k≀ G^Σ_m≀ G (P^k(x)× P^m-k(y)) .

The third equation uses that the composite

    (Σ_m≀ G) / (Σ_k,m-k≀ G)_+  ψ^k_+  {1,2}^m_+∇^(m) (S^0)^(m) =  S^0

sends all of (Σ_m≀ G) / (Σ_k,m-k≀ G) to the non-basepoint.
 



subjectH-infinity@H_∞-ring spectrum|(
[Relation to classical power operations] 
We owe an explanation how power operations for ultra-commutative ring spectra
refine the classical power operations
defined in the (non-equivariant) cohomology theory represented by an
H_∞-ring spectrum. 
We recall from <cit.>
that an H_∞-structure is an algebra structure over the monad

    L :   →

on the stable homotopy category that can be obtained
by deriving the `symmetric algebra' monad

    :   →

on the category of orthogonal spectra (whose algebras are
commutative orthogonal ring spectra).
This is not the full truth, because 
Bruner, May, McClure and Steinberger use a different model for
the stable homotopy category, so strictly speaking one would have 
to translate the relevant parts of <cit.>
to the context of orthogonal spectra.
If we did that, the derived functor of the m-symmetric power of orthogonal spectra
would be modeled by the 
m-th extended powersubjectextended power!of an orthogonal spectrum

    D_m X  =  (EΣ_m)_+_Σ_m X^ m .

Specifying an H_∞-structure on an orthogonal spectrum E 
thus amounts to specifying morphisms,
in the non-equivariant stable homotopy category,

    μ_m  :  D_m E  → E

from the m-th extended power to E;
the algebra structure over the monad L then translates
into a specific collection of relations among the morphisms μ_m
that are spelled out in <cit.>.

For every space A, the H_∞-structure gives rise to power operations

    𝒫_m  :  E^0(A) → E^0(BΣ_m× A)

in E-cohomology 
defined in <cit.> as 
the following composite:

    E^0(A) =  [Σ^∞_+ A   , E]   D_m  [D_m( Σ^∞_+ A), D_m E]  [D_m(Σ^∞_+ A),μ_n] [D_m(Σ^∞_+ A), E] 
        [Σ^∞_+ (D_m A), E]  =  E^0(D_m A)  E^0(EΣ_m×_Σ_mΔ)  E^0(BΣ_m× A) .

Here [-,-] denotes the morphism group in the stable homotopy category ,
D_m A=EΣ_m×_Σ_mA^m is the space level extended power,  
and Δ:A→ A^m is the diagonal.
Depending on the context, the power operations (<ref>)
are often processed further; in favorable cases,
E^0(BΣ_m× A) can be explicitly described as a functor of E^0(A),
and the power operations can be translated into a specific kind of algebraic structure.

Now we let R be an ultra-commutative ring spectrum. 
Then the underlying H_∞-structure
is given by the composite morphism
 
    D_m R  =  (EΣ_m)_+_Σ_m R^ m →Σ_m R^ m mult R

where the first morphism collapses EΣ_m to a point and the second
map is induced by the iterated multiplication R^ m→ R.
The definition (<ref>) of the power operations
on equivariant homotopy groups
directly extends to power operations

    P^m  :  R^0_G(A)  → R^0_Σ_m≀ G(A^m)

in the equivariant cohomology of a G-space A.
The operation P^m takes the class
represented by a based G-map f:S^V A_+→ R(V), for some G-representation V,
to the class of the (Σ_m≀ G)-map

    S^V^m A^m_+   (S^V A_+)^ m  f^ m R(V)^ m  μ_V,…,V  R(V^m)  ,

where μ_V,…,V
is the iterated, (Σ_m≀ G)-equivariant multiplication map of R.
A forgetful homomorphism

    R^0_G(A) → R^0(E G×_G A)

is defined as the composite

    R^0_G(A) =  Σ^∞_+ _G,VA, R  U     [U(Σ^∞_+ _G,VA), R] 
     
    =      [Σ^∞_+ (E G×_G A), R]  =  R^0(E G×_G A)   ,

where U:→ is the forgetful functor from the 
global stable homotopy category to the non-equivariant stable homotopy category.
Then the following diagram commutes:

    @C=12mm@R=8mm 
    R^0(A) [d]_P^m[r]^-D_m   
    [D_m( Σ^∞_+ A), D_m R]  [d]^(μ_m)_*
    
    R^0_Σ_m(A^m) [r]^-U [d]_R^0_Σ_m(Δ)    
    R^0(EΣ_m×_Σ_m A) [d]^R^0(EΣ_m×_Σ_mΔ)
    
    R^0_Σ_m(A) [r]_-U      R^0(BΣ_m× A)

The composite through the upper right corner is the H_∞-power operation _m;
so this diagram makes precise in which way power operations 
for ultra-commu­tative ring spectra refine power operations for H_∞-ring spectra.



[G_∞-ring spectra]
As we recalled in the previous remark, non-equivariant power operations on a ring valued
cohomology theory already arise from a weaker structure 
than a commutative multiplication (or equivalently E_∞-multiplication):
all that is needed is an H_∞-structure,
compare <cit.>.
This suggests a global analog of an H_∞-structure that, for lack of better name,
we call a G_∞-structure.subjectG-infinity@G_∞-ring spectrum
For the formal definition we exploit the 
global model structure for ultra-commutative ring spectra,
to be established in Theorem <ref> below.
This model structure is obtained by lifting the positive global model structure 
on the category of orthogonal spectra 
(see Proposition <ref>)
along the forgetful functor. In particular, the
free and forgetful functor form a Quillen adjoint functor pair:

    @C=10mm :   @<.4ex>[r]     ucom  :  U @<.4ex>[l]

Every Quillen adjoint pair between model categories derives to
an adjoint functor pair between the homotopy categories, 
see <cit.> or <cit.>. 
In our situation this provides the derived adjunction:

    @C=10mm L_ :   @<.4ex>[r]       (ucom)  :  (U) @<.4ex>[l]

Since the forgetful functor is fully homotopical, it does not even have to be derived.
The composite

    =  (U)∘ L_ :   →

is then canonically a monad on the global stable homotopy category, 
whose algebras we call G_∞-ring spectra.

The underlying non-equivariant homotopy type of a G_∞-ring spectrum
comes with an H_∞-structure, so we arrive at a square of 
forgetful functors between categories of structured ring spectra
with different degrees of commutativity:

    (ucom) [r][d]     
    (G_∞-ring spectra) [d] 
    ^π_*-iso(comm. ring spectra) [r]     
    (H_∞-ring spectra)

We emphasize that, like H_∞-ring spectra,
the category of G_∞-ring spectra is not the homotopy category
of any natural model category.

subjectH-infinity@H_∞-ring spectrum|)




[Units of an ultra-commutative ring spectrum]subjectunits!of an ultra-commutative ring spectrum
In Example <ref>subjectunits!of an orthogonal monoid space!naive
we defined the naive units of an orthogonal monoid space.
When R is an orthogonal ring spectrum, then the naive units
of the multiplicative orthogonal monoid space Ω^∙ R satisfy

    π_0^G( (Ω^∙ R)^n×)  =  { x ∈π_0^G(R) |  ^G_e(x) is a unit in π_0^e(R)} ,

the multiplicative submonoid of π_0^G(R) of elements that become
invertible when restricted to the trivial group.
One should beware that these naive units may contain non-invertible elements,
i.e., the orthogonal monoid space (Ω^∙ R)^n× need not be group-like.

When the ring spectrum R is ultra-commutative, then there is a more
refined construction 

    G L_1(R)  =  (Ω^∙ R)^× ,

the global unitssymbolG L_1(R) - global units of the ultra-commutative ring spectrum R 
of R. Indeed, if R is ultra-commutative, then Ω^∙ R is an
ultra-commutative monoid, so we can form the `true' global units,
the homotopy fiber of the multiplication morphism, 
see Construction <ref>.
Then G L_1(R) is a group-like ultra-commutative monoid 
and for every compact Lie group G,

    π_0^G( G L_1(R) )  =  
    (π_0^G(R))^× ,

the multiplicative submonoid of units of the commutative ring π_0^G(R).
Moreover, the power operations in _0(R) correspond to the power operations
in _0(G L_1(R)).


In the non-equivariant context,
G L_1(R) is an infinite loop space, i.e., weakly equivalent
to the 0-th space of an Ω-spectrum of units.
This fact has a global generalization as follows.
As we hope to explain elsewhere, every ultra-commu­tative monoid M has a
global deloopingsubjectglobal delooping M,
an orthogonal spectrum that is a -global Ω-spectrum. 
It also comes with a natural morphism of orthogonal spaces
M→Ω^∙ ( M) that is a -global equivalence 
whenever M is group-like.
Since G L_1(R) is an ultra-commutative monoid, it
has a global deloopingsubjectglobal delooping 

    g l_1(R) =  ( G L_1(R) )  .

Since the ultra-commutative monoid G L_1(R) is group-like, the morphism 

    ξ :   G L_1(R)  → Ω^∙ (  (G L_1(R) ) ) =  Ω^∙ ( g l_1(R) )

is a -global equivalence of orthogonal spaces. 
For every compact Lie group G, this induces a map

    (π_0^G( R))^×     π_0^G(G L_1(R)) 
        π_0^G( ( G L_1(R))^c)   _0(ξ)  π_0^G(Ω^∙ ( g l_1(R) ))  =  π_0^G(g l_1(R))  .

These maps are compatible with restriction
along continuous homomorphisms and they are bijective whenever G is finite.
Moreover, the maps take the multiplication respectively norm operations 
in (_0( R))^× 
to addition respectively finite index transfers in _0(g l_1(R)).






[Picard groups] The global units G L_1(R) of an 
ultra-commu­tative ring spectrum R ought to have an interesting delooping (R)symbol(R) - Picard monoid of an ultra-commutative ring spectrum
subjectPicard group! of an ultra-commutative ring spectrum
that records the information about invertible modules over the
equivariant ring spectra underlying R. At present I have no construction
of this delooping as an ultra-commutative monoid, but I 
describe the evidence for expecting its existence.

For every compact Lie group G the underlying orthogonal G-ring spectrum R_G
of R has a symmetric monoidal model category of modules,
i.e., orthogonal G-spectra with an action by R (where G acts trivially on R).
The equivalences we consider here are R-linear morphisms 
that are _*-isomorphisms of underlying orthogonal G-spectra;
the construction of such a symmetric mo­noidal model 
category can be found in <cit.>.
We let

    (R)(G) =  ((R_G-mod))

be the resulting Picard group, i.e., the set of isomorphism
classes, in the homotopy category of R_G-modules,
of objects that are invertible under the derived smash product.
For a continuous group homomorphism α:K→ G the restriction functor
α^*:R_G-mod→ R_K-mod
derives to a strong symmetric monoidal functor

    Rα^* :  (R_G-mod) → (R_K-mod) .

So Rα^* preserves invertibility and induces a group homomorphism

    α^* :  (R)(G)  → (R)(K) .

For a second homomorphism β:L→ K the functors
(Rβ^*)∘(Rα^*) and R(α∘β)^*
are naturally isomorphic.
Moreover, for every element g∈ G the restriction functor R (c_g)^*
is naturally isomorphic to the identity functor of
(R_G-mod), via left multiplication by g.
So the assignment G↦(R)(G) becomes a functor

    (R) :  ^ →  .

But the ultra-commutativity gives more. For every finite index subgroup H≤ G,
the norm construction of Hill, Hopkins and Ravenel
derives to a strong symmetric monoidal functor

    N_H^G  :  (R_H-mod) → (R_G-mod) ,

compare <cit.>.
So also N_H^G preserves invertibility and induces a group homomorphism

    N_H^G :  (R)(H)  → (R)(G) .

These norm maps are transitive and they extend the abelian -monoid
to a global power monoid (R). 

We expect that there is a `natural' ultra-commutative monoid (R)
such that _0((R))(R) as global power monoids
and such that Ω ((R)) is globally equivalent, as an ultra-commutative
monoid, to G L_1(R). The G-fixed points ((R)(_G))^G
ought to have the homotopy type, as an E_∞-space,
of the nerve of the category of invertible R_G-modules
and _*-isomorphisms. Despite the strong evidence for its existence,
I cannot presently construct (R) as an ultra-commutative monoid
in our formalism.



[Free global power functors]subjectglobal power functor!free  
For a compact Lie group K we construct a 
free global power functor C_K generated by K. 
The underlying global functor is

    C_K  =  ⊕_m≥ 0 (Σ_m≀ K,-)  ,

the direct sum of the global functors represented by the wreath products 
Σ_m≀ K, including the trivial group Σ_0≀ K=e.
The multiplication μ: C_K C_K→ C_K
that makes this into a global Green functor restricted to the (m,n)-summand
is the morphism

    (Σ_m≀ K,-)(Σ_n≀ K,-)  → C_K

that corresponds, via the universal property of the box product, 
to the bimorphism with (G,G')-component

    (Σ_m≀ K,G)(Σ_n≀ K,G')   ×     ((Σ_m≀ K)×(Σ_n≀ K),G× G') 
    (Φ^*_m,n,G× G')    (Σ_m+n≀ K,G× G')  incl C_K(G× G') ;

here Φ^*_m,n
is the restriction map associated to the embedding (<ref>)   

    Φ_m,n :  (Σ_m≀ K)×(Σ_n≀ K)  → Σ_m+n≀ K  .

The multiplication is associative because

    Φ_k+m,n∘(Φ_k,m×(Σ_n≀ K))    = Φ_k,m+n∘((Σ_k≀ K)×Φ_m,n)  : 
       (Σ_k≀ K)×(Σ_m≀ K)× (Σ_n≀ K) → Σ_k+m+n≀ K .

The multiplication is commutative because the group homomorphisms

    Φ_m,n , Φ_n,m∘τ_Σ_m≀ K, Σ_n≀ K : (Σ_m≀ K)×(Σ_n≀ K)  → Σ_m+n≀ K

are conjugate, so they represent the same morphism in
((Σ_m≀ K)×(Σ_n≀ K), Σ_m+n≀ K).
The unit is the inclusion (e,-)→ C_K of the summand indexed by m=0.

The global Green functor C_K can be made into a
global power functor in a unique way such that the relation

    P^m(1_K)  =  1_Σ_m≀ K

holds in the m-th summand of C_K(Σ_m≀ K), 
where 1_K∈(K,K) and 1_Σ_m≀ K∈(Σ_m≀ K,Σ_m≀ K)
are the identity operations.
Indeed, C_K  is generated as a global functor by the classes 
1_Σ_m≀ K for all k≥ 0, so there is at most one such global power structure,
and every morphism of global power functors out of C_K is determined
by its value on the class 1_K.
The existence of a global power structure on C_K with this property
could be justified purely algebraically, but we show it
by realizing C_K by an ultra-commutative ring spectrum.
As we shall make precise in Proposition <ref> (ii)
below, C_K is indeed freely generated, as a global power functor,
by the class 1_K.

The unreduced suspension spectrumsubjectglobal classifying space  

    Σ^∞_+ (B_K )   ⋁_m≥ 0 Σ^∞_+ B_(Σ_m≀ K)

of the free ultra-commutative monoid
(compare Example <ref>)
generated by a global classifying space of K
is an ultra-commutative ring spectrum. 
According to Proposition <ref>, 
its 0-th homotopy group global functor is given additively by

    _0 ( Σ^∞_+ (B_K ) )   ⊕_m≥ 0 _0( Σ^∞_+ B_(Σ_m≀ K) )   ⊕_m≥ 0 (Σ_m≀ K,-)   .

Under this isomorphism, the stable tautological class 
e_K∈π_0^K ( Σ^∞_+ B_K) maps to the 
generator 1_K∈(K,K).
The class e_K is the stabilization of the unstable tautological class
u_K∈π_0^K( B_ K), whose m-th power is [m](u_K)= u_Σ_m≀ K
in π_0^Σ_m≀ K( (B_ K)), see (<ref>).
The stabilization map σ:_0(B_K)→_0(Σ^∞_+ B_K)
commutes with power operations, so this shows that

    P^m(e_K)  =   P^m(σ^K(u_K)) =  σ^Σ_m≀ K([m](u_K))  =  σ^Σ_m≀ K(u_Σ_m≀ K)  =  e_Σ_m≀ K

in the group π_0^Σ_m≀ K(Σ^∞_+ (B_K)).



We warn the reader that the previous example
of the free global power functor C_K is not
the symmetric algebra, with respect to the box product of global functors,
of the represented global functor (K,-). The issue is that the global functors

    (K,-)^ m/Σ_m  (K^m,-)/Σ_m
         and    (Σ_m≀ K,- )

are typically not isomorphic.
The restriction map ^Σ_m≀ K_K^m∈(Σ_m≀ K,K^m)
induces a morphism of represented global functors

    -∘^Σ_m≀ K_K^m :  (K^m,-)  →(Σ_m≀ K,- )

that equalizes the Σ_m-action on the source because
every permutation of the factors of K^m becomes an inner automorphism
in Σ_m≀ K.
So the morphism factors over a morphism of global functors

    (K^m,-)/Σ_m  → (Σ_m≀ K,- )
  
which, however, is generally not an isomorphism
(already for K=e and m=2).
The box product symmetric algebra

    ⊕_m≥ 0 (K,-)^ m/Σ_m   ⊕_m≥ 0 (K^m,-)/Σ_m

also has a universal property: it is the free global Green functor
generated by K. However, this box product symmetric algebra
does not seem to have natural power operations.







§ COMONADIC DESCRIPTION OF GLOBAL POWER FUNCTORS





In this section we show that the category of global power functors 
is both monadic and comonadic over the category of global Green functors.
We introduce the functor of exponential sequences 
and make it into a comonad on the category of global Green functors.
For a global Green functor R and a compact Lie group G,
Construction <ref> introduces the commutative ring exp(R;G) 
of exponential sequences.
Construction <ref> connects these rings for
varying Lie groups G, making the entire data into a new global Green functor exp(R),
compare Proposition <ref>.
Theorem <ref> extends the functor of
exponential sequences to a comonad on the category of global Green functors,
and Theorem <ref> shows that the category of its coalgebras 
is isomorphic to the category of global power functors.
Proposition <ref> shows that the category of
global power functors is also monadic over the category of global Green functors.
A formal consequence is that the category of global power functors
has all limits and colimits, and that they are created in the category of
global Green functors.
So the relationship of global power functors to global Green functors
is formally similar to the situation for λ-rings,
which are both monadic and comonadic over the category of commutative rings.
When restricted to finite groups, most of the results about the comonad
of exponential sequences are contained in the PhD thesis
of J. Singer <cit.>, a former student of the author.
Also for finite groups (as opposed to compact Lie groups),
this comonadic description has independently been obtained by Ganter <cit.>.


Proposition <ref> discusses localization
of global Green functors at a multiplicative subset of the underlying ring;
while the structure of global Green functor always `survives localization',
this is not generally true for power operations.
Theorem <ref>
exhibits a necessary and sufficient condition so that a localization
of a global power functor inherits power operations.
For localization at a set of integer primes this condition is always satisfied
(Example <ref>), so global power functors can in particular 
be rationalized.




We let R be a global Green functor and G a compact Lie group. We let

    exp(R;G)  ⊂ ∏_m≥ 0 R(Σ_m≀ G)

be the set of exponential sequences,subjectexponential sequence!in a global Green functorsymbolexp(R) - global Green functor of exponential sequences
i.e., of those families (x_m)_m that satisfy x_0=1 in R(Σ_0≀ G)=R(e) and

    Φ_k,m-k^*(x_m)  =  x_k× x_m-k

in R((Σ_k≀ G)×(Σ_m-k≀ G)) for all 0 < k < m,
where 

    Φ_k,m-k :  (Σ_k≀ G) × (Σ_m-k≀ G) → Σ_m≀ G

is the monomorphism (<ref>).   
We define a multiplication on the set exp(R;G) by
coordinatewise multiplication in the rings R(Σ_m≀ G), i.e.,

    (x · y )_m   =  x_m· y_m .

We introduce another binary operation ⊕ on exp(R;G) by

    ( x ⊕ y )_m   =  ∑_k=0^m  _k,m-k (x_k× y_m-k)  ,

where x=(x_m), y=(y_m) and
_k,m-k:R( (Σ_k≀ G)×(Σ_m-k≀ G))→ R(Σ_m≀ G) 
is the transfer associated to the monomorphism Φ_k,m-k. 





 
  Let R be a global Green functor and G a compact Lie group.
  Then the addition ⊕ and the componentwise multiplication
  make the set exp(R;G) of exponential sequences into a commutative ring.


If x and y are exponential sequences, then the relation

    Φ_k,m-k^*( x_m · y_m )    =  Φ_k,m-k^*( x_m ) ·Φ_k,m-k^*(  y_m )
       =    (x_k×  x_m-k)·  (y_k× y_m-k) 
    =  (x_k· y_k)× (x_m-k· y_m-k)

holds in R( (Σ_k≀ G)×(Σ_m-k≀ G));
so the product x· y is again exponential.
The product is associative and commutative since 
all the multiplications in the rings R(Σ_m≀ G) 
have this property. The exponential sequence (1)_m ≥ 0 is a multiplicative unit.

Now we show that the sum, with respect to ⊕,
of two exponential sequences is again exponential.
The key step in this verification is an application of the double coset formula,
for which we need to understand
the (Σ_i×Σ_m-i)-(Σ_k×Σ_m-k)-double cosets
inside Σ_m.
We parametrize these double cosets by pairs (a,b) of natural numbers satisfying

    0≤ a ≤ i  ,     0 ≤ b≤ m-i       and     a + b = k  .

For each such pair we define a permutation χ(a,b)∈Σ_m by

    χ(a,b)(j)  =    j          for 1≤ j≤ a, 
      j-a+i      for a+1≤ j≤ a+b, 
      j-b      for a+b+1≤ j≤ i+b, 
      j     for i+b+1≤ j≤ m.

In other words, χ(a,b) is the unique (k,m-k)-shuffle such that

    χ(a,b)({1,…,k})  =  {1,…, a} ∪ {i+1,…, i+b} .

The permutations χ(a,b) form a set of double coset representatives
for the subgroups Σ_i×Σ_m-i and Σ_k×Σ_m-k
inside Σ_m, for all pairs (a,b) subject to (<ref>).

When applying the double coset formula we will need the relations

    (Σ_i×Σ_m-i)^χ(a,b) ∩ (Σ_k×Σ_m-k)  =  Σ_a×Σ_b×Σ_i-a×Σ_m-i-b

and

    (Σ_i×Σ_m-i) ∩ ^χ(a,b) (Σ_k×Σ_m-k) =  Σ_a×Σ_i-a×Σ_b×Σ_m-i-b .

Thus the double coset formula becomes

    Φ_i,m-i^*   (_k,m-k (x× y ))  
       =  ∑_a,b_(Σ_i×Σ_m-i)∩^χ(a,b)(Σ_k×Σ_m-k)^Σ_i×Σ_m-i( 
    χ(a,b)_⋆(
    _(Σ_i×Σ_m-i)^χ(a,b)∩(Σ_k×Σ_m-k)^Σ_k×Σ_m-k(x× y) ))
       =  ∑_a,b_Σ_a×Σ_i-a×Σ_b×Σ_m-i-b^Σ_i×Σ_m-i( 
    χ(a,b)_⋆( ^Σ_k_Σ_a×Σ_b(x)×^Σ_m-k_Σ_i-a,m-i-b( y)))

The two sums run over all pairs (a,b) of natural numbers 
satisfying (<ref>).
Now we consider exponential sequences x,y∈exp(R;G) and calculate

    Φ_i,m-i^*( (x⊕ y)_m)     =  ∑_k=0^m  Φ_i,m-i^*(_k,m-k (x_k× y_m-k)) 
    
     
    _(<ref>)    =  ∑_a,b_Σ_i×Σ_i-a×Σ_b×Σ_m-i-b^Σ_i×Σ_m-i( 
    χ(a,b)_⋆ ( x_a× x_b× y_i-a× y_m-i-b) ) 
       =  ∑_a,b_a,i-a (x_a× y_i-a)×_b,m-i-b (x_b× y_m-i-b) 
       =   (x⊕ y)_i× (x⊕ y)_m-i .

Here the last two sums run over all pairs (a,b) of natural numbers satisfying
0≤ a≤ i and 0≤ b≤ m-i.
This shows that the sequence x⊕ y is again exponential.

The following square of group monomorphisms commutes:

    @C=15mm 
    (Σ_j≀ G)× (Σ_k≀ G)×(Σ_l≀ G) 
    [r]^-(Σ_j≀ G)×Φ_k,l[d]_Φ_j,k× (Σ_l≀ G)   
    (Σ_j≀ G) × (Σ_k+l≀ G) [d]^Φ_j,k+l
    
    (Σ_j+k≀ G)×(Σ_l≀ G) 
    [r]_-Φ_j+k,l   Σ_j+k+l≀ G

So for all x∈ R(Σ_j≀ G), y∈ R(Σ_k≀ G),
and z∈ R(Σ_l≀ G), the relation

    _j,k+l(x×_k,l(y× z))  =  _j+k,l(_j,k(x× y)× z)

holds in the group R(Σ_j+k+l≀ G). 
By unraveling the definitions, this becomes the associativity 
of the operation ⊕.

Also, the following square of group monomorphisms commutes:

    @C=15mm 
    (Σ_k≀ G)×(Σ_l≀ G) 
    [r]^-Φ_k,l[d]_twist   Σ_k+l≀ G [d]^c_χ
    
    (Σ_l≀ G)×(Σ_k≀ G) 
    [r]_-Φ_l,k   Σ_l+k≀ G

Here χ=(χ_k,l;1,…,1), 
for the shuffle permutation χ_k,l∈Σ_k+l.
So for all x∈ R(Σ_k≀ G) and y∈ R(Σ_l≀ G), the relation

    _l,k(y× x)  =  χ_⋆( _k,l(x× y))  =  _k,l(x× y)

holds in the group R(Σ_k+l≀ G). 
By unraveling the definitions, this implies the commutativity 
of the operation ⊕.

The sequence 0 with 0_0=1 and
0_m = 0 for m≥ 1 is a neutral element for ⊕. 
Given an exponential sequence x we define a sequence y inductively by
y_0=1 and by

    y_m  =   - ∑_k=1^m  _k,m-k (x_k× y_m-k)

for m≥ 1.
To see that the sequence y is again exponential,
we show the relation

    Φ_i,m-i^*(  y_m)  =  y_i× y_m-i

by induction on m. The induction starts with m=1, where there is nothing to show.
Now we assume that m≥ 2; then 

    - Φ_i,m-i^*(  y_m)     =  ∑_k=1^m  Φ_i,m-i^*(_k,m-k (x_k× y_m-k)) 
     
    _(<ref>)    =  ∑_ a+b≥ 1_Σ_a×Σ_i-a×Σ_b×Σ_m-i-b^Σ_i×Σ_m-i( 
    χ(a,b)_⋆ ( x_a× x_b× y_i-a× y_m-i-b )) 
       =  ∑_a,b≥ 1_a,i-a (x_a× y_i-a)×_b,m-i-b (x_b× y_m-i-b) 
       +  ∑_a=1^i _a,i-a(x_a× y_i-a) × y_m-i + ∑_b=1^m-i y_i×_b,m-i-b(x_b× y_m-i-b)
       =  
     y_i × y_m-i - y_i× y_m-i -  y_i ×  y_m-i =   
    - y_i× y_m-i .

In the sums, (a,b)  runs over all pairs of natural numbers satisfying
a≤ i and b≤ m-i plus the conditions attached to the summation symbols.
The second equation uses the inductive hypothesis.
This shows that the sequence y is again exponential.
The relation x⊕ y=0 holds by construction,
so y is an inverse of x with respect to ⊕.
Altogether this shows that exp(R;G) is an abelian group under ⊕. 

It remains to show distributivity of multiplication over ⊕.
We let x,y, z∈exp(R;G) be exponential sequences. Then

    ( ( x⊕ y)· z)_m      =  (∑_k=0^m _k,m-k(x_k× y_m-k) )· z_m 
       =  ∑_k=0^m _k,m-k ( (x_k× y_m-k) ·Φ_k,m-k^*(z_m) ) 
       =  ∑_k=0^m _k,m-k ( (x_k× y_m-k) · (z_k×  z_m-k )) 
       =  ∑_k=0^m _k,m-k ( (x_k· z_k ) × (y_m-k· z_m-k) )  = ( ( x· z)⊕ (y· z))_m   .

The second equation is reciprocity in the global Green functor R.



The ring exp(R;G) is covariantly functorial in R:
given a morphism of global Green functors φ:R→ S, we define a map

    exp(φ;G)  :  exp(R;G)  → exp(S;G)    by  ( exp(φ;G)(x) )_m  =  φ(x_m) .

We omit the straightforward verifications that
for x∈exp(R;G) the sequence exp(φ;G)(x) is again exponential,
that exp(φ;G) is a ring homomorphism,
and that exp(-;G) is compatible with identity and composition.

Now we turn to the functoriality of exp(R;G) in the
Lie group variable, which is slightly more subtle. 
The ultimate aim is to make exp(R;-) into another global Green functor. 
Proposition <ref> already provides the ring structures
on the values, so the missing data is to give exp(R;-)
the structure of a global functor. Our approach is indirect and not based on
explicit formulas; rather we exploit that the functor R↦exp(R;G)
is representable by the underlying global Green functor
of the free global power functor C_G discussed in 
Example <ref>.
The next proposition will be used for establishing this representability property,
see Proposition <ref> below.



Let R and S be global power functors and f:R→ S
a morphism of global Green functors.
Then the collection of elements x∈ R(G), for varying compact Lie groups G,
that satisfy

    f(P^m(x)) =  P^m(f(x))

in S(Σ_m≀ G)  for all m≥ 0 form a global power subfunctor of R.


  This is a straightforward consequence of the various relations enjoyed by
  power operations. For the course of this proof we call a pair (G,x)
  consisting of a compact Lie group G and an element x∈ R(G) good
  if the relation f(P^m(x))= P^m(f(x)) holds for all m≥ 0.
  Then the naturality relation P^m∘α^* = (Σ_m≀α)^*∘ P^m
  for a continuous homomorphism α:K→ G implies
  
    f(P^m(α^*(x)))     =  f((Σ_m≀α)^*(P^m(x)))  =  
        (Σ_m≀α)^*(f^*(P^m(x))) 
       =  (Σ_m≀α)^*(P^m(f^*(x)))  =    P^m(α^*(f^*(x)))  
        =    P^m(f^*(α^*(x)))  .

  So if (G,x) is good, then so is (K,α^*(x)).
  By the analogous calculation, the transfer relation
  P^m∘_H^G = _Σ_m≀ H^Σ_m≀ G∘ P^m for a closed subgroup H
  of G implies that if (H,x) is good, then so is (G,_H^G(x)).
  
  The relation P^m(1)=1 and the fact f(1)=1 show that (e,1) is good.
  Since good pairs are closed under restriction, the pair (G,1) 
  is good for every compact Lie group G.
  The multiplicativity P^m(x·  y)= P^m(x)· P^m(y)
  and the hypothesis that f is multiplicative show that
  for all G, the good elements are closed under multiplication in the ring R(G).
  Closure under restriction then implies that good elements
  are also closed under the external pairing
  
    × :  R(G)× R(K) → R(G× K) .

  The additivity relation 
  
    P^m(x+y)  =  ∑_k=0^m _k,m-k (P^k(x)× P^m-k(y))  ,

  the hypothesis that f is additive and the already established 
  closure under transfers and external multiplication then shows that for all G, 
  the good elements are closed under addition in the ring R(G).
  The additivity relation and the fact that P^m(1+(-1))=P^m(0)=0
  imply that
  
    P^m(-1) =   - ∑_k=1^m _k,m-k (P^k(1)× P^m-k(-1)) 
       =   - ∑_k=1^m _k,m-k ( 1× P^m-k(-1))  .

  So induction on m and the previously established closure properties
  show that f(P^m(-1))=P^m(f(-1)), so (e,-1) is good. Closure under restriction
  implies that also (G,-1) is good for every compact Lie group G.
  Since the good elements are closed under multiplication and contain -1,
  they are closed under additive inverses. Since (G,1) is good, so is 
  (G,1-1)=(G,0). This completes the proof that the good elements 
  form a global Green subfunctor of R.
  
  The closure property under power operations is then a consequence of
  the transitivity relation and closure under restriction: if (G,x) is good, then
  
    f(P^k(P^m(x)))    =   f(Ψ_k,m^*(P^k m(x))) =  Ψ_k,m^*(f(P^k m(x)))
       =  Ψ_k,m^*( P^k m(f(x))) =  P^k( P^m(f(x))) =  P^k(f(P^m(x)))

  holds in S(Σ_k≀(Σ_m ≀ G)), for all k≥ 0.
  This proves that the pair (Σ_m≀ G, P^m(x)) is also good.
  Altogether this shows that the good elements form a  
  global power subfunctor of R.



The free global power functor C_K associated to a compact Lie group K 
was introduced in Example <ref>.
The next proposition justifies the adjective `free',
and also shows that the underlying global Green functor of C_K
represents the functor exp(-;K) of exponential sequences.
We recall that

    1_Σ_m≀ K ∈ (Σ_m≀ K,Σ_m≀ K)  ⊂ 
    C_K(Σ_m≀ K)

denotes the identity operation of the functor π_0^Σ_m≀ K.
Two key relations among these elements in the global power functor C_K are

    P^m(1_K) =  1_Σ_m≀ K     and    Φ^*_k,m-k(1_Σ_m≀ K) =  1_Σ_k≀ K× 1_Σ_m-k≀ K .

The second set of relations shows that the tuple (1_Σ_m≀ K)_m≥ 0
is exponential, i.e., an element of exp(C_K;K).
In fact, part (i) of the following proposition shows that it is a universal
exponential element.


Let K be a compact Lie group.


  * For every global Green functor R
the map

    ϵ_K  :  (C_K,R)    → exp(R;K) 
    
    f        ⟼ exp(f;(1_Σ_m≀ K)_m≥ 0) =  ( f(1_Σ_m≀ K))_m≥ 0

is bijective.  

  * For every global power functor R the map

    (C_K,R) → R(K)  ,   
    f  ⟼ f(1_K)

is bijective.  



(i)
The underlying global functor of C_K is the direct sum of the 
represented global functors (Σ_m≀ K,-), for m≥ 0.  
So the enriched Yoneda lemma (see Remark <ref>)
shows that evaluation at the universal elements is a bijection

    (C_K,R) → ∏_m≥ 0  R(Σ_m≀ K) ,   
    f  ⟼ ( f(1_Σ_m≀ K))_m≥ 0 .

A morphism of global functors f:C_K→ R
is a morphism of global Green functors if and
only if it is also multiplicative and unital. Unitality corresponds
to the condition f(1_Σ_0≀ K)=f(_e)=1 in R(e).
Multiplicativity means that the following square of global functors commutes:

    @C=15mm@R=7mm
    C_K C_K[r]^f f[d]_mult    R R[d]^mult
    
    C_K[r]_f     R

Since the box product of global functors is biadditive and
the box product of two represented functors is represented 
(see Remark <ref>), the global functor C_K C_K
is isomorphic to

    ⊕_m,n≥ 0 (Σ_m≀ K,-)(Σ_n≀ K,-) ⊕_m,n≥ 0 ( (Σ_m≀ K)× (Σ_n≀ K),-) .

So commutativity of the above square can be tested by evaluation at the
universal classes, by another application of the enriched Yoneda lemma.
The multiplication of C_K takes the exterior product of
the classes 1_Σ_m≀ K and 1_Σ_n≀ K to the class
Φ^*_m,n( 1_Σ_m+n≀ K), so the multiplicativity condition becomes 
the exponential condition

    Φ^*_m,n( f(1_Σ_m+n≀ K)) =   
    f(1_Σ_m≀ K)× f(1_Σ_n≀ K)  .


(ii)  Every morphism f:C_K→ R of global power functors is in particular
a morphism of global Green functors. So by part (i), f is
determined by the exponential sequence ( f(1_Σ_m≀ K))_m≥ 0. 
The relation P^m(1_K)=1_Σ_m≀ K holds in the global power functor C_K;
so since f also commutes with power operations, it is already determined
by the element f(1_K). This shows that the map in question
is injective.

Now we show that evaluation at 1_K is also surjective. We let y∈ R(K)
be any element. Then the sequence (P^m(y))_m≥ 0 is exponential,
so part (i) provides a unique morphism of global Green functors f:C_K→ R
satisfying

    f(1_Σ_m≀ K)  =  P^m(y)   in  R(Σ_m≀ K)

for all m≥ 0. We must show that f is also compatible with power operations. 
By Theorem <ref>  the abelian group

    C_K(G) =  ⊕_m≥ 0(Σ_m≀ K,G)

is generated by the classes

    _H^G(α^*(P^m(1_K)))

for m≥ 0, and where (H,α)
runs over pairs consisting of a closed subgroup H of G
and a continuous homomorphism α:H→Σ_m≀ K.
In particular, C_K is generated, as a global power functor,
by the single element 1_K in C_K(K).

Proposition <ref> shows that the collection of elements x
of C_K that satisfy f(P^m(x))= P^m(f(x)) for all m≥ 0
form a global power subfunctor of C_K.
The element 1_K is among these because 
f(P^m(1_K))=f(1_Σ_m≀ K)= P^m(y)=P^m(f(1_K)).
Since 1_K generates C_K as a global power functor, all elements of C_K 
have this property.
So f is a morphism of global power functors.


Now we can make the rings exp(R;G) of exponential sequences into
a global functor for varying G.
As an auxiliary tool we introduce a functor

    Γ :  ^ →   .

The functor is given  by Γ(K)=C_K on objects;
on morphisms, the freeness property of Proposition <ref> (ii)
allows us to define

    Γ :  (K,G) → (C_G,C_K)

by the requirement

    Γ(τ)(1_G) =  τ ∈ (K,G) ⊂ C_K(G)  .

The contravariant functoriality is rather formal: for ψ∈(L,K) we have

    Γ(τ∘ψ)(1_G)  =  τ∘ψ =  Γ(ψ)(τ)  =  
     (Γ(ψ)∘Γ(τ))(1_G) .

So Γ(τ∘ψ)=Γ(ψ)∘Γ(τ) by freeness,
i.e., Proposition <ref> (ii).
We emphasize that while  is a preadditive category,
Γ is just a plain functor, i.e., not additive in any sense.


As before we consider a global Green functor R.
For compact Lie groups G and K we define a map

    exp(R;-) :  (G,K)×exp(R;G) → exp(R;K)  , (τ,x) ⟼ exp(R;τ)(x) .

The definition exploits the representability of exp(R;G), i.e.,
that the map

    ϵ_G  :  (C_G,R) → exp(R;G)

sending f to the exponential sequence ( f(1_Σ_m≀ G))_m≥ 0
is bijective, by Proposition <ref> (i).
We define exp(R;τ)
by requiring that for every morphism of global Green functors
f:C_G→ R the following relation holds:

    exp(R;τ)(ϵ_G(f)) =  ϵ_K(f∘Γ(τ))  .

Since ϵ_G is bijective, this is a legitimate definition.
It is rather straightforward to see that the assignments

    G ⟼ exp(R;G)     and    τ ⟼ exp(R;τ)

define a functor from the global Burnside category to the category of sets.
Indeed, for τ∈(K,G), ψ∈(L,K)
and a morphism of global Green functors f:C_G→ R, we have

    exp(R;τ∘ψ)   (ϵ_G(f)) =  ϵ_L(f∘Γ(τ∘ψ))  =  ϵ_L(f∘Γ(ψ)∘Γ(τ)) 
       =  exp(R;τ)(ϵ_K(f∘Γ(ψ)))  
    =  exp(R;τ)(exp(R;ψ)(ϵ_G(f)))  .

Since every exponential sequence in exp(R;G) is of the form
ϵ_G(f), this proves that exp(R;τ∘ψ)=exp(R;τ)∘exp(R;ψ).
It is not completely obvious, though, that this construction is additive
in both variables, but we will show that in the next proposition.

The construction of exponential sequences is in fact a functor in two variables:
for every operation τ∈(G,K) 
and all morphisms of global Green functors f:C_G→ R we have

    ( ( exp(φ;K)∘exp(R;τ))(ϵ_G(f)) )_m    = ( ( exp(φ;K) (ϵ_K(f∘Γ(τ)))) )_m
       = φ( (ϵ_K(f∘Γ(τ)))_m )
       = (φ∘ f∘Γ(τ))(1_Σ_m≀ K)
       =  ( exp(S;τ)(ϵ_G(φ∘ f)) )_m
       =  ( (exp(S;τ)∘exp(φ;G))(ϵ_G(f)) )_m .

Every exponential sequence in exp(R;G) is of the form
ϵ_G(f), so

    exp(φ;K)∘exp(R;τ) = exp(S;τ)∘exp(φ;G)  : exp(R;G) → exp(S;K) .



The next proposition makes the abstract definition of the map
exp(R;τ) more explicit by giving a concrete formula when
τ is a restriction or a transfer.



  Let R be a global Green functor.
  
  
  * For every continuous group homomorphism α:K→ G between
    compact Lie groups the map 
    
    exp(R;α^*) : exp(R;G) → exp(R;K)

    is a ring homomorphism and satisfies
    
    ( exp(R;α^*)(x))_m  =   (Σ_m≀α)^*(x_m)

    in R(Σ_m≀ K) for all x∈exp(R;G).
  
  * For every closed subgroup H of a compact Lie group G 
    the map
    
    exp(R;_H^G) : exp(R;H) → exp(R;G)

    is additive, satisfies reciprocity with respect to restriction from G to H
    and is given by
    
    ( exp(R;_H^G)(x))_m  =  _Σ_m≀ H^Σ_m≀ G(x_m)

    for all x∈exp(R;H).
  
  * For all compact Lie groups G and K, the map 
    
    exp(R;-) :  (G,K)×exp(R;G) → exp(R;K)

    is biadditive, i.e., exp(R;-) becomes a global functor in the Lie group variable.
  
  * As the Lie group G varies, the ring structures and the functoriality
    in the global Burnside category make exp(R) into a global Green functor.
    


(i) We let f:C_G→ R be any morphism of global Green functors. Then

    ( exp(R;α^*)(ϵ_G(f)))_m     =  ( ϵ_K(f∘Γ(α^*)))_m=  (f∘Γ(α^*))(1_Σ_m≀ K) 
       =  f ( Γ(α^*)(P^m(1_K))) =  f ( P^m( Γ(α^*)(1_K))) 
       =  f ( P^m( α^*(1_G))) =  f ( (Σ_m≀α)^*(P^m(1_G))) 
       =  f ( (Σ_m≀α)^*(1_Σ_m≀ G)) =   (Σ_m≀α)^*(f(1_Σ_m≀ G)) 
       =  (Σ_m≀α)^*( ϵ_G(f)_m) .

Since every exponential sequence in exp(R;G) is of the form
ϵ_G(f), this proves the formula for exp(R;α^*).
Since the multiplication in exp(R;G) is coordinatewise
and the original restriction maps for R are ring homomorphisms,
the formula for exp(R;α^*) shows that exp(R;α^*)
is multiplicative and preserves the multiplicative unit.
The additivity of exp(R;α^*) uses the relation

    (Σ_m≀α)^*∘_k,m-k =  _k,m-k∘ ((Σ_k≀α)×(Σ_m-k≀α))^*

as maps from R((Σ_k≀ G)×(Σ_m-k≀ G))
to R(Σ_m≀ K).
To prove (<ref>) we distinguish two cases.
If α is surjective, then so is Σ_m≀α, and 

    (Σ_m≀α)^-1( (Σ_k≀ G)×(Σ_m-k≀ G))  =  
     (Σ_k≀ K)×(Σ_m-k≀ K) .

So for epimorphisms, the relation (<ref>) 
is a special case of compatibility of transfer with inflation.
If H is a closed subgroup of G, then Σ_m≀ G consists of a single
double coset for the left  (Σ_m≀ H)-action
and right ( (Σ_k≀ G)×(Σ_m-k≀ G))-action, and

    (Σ_m≀ H) ∩  ( (Σ_k≀ G)×(Σ_m-k≀ G)) 
       =  
     (Σ_k≀ H)×(Σ_m-k≀ H)   .

So the double coset formula specializes to 

    ^Σ_m≀ G_Σ_m≀ H∘_k,m-k =  _k,m-k∘^ (Σ_k≀ G)×(Σ_m-k≀ G)_(Σ_k≀ H)×(Σ_m-k≀ H) ,

which is precisely the relation (<ref>) 
for the inclusion H→ G. 
Every homomorphism factors as an epimorphism followed by a subgroup
inclusion, so relation (<ref>) follows in general.
We can then conclude that exp(R;α^*) is additive:

    (exp(R;α^*)(x⊕ y) )_m     =  (Σ_m≀α)^*( (x⊕ y)_m)  =  ∑_k=0^m  (Σ_m≀α)^*(_k,m-k (x_k× y_m-k)     )
    
    _(<ref>)    =  ∑_k=0^m  _k,m-k( (  (Σ_k≀α)×(Σ_m-k≀α))^*(x_k× y_m-k))   
       =  ∑_k=0^m  _k,m-k( (Σ_k≀α)^*(x_k)×(Σ_m-k≀α)^*(y_m-k))
       =  ∑_k=0^m  _k,m-k( (exp(R;α^*)(x))_k × (exp(R;α^*)(y))_m-k)
       =  ( exp(R;α^*)(x)⊕exp(R;α^*)(y) )_m  .


Part (ii) is similar to part (i).
We let f:C_H→ R be any morphism of global Green functors. Then

    ( exp(R;_H^G)(ϵ_H(f)))_m     =  ( ϵ_G(f∘Γ(_H^G)))_m=  (f∘Γ(_H^G))(1_Σ_m≀ G) 
       =  f ( Γ(_H^G)(P^m(1_G))) =  f ( P^m( Γ(_H^G)(1_G))) 
       =  f ( P^m( _H^G(1_H))) =  f ( _Σ_m≀ H^Σ_m≀ G(P^m(1_H))) 
       =  f ( _Σ_m≀ H^Σ_m≀ G(1_Σ_m≀ H)) =  _Σ_m≀ H^Σ_m≀ G(f(1_Σ_m≀ H)) =  _Σ_m≀ H^Σ_m≀ G( ϵ_H(f)_m) .

Since every exponential sequence is of the form ϵ_H(f), 
this proves the formula for exp(R;_H^G).
To see that the map exp(R;_H^G) is additive we observe that

    _Σ_m≀ H^Σ_m≀ G∘_k,m-k = _k,m-k∘_(Σ_k≀ H)×(Σ_m-k≀ H)^(Σ_k≀ G)×(Σ_m-k≀ G) ,

by transitivity of transfers. Thus 

    ( exp(R;_H^G)(x ⊕ y) )_m     =  ∑_k=0^m  _Σ_m≀ H^Σ_m≀ G(_k,m-k (x_k× y_m-k)   )
       =  ∑_k=0^m  _k,m-k( _(Σ_k≀ H)×(Σ_m-k≀ H)^(Σ_k≀ G)×(Σ_m-k≀ G)(x_k× y_m-k))   
       =  ∑_k=0^m  _k,m-k( _Σ_k≀ H^Σ_k≀ G(x_k)×_Σ_m-k≀ H^Σ_m-k≀ G(y_m-k))
       =  ( exp(R;_H^G)(x) ⊕exp(R;_H^G)(y) )_m  .

The reciprocity for restriction and transfer is now a direct
consequence of the reciprocity for the global Green functor R:

    ( exp(R;_H^G)(x)· y )_m     =  _Σ_m≀ H^Σ_m≀ G(x_m)· y_m 
       =  _Σ_m≀ H^Σ_m≀ G(x_m·^Σ_m≀ G_Σ_m≀ H( y_m)) =  ( exp(R;_H^G)(x ·_H^G(y) ))_m  .


(iii)
We start with additivity of the map exp(R;-) in the variable τ.
We let τ'∈(G,K) be another natural transformation.
Then the morphism of global power functors Γ(τ+τ'):C_K → C_G
satisfies

    Γ(τ+τ')(1_Σ_m≀ K)     =  Γ(τ+τ')(P^m(1_K))  =    P^m( Γ(τ+τ')(1_K)) 
       =    P^m( τ+τ')  =∑_k=0^m _k,m-k(P^k(τ)× P^m-k(τ')) 
       =  ∑_k=0^m _k,m-k( P^k( Γ(τ)(1_K))× P^m-k(Γ(τ')(1_K))) 
       =  ∑_k=0^m _k,m-k(  Γ(τ)(1_Σ_k≀ K)×Γ(τ')(1_Σ_m-k≀ K))  .

For a morphism of global Green functors f:C_G→ R this implies

    (f∘Γ(τ+τ'))(1_Σ_m≀ K)     =  ∑_k=0^m _k,m-k(  f(Γ(τ)(1_Σ_k≀ K))× f(Γ(τ')(1_Σ_m-k≀ K))) .

This establishes the m-th component of the relation

    ϵ_K(f∘Γ(τ+τ')) =  ϵ_K(f∘Γ(τ)) ⊕ϵ_K(f∘Γ(τ')) ,

and this in turn proves

    exp(R;τ+τ')(ϵ_G(f)) =  exp(R;τ)(ϵ_G(f)) ⊕ exp(R;τ')(ϵ_G(f))
     .

Now we come to additivity of the map in the exponential sequence,
i.e., that for fixed τ∈(G,K) the map

    exp(R;τ) :  exp(R;G) →exp(R;K)

is additive with respect to ⊕.
By Theorem <ref> the abelian group (G,K) 
is generated by operations of the form _L^K∘α^*
where L is a closed subgroup of K and α:L→ G
a continuous homomorphism.
Since we have already established additivity in τ, we can assume
that τ=_L^K∘α^* is one of the generating operations.
Since

    exp(R;_L^K∘α^*) =  exp(R;_L^K)∘exp(R;α^*)  ,

this in turn follows from additivity for restrictions and transfers,
which we showed in part (i) respectively (ii).

Part (iv) is just the summary of the properties in parts (i), (ii) and (iii).





The previous proposition establishes the functor exp
of exponential sequences as an endofunctor of the category of global Green functors.
When iterating the construction, 
we encounter iterated wreath products, and we will save a 
substantial number of parentheses by using the short hand notation

    Σ_k≀Σ_m≀ G  =  Σ_k≀(Σ_m≀ G) .

Now we make this endofunctor into a comonad.
A natural transformation of global Green functors

    η_R :  exp(R)  → R

is given by η(x)=x_1, using the identification GΣ_1≀ G
via g↦ (1;g).
A natural transformation 

    κ_R :  exp(R)  → exp(exp(R))

is given at a compact Lie group G by 

    ( κ(x)_m )_k  =  Ψ_k,m^*(x_k m)  ∈  R(Σ_k≀Σ_m≀ G) ;

here the restriction is along the monomorphism (<ref>) 

    Ψ_k,m :  Σ_k≀Σ_m≀ G        →    Σ_k m≀ G 
    
     (σ;  (τ_1;  h^1),…,(τ_k;  h^k))     ⟼(σ♮(τ_1,…,τ_k);  h^1+…+h^k) .



The analog of the following for finite groups is Satz 2.17 in <cit.>.




  
    Let R be a global Green functor.
  
  * For every compact Lie group G 
    and for every exponential sequence x∈exp(R;G),
    the sequence κ(x) is an element of exp(exp(R);G).
  
  * As the group varies, the maps κ form a morphism of global Green functors
    κ_R:exp(R)→exp(exp(R)), natural in R.
  
  * The natural transformations 
    
    η :  exp →      and    κ : exp → exp∘exp

    make the functor exp into a comonad on the category of global Green functors.
  



(i) Because the square of group homomorphisms

    @C=13mm@R=7mm 
    (Σ_j≀Σ_m≀ G) × (Σ_k-j≀Σ_m≀ G) 
    [r]^-Φ_j,k-j[d]_Ψ_j,m×Ψ_k-j,m   Σ_k≀Σ_m≀ G [d]^Ψ_k,m
    
    (Σ_j m≀ G) × (Σ_(k-j)m≀ G) [r]_-Φ_j m,(k-j)m   Σ_k m≀ G

commutes, the elements Ψ_k,m^*(x_k m) satisfy

    Φ_j,k-j^*( Ψ_k,m^*(x_k m) )    =   (Ψ_j,m×Ψ_k-j,m)^*(Φ_j m,(k-j)m^*(x_k m))
       =   (Ψ_j,m×Ψ_k-j,m)^*(x_j m× x_(k-j)m)
       =  Ψ_j,m^*(x_j m)×Ψ_k-j,m^*(x_(k-j)m) .

This shows that for fixed m≥ 0, the sequence
κ(x)_m=(Ψ_k,m^*(x_k m))_k≥ 0 is exponential, i.e., an element
of the ring exp(R;Σ_m≀ G).

The square of group homomorphisms

    @C=13mm@R=7mmΣ_k≀( (Σ_i≀ G) × (Σ_m-i≀ G) ) 
    [r]^-Σ_k≀Φ_i,m-i[d]_Δ   Σ_k≀Σ_m≀ G [dd]^Ψ_k,m
    
    (Σ_k≀Σ_i≀ G) × (Σ_k≀Σ_m-i≀ G ) 
    [d]_Ψ_k,i×Ψ_k,m-i   
    
    (Σ_k i≀ G) × (Σ_k(m-i)≀ G) [r]_-Φ_k i,k(m-i)   Σ_k m≀ G

does not commute; we invite the reader to check
the case k=m=2, i=1 and G=e, where the phenomenon is already visible
in the fact that the square

    @C=13mm@R=7mmΣ_2 [r]^-Σ_2≀Φ_1,1[d]_Δ   Σ_2≀Σ_2 [d]^♮
    Σ_2 ×Σ_2[r]_-+   Σ_4

does not commute.
However, the square (<ref>) 
does commute up to conjugation by an element of Σ_k m≀ G.
Since inner automorphisms are invisible through the eyes of a Rep-functor,
we conclude that the relation

    (  Φ_i,m-i^*( κ(x)_m ) )_k     =  
    (Σ_k≀Φ_i,m-i)^*( Ψ_k,m^*(x_k m) ) 
       =  Δ^*( ( Ψ_k,i×Ψ_k,m-i)^*(Φ_k i, k(m-i)^*( x_k m)))
       =  Δ^*( Ψ_k,i^*(x_k i) ×Ψ_k,m-i^*(x_k(m-i)))
       =  Δ^*( (κ(x)_i )_k × (κ(x)_m-i)_k)=  (κ(x)_i ×κ(x)_m-i)_k

holds in R(Σ_k≀ ((Σ_i≀ G)× (Σ_m-i≀ G))).
For varying k≥ 0, this shows that

    Φ_i,m-i^*( κ(x)_m )  =  κ(x)_i×κ(x)_m-i     in  exp(R;(Σ_i≀ G)× (Σ_m-i≀ G)).

In other words, the sequence κ(x)=(κ(x)_m)_m≥ 0 is 
itself exponential.


(ii) The relations κ(1)=1 and
κ(x· y)=κ(x)·κ(y)
are straightforward from the definitions, using that multiplication
is defined coordinatewise and that the restriction map Ψ_k,m^*
is multiplicative and unital.
The verification of the additivity of κ_R(G):exp(R;G)→exp(exp(R);G)
is more involved,
and forces us to confront the double coset formula for the subgroups

    Σ_k≀'Σ_m  =  Ψ_k,m(Σ_k≀Σ_m≀ G) 
       and  Σ_i×'Σ_k m-i =  Φ_i,k m-i( (Σ_i≀ G)×(Σ_k m-i≀ G))

of the wreath product Σ_k m≀ G. To a large extent, the group G
acts like a dummy, which is why we omit it from the notation for
Σ_k≀'Σ_m and Σ_i×'Σ_k m-i.
We specify a bijection between the set of double cosets 

    Σ_k≀'Σ_m Σ_k m≀ G / Σ_i×'Σ_k m-i

and the set of tuples (a_0,…,a_m) of natural numbers that satisfy 

    a_0+…+a_m  =  k      and    ∑_j=0^m  j· a_j  =  i  .

For 1≤ b≤ k we let j_b∈{1,…, m} be the unique number such that

    a_0+…+a_j_b-1 <  b ≤  a_0+…+a_j_b-1+a_j_b .

We define

    A(a_0,…,a_m) =  ⋃_b=1^k { (b-1)m+1,…, (b-1)m+ j_b } ⊆ {1,…, k m} ;

because

    ∑_b=1^k   j_b  =  ∑_j=1^m   a_j· j =   i ,

the set A(a_0,…,a_m) has exactly i elements.
We let σ̅=σ̅(a_0,…,a_m)∈Σ_k m
be any permutation such that

    {σ̅(1),…,σ̅(i)} =  A(a_0,…,a_m) .

Then the elements

    σ(a_0,…,a_m) =  (σ̅(a_0,…,a_m); 1,…,1)  ∈ Σ_k m≀ G

are a complete set of double coset representatives as (a_0,…,a_m)
ranges over those tuples that satisfy (<ref>).
Moreover,

    (Σ_k≀'Σ_m)^σ(a_0,…,a_m)   ∩ (Σ_i×'Σ_k m-i) 
     =  ( ∏'_j=0^m Σ_a_j≀'Σ_j )
    ×'
    ( ∏'_j=0^mΣ_a_j≀'Σ_m-j)

and

    (Σ_k≀'Σ_m)   ∩^σ(a_0,…,a_m)(Σ_i×'Σ_k m-i) 
     =  ∏'_j=0^m (Σ_a_j≀'Σ_j)×' (Σ_a_j≀'Σ_m-j) .

For x_i∈ R(Σ_i≀ G) and y_k m-i∈ R(Σ_k m-i≀ G),
the double coset formula thus becomes the relation

    Ψ_k,m^*   ( _i,k m-i( x_i × y_k m-i ))
     =   
       ∑^Σ_k≀Σ_m≀ G_∏' (Σ_a_j≀'Σ_j)×' (Σ_a_j≀'Σ_m-j)( σ(a_0,…,a_m)_⋆( ^(Σ_i≀ G)×(Σ_k m-i≀ G)_( ∏' Σ_a_j≀'Σ_j )
    ×'
    ( ∏'Σ_a_j≀'Σ_m-j)( x_i × y_k m-i ) ))

where the sum ranges over all tuples (a_0,…,a_m) 
that satisfy (<ref>).
If x_i and y_k m-i are the respective components of two exponential sequences 
x, y∈exp(R;G), then

    σ(a_0,…,a_m)_⋆   (^(Σ_i≀ G)×(Σ_k m-i≀ G)_( ∏'Σ_a_j≀'Σ_j )
    ×'
    ( ∏'Σ_a_j≀'Σ_m-j)( x_i × y_k m-i )) 
       =  σ(a_0,…,a_m)_⋆(^Σ_i≀ G_∏' Σ_a_j≀'Σ_j ( x_i) ×^Σ_k m-i≀ G_∏'Σ_a_j≀'Σ_m-j( y_k m-i ))
       =  σ(a_0,…,a_m)_⋆( ∏_j=0^m Ψ_a_j,j^*(x_ a_j j) ×∏_j=0^m Ψ_a_j,m-j^*(y_ a_j(m-j)))
       =  ∏_j=0^m ( Ψ_a_j,j^*(x_a_j j)×Ψ_a_j,m-j^*(y_a_j(m-j))) .

Given x,y∈exp(R;G) we have

    ( κ(x) ⊕κ(y) )_m   =  ⊕_j=0^m _j,m-j( κ(x)_j ×κ(y)_m-j ) ,

where the sum on the right is taken in the group exp(R;Σ_m≀ G)
under ⊕. Expanding this further we arrive at the expression

    ( ( κ(x)    ⊕κ(y) )_m)_k   =  ∑_a_0+…+a_m=k_a_0,…,a_m(
    ∏_j=0^m _j,m-j( κ(x)_j ×κ(y)_m-j )_a_j)  
       =  ∑_a_0+…+a_m=k_a_0,…,a_m(
    ∏_j=0^m _Σ_a_j≀Σ_j,m-j^Σ_a_j≀Σ_m( Ψ_a_j,j^*( x_a_j j) ×Ψ^*_a_j,m-j(y_a_j(m-j) ))) 
       =  ∑_a_0+…+a_m=k^Σ_k≀Σ_m≀ G_∏' (Σ_a_j≀'Σ_j)×' (Σ_a_j≀'Σ_m-j)(  ∏_j=0^m Ψ_a_j,j^*(x_a_j j)×Ψ_a_j,m-j^*(y_a_j(m-j)))
       =  ∑_a_0+…+a_m=k^Σ_k≀Σ_m≀ G_∏' (Σ_a_j≀'Σ_j)×' (Σ_a_j≀'Σ_m-j)(  ∏_j=0^m Ψ_a_j,j^*(x_a_j j)×Ψ_a_j,m-j^*(y_a_j (m-j)))
       =  ∑_i=0^k mΨ_k,m^*( _i,k m-i( x_i × y_k m-i )) =  ( κ(x ⊕ y)_m )_k

in the group R(Σ_k≀Σ_m≀ G).
Here _a_0,…,a_m is shorthand notation for the transfer along the monomorphism

    Φ_a_0,…,a_m :  (Σ_a_0≀Σ_m≀ G)×…×
     (Σ_a_j≀Σ_m≀ G)×…× (Σ_a_m≀Σ_m≀ G)
     → Σ_k≀Σ_m≀ G   ,

the analog of the monomorphism (<ref>)
with multiple inputs (and for the group Σ_m≀ G instead of G).   
So the map κ_R(G):exp(R;G)→exp(exp(R);G) is additive,
and hence a ring homomorphism.

It remains to check that the maps κ_R commute with restrictions
and transfers.
For every continuous homomorphism α:K→ G the relations

    ( κ(α^*(x))_m )_k     =  Ψ_k,m^*((Σ_k m≀α)^* (x_k m))   =    (Σ_k≀Σ_m≀α)^* (Ψ_k,m^*(x_k m)) 
       =    (Σ_k≀Σ_m≀α)^* ((κ(x)_m)_k)  =  ( ( α^*(κ(x)))_m )_k

hold in R(Σ_k≀Σ_m≀ K). 
So κ_R(K)∘exp(R;α^*)=exp(exp(R);α^*)∘κ_R(G).

The compatibility of κ with transfers needs another application 
of a double coset formula.
Indeed, for every closed subgroup H of G, the group 
Σ_k m≀ G consists of a single
double coset for the left (Σ_k≀Σ_m≀ G)-action
and right (Σ_k m≀ H)-action, and

    (Σ_k≀Σ_m≀ G)  ∩ (Σ_k  m≀ H)   =  Σ_k≀Σ_m≀ H   .

So for every x∈exp(R;H), the relations

    ( κ(_H^G(x))_m )_k =  Ψ_k,m^* (_Σ_k m≀ H^Σ_k m≀ G (x_k m) )     =  _Σ_k≀Σ_m≀ H^Σ_k≀Σ_m≀ G (Ψ_k,m^*(x_k m)) 
       =  _Σ_k≀Σ_m≀ H^Σ_k≀Σ_m≀ G ((κ(x)_m)_k)  = ( ( _H^G(κ(x)))_m )_k

holds in R(Σ_k≀Σ_m≀ G). Hence κ∘_H^G=_H^G∘κ.
Altogether this shows that the maps κ_R(G) are
ring homomorphisms and compatible with restrictions and transfers,
so they form a morphism of global Green functors.


(iii)
We have to show that the transformation κ is coassociative, 
and counital with respect to η, and 
these are all straightforward from the definitions.
The counitality relations

    η_exp(M)∘κ_M  =  _exp(M) =  exp(η_M)∘κ_M

come down to the facts that the homomorphism Ψ_k,1
is the result of applying Σ_k≀ -
to the preferred isomorphism Σ_1≀ G G,
and that the homomorphism Ψ_1,n is 
the preferred isomorphism Σ_1≀Σ_n≀ GΣ_n≀ G.
The coassociativity relation

    exp(κ_M)∘κ_M  =  κ_exp(M)∘κ_M

ultimately boils down to the observation that the following square of monomorphisms
commutes:

    [b]
    @C=15mm@R=7mmΣ_k≀Σ_m≀Σ_n≀ G[r]^-Σ_k≀Ψ_m,n[d]_Ψ_k,m   Σ_k≀Σ_m n≀ G[d]^Ψ_k, m n
    Σ_k m≀Σ_n≀ G[r]_-Ψ_k m, n   Σ_k m n≀ G




Now we can finally get to the main result of this section,
identifying global power functors with coalgebras over the comonad
of exponential sequences.
We suppose that R is a global Green functor and P:R→exp(R)
a morphism of global Green functors. 
For every compact Lie group G, a sequence of operations P^m:R(G)→ R(Σ_m≀ G)
is then defined by

    P^m(x)  =  ( P(x) )_m  ,

i.e., P^m(x) is the m-th component of the exponential sequence P(x).


  
  
  
  * Let R be a global Green functor and
    P:R→exp(R) a morphism of global Green functors that makes R into
    a coalgebra over the comonad (exp,η,κ).
    Then the operations  P^m:R(G)→ R(Σ_m≀ G)
    make R into a global power functor.
  
  * The functor
    
    (exp-coalgebras)     →  ,  
        (R, P)  ⟼ (R, {P^m}_m≥ 0)

    is an isomorphism of categories.
  


(i) The fact that P:R→exp(R) takes values in exponential sequences
is equivalent to the restriction condition of the power operations.
The fact that P:R→exp(R) is a morphism of global Green functors
encodes simultaneously the unit, contravariant naturality, transfer,
multiplicativity and additivity relations of a global power functor.
The identity relation P^1= is equivalent to the counit condition of a coalgebra,
i.e., that the composite

    R   P  exp(R)   η_R  R

is the identity. The transitivity relation is equivalent to 

    exp(P)∘ P   =  κ_R ∘ P  ,

the coassociativity condition of a coalgebra.

Part (ii) is essentially reading part (i) backwards, and we omit the details.




The interpretation of global power functors as coalgebras over
a comonad has some useful consequences.
In general, the forgetful functor from any category of coalgebras 
to the underlying category has a right adjoint `cofree' functor. In particular,
colimits in a category of coalgebras are created in the underlying category. 
In our situation that means:


  
  
  * Colimits in the category of global power functors
    exist and are created in the underlying category of global Green functors.  
  
  * For every global Green functor R, the maps
    
    P^m  :  exp(R;G)  →exp(R;Σ_m≀ G) ,    P^m(x) =  κ(x)_m=(Ψ_k,m^*(x_k m))_k≥ 0

    make the global Green functor exp(R)
    into a global power functor.
  
  * When viewed as a functor to the category of
    global power functors as in (ii), the functor exp is right adjoint to the
    forgetful functor.
  



[Coproducts]
We let R and S be two global Green functors. Global Green functors
are the commutative monoids, with respect to the box product,
in the category of global functors. 
So the box product R S is the coproduct in the category of global
Green functors, with multiplication defined as the composite

    R S R S  Rτ_S,R SR R S S  μ_Rμ_S R S  .

If P:R→exp(R) and P':S→exp(S)
are global power structures on R and S, 
then R S has preferred power operations specified by
the morphism of global Green functors

    R S   P P' exp(R)exp(S)  → exp(R S)  ,

where the second morphism is the canonical one from the coproduct
of exp to the values of exp at a coproduct.
With these power operations, R S becomes a coproduct 
of R and S in the category of global power functors,
by Corollary <ref> (i).

This abstract definition of the power operations on R S 
can be made more explicit. Indeed, the power operations on R S 
are determined by  the formula

    P^m(x× y)  =  Δ^*(P^m(x)× P^m(y))
 
for all compact Lie groups G and K and classes x∈ R(G) and y∈ S(K), 
and by the relations of the power operations.
Here Δ:Σ_m≀(G× K)→(Σ_m≀ G)×(Σ_m≀ K)
is the diagonal monomorphism (<ref>).

The coproduct of global power functors is realized by the coproduct
of ultra-commutative ring spectra in the following sense.
If E and F are ultra-commutative ring spectra, then the ring spectra
morphisms E→ E F and F→ E F induce morphisms
of global power functors _0(E) →_0(E F)
and _0 (F) →_0(E F), and together they define a morphism
from the coproduct of global power functors

    _0 (E)  _0 (F)  → _0(E F)  .

If E and F are globally connective and at least one of them is flat
as an orthogonal spectrum, then this is an isomorphism of
global functors by Proposition <ref>, 
hence an isomorphism of global power functors.




[Localization of global power functors]
Now we discuss localizations of global power functors.
subjectlocalization!of global Green functors
We first consider a global Green functor R and a multiplicative
subset S⊆ R(e) in the `underlying ring', i.e., the value
at the trivial group. We define a global Green functor R[S^-1]
and a morphism of global Green functors i:R→ R[S^-1].
The value at a compact Lie group G is the ring

    R[S^-1](G) =  R(G)[p_G^*(S)^-1] ,

the localization of the ring R(G) at the multiplicative subset
obtained as the image of S under the inflation homomorphism p_G^*:R(e)→ R(G).
The value of the morphism i at G
is the localization map R(G)→ R(G)[p_G^*(S)^-1].
If α:K→ G is a continuous homomorphism, the relation
p_G∘α=p_K implies that the ring homomorphism

    α^*  :  R(G) → R(K)

takes the set p_G^*(S) to the set p_K^*(S).
So the universal property of localization provides a unique ring homomorphism

    α[S^-1]^* :    R[S^-1](G) =  R(G)[p_G^*(S)^-1]  → R(K)[p_K^*(S)^-1]
     =  R[S^-1](K)

such that α[S^-1]^*∘ i(G)=i(K)∘α^*.
Again by the universal property of localizations, this data produces 
a contravariant functor from the category  to the category of commutative rings.

Now we let H be a closed subgroup of G. We consider R(H)
as a module over R(G) via the restriction homomorphism ^G_H:R(G)→ R(H).
Because ^G_H(p_G^*(S))=p_H^*(S),
the localization R(H)[p_H^*(S)^-1]=R[S^-1](H)
is also a localization of R(H) at p_G^*(S) as an R(G)-module.
The reciprocity formula means that the transfer map _H^G:R(H)→ R(G)
is a homomorphism of R(G)-modules. The composite R(G)-linear map

    R(H)  _H^G  R(G) i(G) R(G)[p_G^*(S)^-1] =  R[S^-1](G)

thus extends over a unique R[S^-1](G)-linear map

    [S^-1]_H^G  : R[S^-1](H) =  R(H)[p_G^*(S)^-1] → R[S^-1](G)

such that [S^-1]_H^G∘ i(H)=i(G)∘_H^G.
Reciprocity for [S^-1]_H^G is equivalent to R[S^-1](G)-linearity.
The other necessary properties of transfers, such as transitivity, 
compatibility with inflations, vanishing for infinite Weyl groups, 
and the double coset formula all follow from corresponding properties
in the global Green functor R and the universal property of localization.
So altogether this shows that the objectwise localizations assemble into
a new global Green functor R[S^-1]; the homomorphisms i(G)
altogether form a morphism of global Green functors i:R→ R[S^-1]
by construction. The following universal property is also straightforward
from the universal property of localizations of commutative rings and modules.


subjectlocalization!of global power functors
Let R be a global Green functor and S a multiplicative subset of
the underlying ring R(e). Let f:R→ R' be a morphism of global Green functors
such that all elements of the set f(e)(S) are invertible in the ring R'(e).
Then there is a unique homomorphism of global Green functors f̅:R[S^-1]→ R'
such that f̅ i=f.


Now we let R be a global power functor. If we want the localization
R[S^-1] to inherit power operations, then we need an extra hypothesis
on the multiplicative subset S.


Let R be a global power functor and S a multiplicative subset of
the underlying ring R(e). Suppose that the multiplicative subset

    P^m(S)  ⊂ R(Σ_m)

becomes invertible in the ring R[S^-1](Σ_m), for every m≥ 1.  


  * There is a unique extension of the global Green functor R[S^-1]
to a global power functor such that the morphism i:R→ R[S^-1]
is a morphism of global power functors.

  * Let f:R→ R' be a morphism of global power functors
such that all elements of the set f(e)(S) are invertible in the ring R'(e).
Then there is a unique homomorphism of global power functors f̅:R[S^-1]→ R'
such that f̅ i=f.



(i) We use the comonadic description of global power functors given in  
Theorem <ref>. This exhibits the power operations of R
as a morphism of global Green functors P:R→exp(R).

The multiplication in the
ring exp(R;e) is componentwise, and for every s∈ S
the element P^m(s) has a multiplicative inverse t_m∈ R[S^-1](Σ_m) 
by hypothesis.
We claim that the sequence t=(t_m)_m≥ 0 is again exponential,
and hence an element of the ring exp(R[S^-1],e). Indeed,
for 0<i<m we have

    Φ_i,m-i^*(t_m)· (P^i(s)× P^m-i(s))    =  Φ_i,m-i^*(t_m)·Φ_i,m-i^*(P^m(s))
       =  Φ_i,m-i^*(t_m· P^m(s)) =  Φ_i,m-i^*(1) =    1 .

Since t_i× t_m-i is also inverse to P^i(s)× P^m-i(s)
and inverse are unique, we conclude that Φ^*_i,m-i(t_m)=t_i× t_m-i.
This shows that the inverses form another exponential sequence.

The relation

    P(s)· t  =  (P^m(s)· t_m)_m =  1

holds in the ring exp(R[S^-1];e), by construction. So the composite morphism

    R   P   exp(R) exp(i) exp(R[S^-1])

takes S to invertible elements. The universal property of 
Proposition <ref> thus provides 
a unique morphism of global Green functors

    P[S^-1]  :  R[S^-1] → exp(R[S^-1])

such that P[S^-1]∘ i=exp(i)∘ P.
Now we observe that

    η_R[S^-1]∘ P[S^-1]∘ i  =  η_R[S^-1]∘exp(i)∘ P  =  
     i∘η_R∘ P  =  i  ;

the uniqueness clause in the universal property of 
Proposition <ref> then implies that
η_R[S^-1]∘ P[S^-1] is the identity of R[S^-1].
Similarly,

    exp(P[S^-1])∘ P[S^-1]∘ i     =  exp(P[S^-1])∘exp(i)∘ P=  exp(P[S^-1]∘ i)∘ P
       =  exp(exp(i)∘ P)∘ P=  exp(exp(i))∘exp(P)∘ P
       =  exp(exp(i))∘κ_R∘ P=  κ_R[S^-1]∘exp(i)∘  P
       =  κ_R[S^-1]∘ P[S^-1]∘ i .

The uniqueness clause then implies that

    exp(P[S^-1])∘ P[S^-1] =  κ_R[S^-1]∘ P[S^-1]  .

So the morphism P[S^-1] is a coalgebra structure over the exp comonad.
The relation P[S^-1]∘ i=exp(i)∘ P then says that i is a morphism
of exp-coalgebras, so this completes the proof.

(ii) Since morphisms of global power functors are in particular morphisms
of global Green functors, the uniqueness clause follows from
Proposition <ref>.
If f:R→ R' is a morphism of global power functors
such that all elements of f(e)(S) are invertible in the ring R'(e),
then Proposition <ref> provides a
homomorphism of global Green functors f̅:R[S^-1]→ R'
such that f̅ i=f. We need to show that f̅ is also compatible
with the power operations.
We let

    P[S^-1]  :  R[S^-1] → exp(R[S^-1])      and    
     P'  :  R' → exp(R')

be the morphisms of global Green functors that encode the exp-coalgebra structures.
We observe that

    exp(f̅)∘ P[S^-1]∘ i     =  exp(f̅)∘exp(i)∘ P
       =  exp(f)∘ P =     P'∘ f =  P'∘f̅∘ i ;

the third equation is the hypothesis that f is a morphism of global power functors.
This is an equality between morphisms of global Green functors, so the 
uniqueness of Proposition <ref>
shows that exp(f̅)∘ P[S^-1]=P'∘f̅, i.e.,
f̅ is a morphism of exp-coalgebras.




[Localization at a subring of ]
We use Theorem <ref>
to show that global power functors can always be rationalized;
more generally, power operations `survive' localization 
at any subring of the ring  of rational numbers.
We consider a global power functor R and a natural number n≥ 2.
We claim that for every m≥ 1 the element P^m(n) of R(Σ_m) becomes
invertible in the ring R(Σ_m)[1/n]. We argue by induction over m;
we start with m=1, where the relation P^1(n)=n shows the claim.
Now we consider m≥ 2 and assume that for all 0≤ i < m
the element P^i(n) has an inverse t_i in the ring R(Σ_i)[1/n]. 
Then for every n-tuple (i_1,…,i_n) that
satisfies 0≤ i_j <m and i_1+…+i_n=m we get the relation

    ^Σ_m_Σ_i_1×…×Σ_i_n (P^m(n))·(t_i_1   ×…× t_i_n ) 
    =  (P^i_1(n)×…× P^i_n(n)) ·(t_i_1×…× t_i_n )
       =  (P^i_1(n)· t_i_1)×…× (P^i_n(n)· t_i_n )  
    =  1×…× 1

in the ring R(Σ_i_1×…×Σ_i_n)[1/n]. Thus 

    P^m(n)    =  P^m(1+…+1_n) =  ∑_i_1+…+i_n=m_Σ_i_1×…×Σ_i_n^Σ_m( P^i_1(1)×…× P^i_n(1)) 
       =  
    n· P^m(1) + ∑_i_1+…+i_n=m, i_j<m_Σ_i_1×…×Σ_i_n^Σ_m( 1×…× 1) 
       =  
    n + ∑_i_1+…+i_n=m, i_j<m_Σ_i_1×…×Σ_i_n^Σ_m(^Σ_m_Σ_i_1×…×Σ_i_n (P^m(n))·(t_i_1×…× t_i_n )) 
       =  
    n + ∑_i_1+…+i_n=m, i_j<m P^m(n)·_Σ_i_1×…×Σ_i_n^Σ_m(t_i_1×…× t_i_n )

Rearranging the terms gives

    P^m(n)·   (1- ∑_i_1+…+i_n=m, i_j< m_Σ_i_1×…×Σ_i_n^Σ_m( t_i_1×…× t_i_n))  =  n  .

So P^m(n) has an inverse in the ring R(Σ_m)[1/n],
and this completes the inductive step.

Now we let S be any multiplicative subset of the ring of integers.
By the previous paragraph, Theorem <ref>
applies and provides a unique structure of global power functor on the
global Green functor R[S^-1] such that the morphism i:R→ R[S^-1]
is a morphism of global power functors.subjectrationalization!of global power functors
In particular, if we let S be the set of all positive integers, 
we can conclude that the rationalization R of R
has a unique structure of global power functor such that the localization
map R→ R is a  morphism of global power functors.  




[Monadic description of global power functors] 
Now we explain that the category of global power functors in not only comonadic,
but also monadic over the category of global Green functors.
In fact, both categories are examples of algebras over 
multisorted algebraic theories (also called colored theories).
The `sorts' (or `colors') are the compact Lie groups and the 
content of this claim is that the structure of
global Green functors respectively global power functors
can be specified by giving the values R(G) at every compact Lie group,
together with n-ary operations for different n≥ 0 and
varying inputs and output, and relations between composites of
those operations.

In the case of global Green functors, the operations to be specified are


  * the constants given by the additive and multiplicative units in the rings R(G),

  * the unary operations given by the additive inverse map in R(G),
the restriction maps and transfers,

  * and the binary operations specifying the addition and multiplication in
the rings R(G).

The relations include, among others, the neutrality, associativity and
commutativity of addition and multiplication; the distributivity
in the rings R(G); the additivity of restriction and transfers;
the functoriality of restrictions and transitivity of transfers;
and the double coset and reciprocity formulas.

Global power functors have additional unary operations, the power operations,
and additional relations as listed in Definition <ref>.



The forgetful functor from the category of global power functors
to the category of global Green functors has a left adjoint.  
The category of global power functors is isomorphic to the
category of algebras over the monad of this adjunction.


For the existence of the left adjoint we have to show that for every
global Green functor R the functor

    → (sets) ,   
    S  ⟼ (R, S)

is representable. This is a formal consequence of the existence of free
global power functors, colimits of global power functors and the fact
that global power functors are a multi-sorted theory. 
We explain this in more detail, without completely formalizing the argument. 

We choose a set of compact Lie groups {K_i}_i∈ I
that contains one compact Lie group from every isomorphism class.
We form the global power functor

    L  =  _ i∈ I, x∈ R(K_i)  C_i,x ,

a coproduct, indexed by all pairs (i,x)
consisting of an index i∈ I and an element x∈ R(K_i),
of free global power functors

    C_i,x =  C_K_i

generated by the compact Lie groups K_i.
On this free global power functor we impose the minimal amount of relations
so that the maps R(K_i)→ L(K_i) that send x∈ R(K_i)
to the generator indexed by (i,x) becomes a morphism of global Green functors.
Here `imposing relations' means that we form 
another box product L'  of free global power functors,
with one box factor for each relation between elements in the various sets R(K_i).
For example, we include one factor for the sum of each pair 
of elements in the same set R(K_i),
another factor for the product of each pair 
of elements in the same set R(K_i),
and more factors for zero elements, multiplicative units, 
all restriction relations  and all transfers relations. 
Then we form a coequalizer,
in the category of global power functors

    L' @<.4ex>[r] @<-.4ex>[r]     L [r]     F

where the two morphisms from L' to L restrict, on each box factor,
to the morphism that represents the respective relation.
The resulting global power functor then represents 
the functor (R,-),
so we can take F as the value of the left adjoint on R.

Since global power functors are equivalent to
the coalgebras over the exp-comonad, the forgetful functor
creates all colimits, in particular coequalizers.
So by Beck's monadicity theorem (see for example <cit.>),
the tautological functor from global power functors to algebras over
the adjunction monad is an isomorphism of categories.



[Limits]
The category of global power functors has limits, and they are 
defined `groupwise'. 
A special case of a limit is the product of global power functors,
which is realized by the product of ultra-commutative ring spectra.
Indeed, if E and F are ultra-commutative ring spectra, then so is
the product E× F of the underlying orthogonal spectra,
and the canonical map

    _0(E× F)  → _0( E) ×_0( F)

is an isomorphism of global power functors
(by Corollary <ref> (ii)).






§ EXAMPLES



In this section we discuss various examples of 
and constructions with global power functors, and how these are
realized topologically by ultra-commutative ring spectra.
These examples include the Burnside ring global power functor
(Example <ref>), the global functor represented
by an abelian compact Lie group (Proposition <ref>),
and constant global power functors (Example <ref>).
The orthogonal spectrum A consisting of the A-linearizations
of spheres (Construction <ref>) tries to be an Eilenberg-Mac Lane spectrum
for the constant global functor, and it is so on finite groups.
Closely related to  is the infinite symmetric product spectrum ,
see Example <ref>; this ultra-commutative
ring spectrum comes with a filtration by finite symmetric products,
realizing an interesting filtration of the Burnside ring global
functor on _0, see (<ref>). 
We close this section with a discussion of
the complex representation ring global functor (Example <ref>),
and a global view on `explicit Brauer induction' (Remark <ref>).




[Burnside ring global functor]subjectBurnside ring global functor
The Burnside ring global functor =(e,-) 
is the unit object for the box product of global functors,
and hence an initial object in the category of global Green functors.
Initial objects are examples of colimits,
so Corollary <ref> (i) 
implies that  has a unique structure of global power functor. 
Indeed, there is a unique morphism P:→exp() of global Green functors
(since  is initial), and the coalgebra diagrams commute
(again since  is initial).
With these power operations,  is also an initial global power functor.

We can make the power operations in the Burnside ring global functor more explicit. 
Indeed, the group (G) is free abelian with a basis
given by the elements t_H=_H^G(p_H^*(1)) for every conjugacy class
of closed subgroups H≤ G with finite Weyl group, where p_H:H→ e
is the unique homomorphism. On these generators,
the naturality properties of a global power functor force the power operations to be 

    P^m(t_H)     =  P^m(_H^G(p_H^*(1))) =  _Σ_m≀ H^Σ_m≀ G((Σ_m≀ p_H)^*(P^m(1)))
       =  _Σ_m≀ H^Σ_m≀ G((Σ_m≀ p_H)^*(p_Σ_m^*(1)))=  _Σ_m≀ H^Σ_m≀ G(p_Σ_m≀ H^*(1)) =    
    t_Σ_m≀ H .

This determines the power operations in general by the additivity property,
and also shows the uniqueness.

When restricted to finite groups, the ring (G) is isomorphic to 
the Grothen­dieck group of finite G-sets, and in this description 
the power operations are given by raising a finite G-set to a power, i.e.,
the power map

    P^m  :  (G)  → (Σ_m≀ G)

takes the class of a finite G-set S to the
class of the (Σ_m≀ G)-set S^m.
Indeed, for the additive generator [G/H]=t_H
of (G) this is the relation (<ref>),
and for general finite G-sets it follows from the additivity formula
for power operations and the fact that for two finite G-sets S and T
the power (S⨿ T)^m is (Σ_m≀ G)-equivariantly isomorphic to
the coproduct 

    ∐_k=0^m  
    (Σ_m≀ G)×_(Σ_k≀ G)× (Σ_m-k≀ G) (S^k× T^m-k) .

The canonical power operations in the Burnside ring global
functor correspond to the  homotopy theoretic power operations 
for the global sphere spectrum.subjectsphere spectrum
Indeed, since  is initial in both the category
of global Green functors and in the category of global power functors,
any isomorphism of global Green functors is automatically
compatible with power operations. In other words, we can conclude that the square

    (G) [r]^-P^m[d]_   (Σ_m≀ G) [d]^
    π_0^G() [r]_-P^m   π_0^Σ_m≀ G ()

commutes for all G and m without having to go back to the
definition of the operations in _0();
the vertical maps are the action on the multiplicative units.




The previous example generalizes to the represented global functor
_A=(A,-) for every abelian compact Lie group,
which has a preferred structure of a global power functor.
The following proposition is a stable analog of 
Proposition <ref>, which describes
a structure  of global power monoid on the -functor represented by A.



subjectcompact Lie group!abelian
Let A be an abelian compact Lie group.
The represented global functor _A
has a unique structure of global power functor subject to the following two
conditions:


  * The multiplication is the composite

    _A_A   _A× A (μ^*,-) _A ,

where μ:A× A→ A is the multiplication
and μ^*∈(A,A× A) is the associated restriction morphism.

  * The power operations satisfy

    P^m(1_A) =  p_m^*    in  (A,Σ_m≀ A)  ,

the inflation operation of the continuous homomorphism 

    p_m : Σ_m≀ A →  A  ,   
    (σ; a_1,…,a_m) ⟼ a_1·…· a_m  .


Moreover, for every global power functor R the evaluation map  

    (_A,R)  → R(A) ,    f  ⟼ f_A(1_A)

is injective with image the set of those x∈ R(A)
such that P^m(x)=p_m^*(x) for all m≥ 1.


Since the convolution product of two represented functors is again represented,
morphisms of global functors _A_A→_A
are determined by their effect on 
the element 1_A 1_A∈ (_A_A)(A× A).
The first condition fixes this to be the operation μ^*.
So there is at most one structure of global Green functor on _A
that satisfies the first condition.
Since this representable global functor is freely generated by the identity
1_A in (A,A), the power operations are all determined by naturality
from the effect on this generator. So together we have shown that there is at most
one structure of global power functor that satisfies the two conditions.

Now we construct the desired structure.
The associativity, commutativity and unitality 
of the multiplication μ:A× A→ A readily imply that
the associated multiplication of _A is 
associative, commutative and unital. So this multiplication makes _A
into a global Green functor. The relations

    p_m∘Φ_k,m-k =  p_k· p_m-k :  
    (Σ_k≀ A)×(Σ_m-k≀ A)  → A

imply that the sequence (p_m^*)_m≥ 0 is exponential.
So the Yoneda lemma provides a unique morphism of global functors

    P  = (P^m)_m≥ 0 :  _A → exp(_A)      such that    
    P^m(1_A) =   p_m^*  .

We claim that P is a morphism of global Green functors.
The unitality follows from the fact that the restriction of p_m
to Σ_m≀ e is the trivial homomorphism.
Since the global functor _A_A
is representable by A× A, the Yoneda lemma reduces 
the multiplicativity property to the relation

    exp(_A,μ^*)((p^*_m)_m)  =  exp(_A,q_1^*)((p_m^*)_m) · exp(_A,q_2^*)((p_m^*)_m)

in the ring exp(_A,A× A), where q_1,q_2:A× A→ A
are the two projections. This relation, in turn,
is a consequence of the commutativity of
the following diagram of group homomorphisms:

    @C=12mmΣ_m≀(A× A)[d]_Δ[r]^-Σ_m≀μ   Σ_m≀ A[d]^p_m
    
    (Σ_m≀ A)× (Σ_m≀ A)[r]_-p_m· p_m    A

Here the left vertical map is a diagonal, given by

    Δ(σ; (a_1,a̅_1),…,(a_m,a̅_m)) =((σ; a_1,…,a_m),(σ; a̅_1,…,a̅_m))  .

We conclude the construction of the global power structure by showing 
that the morphism P:_A→exp(_A) provides a coalgebra structure
over the exp-comonad.
It suffices to show
the relations η__A∘ P= and exp(P)∘ P=κ__A∘ P 
on the universal element 1_A in _A(A),
one more time by the Yoneda lemma.
For the first one, this boils down to the fact 
that p_1:Σ_1≀ A→ A is the preferred isomorphism.
The second one follows from the commutativity of the following square:

    [b]
     @C=12mmΣ_k≀Σ_m≀ A[r]^-Ψ_k,m[d]_Σ_k≀ p_m   Σ_k m≀ A[d]^p_k m
    Σ_k≀ A[r]_-p_k    A



The global power functor _A described
in Proposition <ref> is realized by 
an ultra-commutative ring spectrum.
Indeed, Construction <ref>
provides a global classifying space R(B A) for the
abelian compact Lie group A with an ultra-commutative multiplication.subjectglobal classifying space!of an abelian compact Lie group  
The associated unreduced suspension spectrum
Σ^∞_+ R(B A) is then an ultra-commutative ring spectrum.
Theorem <ref>
shows that _0(Σ^∞_+ R(B A)) is 
freely generated, as a global functor,
by the stable tautological class e_A∈π_0^A(Σ^∞_+ R(B A)),
the stabilization of the unstable tautological class u_A.
The characterization of the multiplication and power operations on _A
only involve restriction maps (but no transfers); so the fact that
_0(Σ^∞_+ R(B A)) 
realizes the global power structure of
Proposition <ref> follows from the
corresponding unstable relations in the global power monoid
_0( R(B A)), as explained in Example <ref>. 



For a compact Lie group G we define

    u_G^ucom ∈ π_0^G(Σ^∞_+(B_G))

as the image of the unstable tautological class
u_G ∈ π_0^G( B_B), defined in (<ref>),
under the two composites in the commutative square

    @C=15mmπ_0^G( B_B)[r]^-π_0^G(η)[d]_σ^G   π_0^G( (B_B))[d]^σ^G
    π_0^G(Σ^∞_+ B_B) [r]_-π_0^G(Σ^∞_+ η)   π_0^G(Σ^∞_+ (B_B))

where η:B_ G→(B_ G) is the adjunction unit, i.e., the
inclusion of the homogeneous summand for m=1. 
The stabilization map σ^G was defined in (<ref>).
Here, as usual, we have made an implicit choice of non-zero faithful 
G-representation V, and the global classifying space is B_G=_G,V.
We already know that


  * the class u_G freely generates _0(B_G) as a Rep-functor
  (Proposition <ref> (ii)),

  * the class u_G^umon=π_0^G(η)(u_G) freely generates
  _0((B_G)) as a global power monoid (Theorem <ref> (ii)), and

  * the class e_G=σ^G(u_G) freely generates
  _0(Σ^∞_+ B_G) as a global functor
  (Proposition <ref>).subjectglobal classifying space



  * _0(B_G) is the freely generated by u_G as a  Rep-functor
  (Proposition <ref> (ii)),

  * _0((B_G)) is the freely generated by u_G^umon=π_0^G(η)(u_G) 
  as a global power monoid (Theorem <ref> (ii)), and

  * _0(Σ^∞_+ B_G) is the freely generated by e_G=σ^G(u_G) 
  as a global functor
  (Proposition <ref>).subjectglobal classifying space

The next theorem complements these results and shows that
the global power functor _0(Σ^∞_+(B_B))
is freely generated by the class u_G^ucom.
The free global power functorsubjectglobal power functor!free
C_G generated by the compact Lie group G 
was introduced in Example <ref>.
The underlying global functor is

    C_G  =  ⊕_m≥ 0 (Σ_m≀ G,-)  ,

and there is a preferred element 1_G∈ C_G(G), the identity operation
of π_0^G in the summand indexed by m=1.
The adjective `free' is justified by Proposition <ref> (ii),
which says that for every global power functor R and every element x∈ R(G)
there is a unique morphism of global power functors f:C_G→ R such that f(1_G)=x.


  Let G be a compact Lie group.
  
  
  * The unique morphism of global power functors

    φ :  C_G  → _0(Σ^∞_+ (B_ G))

satisfying φ(1_G)=u_G^ucom is an isomorphism.
  
  * For every global power functor R and every element x∈ R(G)
  there is a unique morphism of global power functors
  f:_0(Σ^∞_+ (B_ G))→ R such that f(u_G^ucom) =  x.
  


  (i)
  The ultra-commutative monoid (B_G) is the disjoint union
  of the orthogonal spaces ^m(B_G) for m≥ 0.
  Moreover, ^m(B_G) is a global classifying space for the group
  Σ_m≀ G, in such a way that the class [m](u_G)∈π_0^G(^m(B_G))
  matches the unstable tautological class u_Σ_m≀ G, 
  by Example <ref>.
  So by Proposition <ref> the morphism of global functors
  
    ψ_m  :  (Σ_m≀ G,-) → _0(Σ^∞_+^m(B_G))

  classified by σ^Σ_m≀ G([m](u_G))=P^m(σ^G(u_G)) in 
  π_0^Σ_m≀ G(Σ^∞_+^m(B_G))
  is an isomorphism.
  The functor Σ^∞_+ takes a disjoint union of orthogonal
  spaces to a wedge of orthogonal spectra, and
  equivariant stable homotopy groups take wedges to sums.
  Hence the morphism  
  
    ψ =  ⊕_m≥ 0ψ_m  :  
      C_G  =     ⊕_m≥ 0 (Σ_m≀ G,-)
    →    ⊕_m≥ 0  _0(Σ^∞_+^m(B_G)) =  _0(Σ^∞_+(B_G))

  is an isomorphism of global functors.

  The power operations in C_G satisfy [m](1_G)=1_Σ_m≀ G;
  since φ is a morphism of global power functors, it also satisfies
  
    φ(1_Σ_m≀ G) =  φ([m](1_G))  =  [m](φ(1_G))
        =   [m](u^umon_G) =  σ^G([m](u_G)) .

  So the morphisms φ and ψ coincide on the classes 1_Σ_m≀ G
  for all m≥ 0. Since these classes generated C_G as a global functor,
  we conclude that φ=ψ. Since ψ is an isomorphism, this proves the claim.
  
  Part (ii) is the combination of (i) and the freeness property of C_G
  from Proposition <ref> (ii).





[Monoid rings] 
We let R be a global Green functor and M a commutative monoid.
We denote by R[M] the monoid ring functor; its value at
a compact Lie group G is given by

    (R[M])(G)  =  R(G)[M]  ,

the monoid ring of M over R(G). The structure as global functor
is induced from the structure of R and constant in M.
The multiplication and unit are
induced from the multiplication and units of R and M.
The global Green functor R[M] can be characterized as follows
by the functor that it represents:
for every global Green functor S, morphisms R[M]→ S biject with pairs consisting 
of a morphism of global Green functors R→ S and a monoid homomorphism
M→ (S(e),·) to the multiplicative monoid of the underlying ring S(e).

Now suppose that R is even a global power functor.
Then R[M] inherits a natural structure as global power functor:
we define the power operation

    P^m  :   R(G)[M]  → R(Σ_m≀ G)[M]

by P^m(r· x)=P^m(r)· x^m for r∈ R(G) and x∈ M, 
and then we extend this
by additivity to general elements in R(G)[M].
In this upgraded setting, the global power functor R[M] 
has a similar characterization as in the previous paragraph:
for a global power functor S, we let 

    (S)  =  { x∈ S(e) |  P^m(x)=(p_Σ_m)^*(x) for all m≥ 1}

be the set of monoid-like elements of S;
here p_Σ_m:Σ_m→ e is the unique homomorphism.
The set (S) is a submonoid of the multiplicative monoid of S(e).
Then morphisms R[M]→ S biject with pairs consisting 
of a morphism of global power functors R→ S and a monoid homomorphism
M→(S) to the monoid-like elements of S.

Now we let E be an ultra-commutative ring spectrum.
Then the monoid ring spectrum E[M]=E M_+subjectmonoid ring spectrum
is another ultra-commutative ring spectrum, 
see Construction <ref>.
The unit of M induces a morphism of ultra-commutative ring spectra E→ E[M]
that induces a morphism of global power functors 
_0(E)→_0(E[M]). Similarly, the unit of E induces a monoid homomorphism
M→ E(0) M_+, and this induces a monoid homomorphism M→(_0(E[M]))
to the  monoid-like elements in the multiplicative monoid of the ring π_0^e(E[M]).
The universal property of the algebraic monoid ring combines these
two pieces of data into a morphism of global power functors

    (_0 E)[M]   → _0(E[M])  .

Additively, the left hand side is a direct sum of copies
of _0(E), indexed by the elements of M.
Similarly, the orthogonal spectrum E[M] is a wedge of copies
of E, indexed by the elements of M.
So the right hand side _0(E[M]) is also a direct sum of copies of _0(E)
(by Corollary <ref> (i)).
We conclude that the morphism of global power functors (<ref>) is an isomorphism. 




[Constant global power functors and Eilenberg-Mac Lane spectra]subjectEilenberg-Mac Lane spectrum!of a commutative ring
We let B be a commutative ring. Then the constant globalsubjectglobal power functor!constant  
functor B (see Example <ref> (iii)) 
is naturally a global Green functor, via the ring structure of B.
For every compact Lie group G, every exponential sequence x∈exp(B;G) 
satisfies x_n=(x_1)^n; so an exponential sequence
is completely determined by the element x_1, which can be chosen arbitrarily. 
Hence the morphism

    η_B :  exp(B)  → B

is an isomorphism of global Green functors. So when restricted
to constant global Green functors, the exp comonad is isomorphic to the
identity. Thus B has a unique structure of coalgebra
over the comonad exp, with structure morphism the inverse of η_B.

The preferred global power structure on the constant global functor B
can also be derived more directly.
In fact, since the power operation P^m has to be an equivariant refinement
of the m-th power map in R(G) (compare Remark <ref>), 
the only possibility to define P^m is as

    P^m :  B(G) = B  → B = B (Σ_m≀ G) ,   
    b  ⟼ b^m   ,

the m-th power in the ring B.


Since B is constant, 
every morphism R→B of global functors is determined
by the map R(e)→B(e)=B. Moreover, every ring homomorphism
ψ:R(e)→ B extends uniquely to a morphism of global power functors
ψ̂:R→B by defining its value at G as the composite

    R(G)  ^G_e  R(e)  ψ  B .

In other words, the functor 

    (commutative rings) →  ,   
    B  ⟼ B

is right adjoint to the functor that takes a global power functor R 
to the underlying ring R(e).



The author does not know an explicit pointset level model
for an ultra-commutative ring spectrum that realizes the  constant global power functor
B of a commutative ring B. 
The Eilenberg-Mac Lane spectrum B, 
discussed in the next construction,
is an ultra-commutative ring spectrum and tries to realize B:
the global power functor _0( B)
is indeed constant on finite groups, but the restriction maps
are not generally isomorphisms, compare Example <ref>.
The morphism of global power functors
_0( B)→B adjoint to the identification
π^e_0( B) B is thus an isomorphism at finite groups
(and some other compact Lie groups), but not an isomorphism in general.


subjectEilenberg-Mac Lane spectrum!of an abelian group|( 
subjectEilenberg-Mac Lane spectrum!of an abelian group 


Let A be an abelian group.
The Eilenberg-Mac Lane spectrum
AsymbolA - Eilenberg-Mac Lane spectrum of an abelian group 
is defined at an inner product space V by

    ( A)(V)  =  A[S^V]  ,

the reduced A-linearizationsubjectlinearization!of a space 
of the V-sphere. 
The orthogonal group O(V) acts through the action on S^V and
the structure map
σ_V,W:S^V ( A)(W)→ ( A)(V⊕ W) is given by

    S^V A[S^W]  →  A[S^V⊕ W]  ,   
    v  ( ∑_i m_i· w_i )  ⟼∑_i m_i·  (v w_i)   .

Non-equivariantly, A[S^V] is an Eilenberg-Mac Lane space of type (A,n), 
where n=(V). 
Hence the underlying non-equivariant homotopy type of A is that of
an Eilenberg-Mac Lane spectrum for A.


As we shall now discuss, the equivariant homotopy groups of A are 
in general not concentrated in dimension 0, and hence A is 
not the Eilenberg-Mac Lane spectrum of a global functor.
However, on finite groups, the equivariant behavior of A is as expected.
We recall from Definition <ref>
that an orthogonal spectrum is
left inducedsubjectleft induced 
from the global family subjectglobal family!of finite groups 
of finite groups if it is in the essential image of the left adjoint
L_:_→ from the -global homotopy category.
The inclusion of generators is a homeomorphism A A[S^0]=( A)(0)
and this induces a bijection

    A   π_0(A)  =  [S^0, ( A)(0)]  .

Since A is in particular a non-equivariant Ω-spectrum
(for example by Proposition <ref> below),
the composite of this bijection with the stabilization map is an 
isomorphism of abelian groups

    A   π_0^e( A)  .

The restriction maps

    ^G_e  :  π_0^G( A)  → π_0^e( A)   A  =  A(G)

form a morphism of global functors _0( A)→A
to the constant global functor with value A.
Since A is globally connective (by the next proposition), 
there is a unique morphism 

    ρ :   A  → HA

in the global stable homotopy category that realizes the morphism on π_0^e.




For every abelian group A the Eilenberg-Mac Lane spectrum  A is 
globally connective, 
left induced from the global family  of finite groups 
and a -Ω-spectrum. 
The morphism ρ:  A → HA is a -global equivalence.


The orthogonal spectrum A is obtained by evaluation of a 
-spacesubjectGamma-space@-space A on spheres,
where A(T)=A[T] is the reduced A-linearization of a finite based set T.
So A is left induced from the global family 
by Proposition <ref> (ii). 

The -space A is actually a -set,
i.e., all its values have the discrete topology.
The latching map l_n:_U⊊{1,…,n}  A[U] →  A[{1,…,n}]
defined in (<ref>) is thus an injective map 
(by Proposition <ref> (ii)) 
between discrete Σ_n-spaces, hence a Σ_n-cofibration.
It is then a (G×Σ_n)-cofibration
if we let a compact Lie group G act trivially.
Proposition <ref> (ii) then shows that
the orthogonal G-spectrum A()= A (with trivial G-action)
is equivariantly connective. Since G was arbitrary, this proves that A
is globally connective.

Dos Santos shows in <cit.> that for every representation V 
of a finite group G, the G-space A(V)=A[S^V] 
is an equivariant Eilenberg-Mac Lane space of type (A,V), i.e., the G-space
_*(S^V, A[S^V]) has homotopically discrete fixed points for 
all subgroups H of G and the natural map

    A  → [S^V,A[S^V]]^H = π_0( _*^H(S^V,A[S^V]) )

sending m∈ A to the homotopy class of m· -:S^V→ A[S^V]
is an isomorphism.
This shows that A is a -Ω-spectrum
for the constant global functor A.

We offer an independent proof of the -Ω-property 
via the -G-space techniques of Segal and Shimakawa <cit.>,subjectGamma-space@-space!equivariant 
in our adaptation of Theorem <ref>.
We let G act trivially on the -space A.
The G-maps 

    P_S  :  A(S_+)  → (S,A(1_+))  =  A^S  ,   
    P_S(x)(s) =  A(p_s)(x)  ,

are homeomorphisms, so in particular G-weak equivalences.
So A is a special -G-space in the sense 
of Shimakawa <cit.>, 
see also Definition <ref> below. 
Since π_0(A(1_+)) is isomorphic to A, 
and hence a group (as opposed to a monoid only),
the -G-space A is very special.
Since A is also G-cofibrant,
 Theorem <ref> shows that
for every finite group G and every pair of
G-representations V and W, the adjoint structure maps 
σ̃_V,W: A[S^W]→_*(S^V, A[S^V⊕ W])
is a G-weak equivalence. 



The properties mentioned in the previous proposition
do not generalize to compact Lie groups of positive dimension,
i.e., contrary to what one may expect at first, 
( A)_G is not generally a G-Ω-spectrum,
not all restriction maps in dimension 0 are isomorphisms 
(see Example <ref>),
and the groups π_*^G (  A ) need not be concentrated in dimension 0
(see Theorem <ref>).





We now consider the important special case A=.
The equivariant homotopy group π_0^G(  ) 
may be larger thansymbol - Eilenberg-Mac Lane spectrum of the integers 
a single copy of the integers, and we are now going to give a presentation
of π_0^G(  ). Before we do so, we compare  to
the `infinite symmetric product of the sphere spectrum'.


[Infinite symmetric product]subjectinfinite symmetric product spectrum
There is no essential difference if we consider
the infinite symmetric product S p^∞ (i.e., the reduced free abelian monoid)  
instead of the reduced free abelian group [-] generated by representation spheres:
the levelwise inclusions of the free abelian monoids into the free abelian groups
provide a morphism of ultra-commutative ring spectrasymbol - infinite symmetric product

    =  {(S^V) }_V →{[S^V] }_V =   ,

and this morphism is a global equivalence by the following proposition.

In his unpublished preprint <cit.>,
Segal argues that for every finite group G 
and every G-representation V with V^G 0 the map

    (S^V)  → [S^V]

is a G-weak equivalence. A published proof of this fact
appears as Proposition A.6 in Dugger's paper <cit.>. 
This generalizes to compact Lie groups:
subjectEilenberg-Mac Lane spectrum!of the integers


Let G be a compact Lie group.

  
  * For every G-representation V such that V^G 0, 
    the natural map (S^V)→[S^V] is a G-weak equivalence.
  
  * The morphism (<ref>) is a global equivalence 
    of orthogonal spectra from S p^∞ to .



(i) By application to all closed subgroups of G, it suffices to show 
that the fixed point map (S p^∞(S^V))^G→ ([S^V])^G is a 
non-equivariant weak equivalence. 
We let G^∘≤ G denote the connected component of the identity element and
G̅=G/G^∘ the finite group of components of G.
To calculate G-fixed points we can first take G^∘-fixed points and
then G̅-fixed points. 
Proposition <ref> provides homeomorphisms

    (S p^∞(S^V))^G    (S p^∞(S^V^G^∘))^G̅     and    
    ([S^V])^G    ([S^V^G^∘])^G̅ .

Since V^G^∘ is an orthogonal representation of
the finite group G̅ the map

    (S p^∞(S^V^G^∘))^G̅ →  ([S^V^G^∘])^G̅ ,

is a weak equivalence by <cit.>. 
Strictly speaking, Dugger's proposition is stated only for
geometric realizations of G̅-simplicial sets; 
since the representation sphere S^V^G^∘ 
admits the structure of a finite G̅-CW-complex, 
it is G̅-homotopy equivalent to the realization of a G̅-simplicial set,
compare Proposition <ref> (ii).
So we can also apply Dugger's result in our situation. 
Part (ii) is then immediate from (i).




The infinite symmetric product of any based space X has a natural filtration

    X = S p^1(X) ⊆ S p^2(X) ⊆ … ⊆ S p^n(X) ⊆ …

by the finite symmetric products. Evaluation on spheres
provides orthogonal spectra S p^n  =  S p^n() and a filtration

    =S p^1 ⊆ S p^2 ⊆ … ⊆S p^n ⊆ …

of S p^∞ by orthogonal subspectra. We study the equivariant
and global properties of this filtration in <cit.>.
The 0-th homotopy group global functor of the orthogonal spectrum S p^n
has a very compact description as follows.
We recall that the Burnside ring (G) of a compact Lie group G is
freely generated by the classes t_H^G=_H^G(p_H^*(1))
where H runs through a set of representatives of the conjugacy classes
of subgroups of G with finite Weyl group.
Theorem 3.12 of <cit.> shows that
the global functor _0(S p^n) is the quotient
of the Burnside ring global functor by the global subfunctor generated by
the element n· 1- t_Σ_n-1^Σ_n in (Σ_n),

    _0(S p^n)   /n· 1  -  t_Σ_n-1^Σ_n .

Letting n go to infinity and combining this with part (ii)
of Proposition <ref> gives the following calculation of
the global functor _0().
We let I_∞ denote the global subfunctor of the Burnside ring
global functor  generated by the classes 
n· 1- t_Σ_n-1^Σ_n for all n≥ 1.
We show in <cit.>
that the unit map → S p^∞ induces an isomorphism of global functors

    /I_∞  _0(S p^∞)  _0() .

This calculation can be made even more explicit.
Elementary algebra (see <cit.>)
identifies the value I_∞(G) at a compact Lie group G
as the subgroup of (G) generated by the classes

    [H:K]· t_H^G  -  t_K^G

for all nested sequences of 
closed subgroup K≤ H≤ G such that W_G H is finite
and K has finite index in H. The possibility that K has infinite Weyl group
in G is allowed, in which case t_K^G=0.
This presents the equivariant homotopy group
as an explicit quotient of the Burnside ring of G:

    π_0^G()  π_0^G(S p^∞)  (G)/I_∞(G)






For every finite group G, the group π_0^G() is free of rank 1,
generated by the multiplicative unit, 
and the inflation map p_G^*:π_0^e()→π_0^G() is an isomorphism.
This does not persist to general compact Lie groups of positive dimension.
An explicit example for which π_0^G(  ) has rank bigger than one 
is G=S U(2).subjectspecial unitary group!S U(2)
Then the classes 1 and _N^S U(2)(1) are a -basis for
π_0^S U(2)(  ) modulo torsion,
see <cit.>,
where N is a maximal torus normalizer. 



To illustrate that the orthogonal spectrum A
is typically not an Eilenberg-Mac Lane spectrum of any global functor,
we calculate the U(1)-equivariant homotopy group π_1^U(1)( A).


As we shall now see, all classes in the group π_1^U(1)( A) 
arise in a systematic way via dimension shifting transfers 
from finite subgroups of U(1).subjectunitary group!U(1)
We organize these transfers into a natural isomorphism from A
to π_1^U(1)( A).
 
We let C be any finite subgroup of U(1).
We identify  with the tangent space 
of the distinguished coset e C in U(1)/C via the differential of the smooth curve

    → U(1)/C  ,    t ⟼ e^2π i t· C .

The upshot of this identification T_e C( U(1)/C)
is that we can view the dimension shifting transfer (<ref>)
as a homomorphism

    _C^U(1) :  π_1^C(E S^1)  → π_1^U(1)(E) .

We can then define an additive map ψ_C:A→π_1^U(1)( A) as the composite

    A   π_0^e( A) [] p_C^*  π_0^C( A) []- S^1π_1^C( A S^1)  _C^U(1)  π_1^U(1)( A)

of inflation, the suspension isomorphism and the dimension shifting transfer.

Now we consider a subgroup C' of C of index m. 
Since the restriction of _0( A) to finite groups is a 
constant global functor, the relation

    _C'^U(1)∘ (- S^1)   ∘ p_C'^*  =  _C^U(1)∘_C'^C∘ (- S^1)∘ p_C'^* 
       =  _C^U(1)∘ (- S^1)∘_C'^C∘ p_C'^*  
    =    m·_C^U(1)∘ (- S^1)∘ p_C^*

holds as homomorphisms π_0^e( A)→π_1^U(1)( A),
by transitivity of transfers (Proposition <ref>).
Hence ψ_C'=m·ψ_C and we obtain a well-defined homomorphism

    ψ :  A → π_1^U(1)( A)     by    ψ(a r/s )  =  r·ψ_C_s(a) ,

where C_s⊂ U(1) is group of s-th roots of unity.


 
The homomorphism ψ : A→π_1^U(1)( A) is an isomorphism
for every abelian group A.


Since the orthogonal spectrum A is obtained from a -space
by evaluation on spheres, the inflation homomorphism

    p^* :  π_*^e( A) =  Φ_*^e( A) → Φ_*^U(1)( A)

is an isomorphism of geometric fixed point homotopy groups,
by Proposition <ref> (ii).
The non-equivariant homotopy groups of A are concentrated in dimension 0,
so the U(1)-equivariant geometric fixed point homotopy groups Φ_k^U(1)( A)
vanish for all k 0.
The isotropy separation sequence (<ref>)
thus shows that the map E→∗ induces an isomorphism 

    π_1^U(1)( A E_+)  π_1^U(1)( A)  ,

where E is a universal U(1)-space for the family of proper closed (i.e., finite)
subgroups of U(1).

We let X be mapping telescope of the sequence of projections

    U(1) → U(1)/C_2 → U(1)/C_6 → … → U(1)/C_n ! → … .

This is a U(1)-CW-complex without U(1)-fixed points and
such that the fixed point space X^C is path connected for every finite subgroup C
of U(1). So we can build a universal space E by attaching
equivariant cells with finite isotropy groups of dimension 2 and larger to X.
The Wirthmüller isomorphisms

    π_k^U(1)(  A (U(1)/C)_+ S^n)   π_k^C( A S^1 S^n)   π_k-1-n^C( A)

and the fact that the C-equivariant homotopy groups of A
are concentrated in dimension 0 mean that the group
π_k^U(1)(  A E/X) is trivial for all k≤ 2. 
Hence the inclusion X→ E induces an isomorphism

    π_1^U(1)( A  X_+)     π_1^U(1)( A  E_+)  .


We let f^n:U(1)/C_n !→ X be the inclusion.
The dimension shifting transfer map _C^U(1) factors as the composite

    π_1^C( A S^1)  U(1)⋉_C -    π_1^U(1)( A (U(1)/C_n !)_+)
       f^n_*       π_1^U(1)( A X_+)  → π_1^U(1)( A)  .

So the map ψ factors through π_1^U(1)( A X_+).
Since A X_+ is a mapping telescope, its equivariant homotopy
groups can be calculated as the colimit of the sequence:

    π_1^U(1)( A U(1)_+)
     → … → π_1^U(1)( A (U(1)/C_n !)_+)  → …

We rewrite this using the Wirthmüller and suspension isomorphisms as

    π_1^U(1)( A (U(1)/C_k)_+)   π_1^C_k( A S^1)
      π_0^C_k( A)  A  .

Example <ref> 
shows that for all k,m≥ 1 the following diagram commutes:

    @C=20mmπ_1^C_k( A S^1)[r]^-_C_k^C_k m[d]_U(1)⋉_C_k -^   π_1^C_k m( A S^1)[d]^U(1)⋉_C_k m -_
    π_1^U(1)( A (U(1)/C_k)_+)[r]_- ( Aproj)_*   π_1^U(1)( A (U(1)/C_k m)_+)

Transfers commute with the suspension isomorphism and 
in the global functor _0( A), 
the transfer _C_k^C_k m is multiplication by
the index [C_k m:C_k]=m;
so the above homotopy group sequence becomes the sequence

    A   · 2  A  · 3  ⋯ → A 
     · n  A  → ⋯  .

The colimit of this sequence is isomorphic to A,
compatible with the map ψ:A→π_1^U(1)( A X_+). 
This proves the claim.

subjectEilenberg-Mac Lane spectrum!of an abelian group|) 


subjectglobal Borel theory  
The right adjoint R:→ to the forgetful functor
from the global to the non-equivariant stable homotopy category
is modeled on the pointset level by the functor
b:→ discussed in Construction <ref>.
The global homotopy type of b E is
that of a Borel cohomology theory, and in particular,

    π_0^G( b E)   E^0(B G) ,

natural in G for transfers and restriction maps.
The functor b is lax symmetric monoidal, so it takes
an ultra-commutative ring spectrum R to an ultra-commutative
ring spectrum b R; the power operations

    P^m :  π_0^e(b E)  → π_0^Σ_m(b E)

correspond to the classical power operations

    _m :  E^0(∗)  → E^0(BΣ_m)  ,

compare the more general Remark <ref>.



[Representation ring global power functor]
As G varies over all compact Lie groups, the unitary representation rings (G)
form the unitary representation ring global functorsubjectrepresentation ring!unitary , compare Example <ref> (iv).
This is classical in the restricted realm of finite groups, but somewhat less familiar
for compact Lie groups in general.
The restriction maps α^*:(G)→(K)
are induced by restriction of representations along a homomorphism α:K→ G.
The transfer maps _H^G:(H)→(G) along a closed subgroup inclusion
H≤ G are given by the smooth induction 
of Segal <cit.>.
If H is a subgroup of finite index of G, then this induction 
sends the class of an H-representation V to the induced G-representation
^H(G,V);
in general, induction sends actual representations to virtual representations.
In the generality of compact Lie groups,
the double coset formula for  was proven by Snaith <cit.>.

The representation rings also have well-known power operations

    P^m  :  (G) → (Σ_m≀ G)  ;

on the class of a G-representation V, the power operation is
represented by the tensor power

    P^m[V]  =  [V^ m] ,

using the canonical action of Σ_m≀ G on V^ m.
Since power operations are not additive, one has to argue why
this assignment extends to virtual representations.
The standard way is to assemble all power operations 
on representations into a map

    P  :  ^+(G) → exp(;G)  ,   
    P( [V] ) =  ( [V^ m])_m≥ 0

from the monoid of isomorphism classes of G-representations.
If W is another G-representation, then

    (V⊕ W)^ m     and    ⊕_k=0^m  _ (Σ_k≀ G)×(Σ_m-k≀ G)^Σ_m≀ G 
    (V^ k W^ (m-k))

are isomorphic as (Σ_m≀ G)-representations,
because tensor product distributes over direct sum.
This means that the total power map P is a monoid homomorphism
from ^+(G) to the group exp(;G) under ⊕.
So the total power operation extends uniquely to a group homomorphism

    P  :  (G) → exp(;G)

on the representation ring. The operation P^m is then the m-th factor
of this extension.
The representation ring global power functor  is
realized by the ultra-commutative ring spectrum ,
the unitary global K-theory spectrum,
see Theorem <ref> below.subjectK-theory@K-theory!periodic global





[Brauer induction]
By Brauer's theorem <cit.> 
the complex representation ring of a finite group
is generated, as an abelian group, by representations that are induced from
1-dimensional representations of subgroups.
Segal generalized this result 
to compact Lie groups in <cit.>, 
where `induction' refers to the smooth induction. 
In fact, in the world of compact Lie group, Segal's smooth induction
for not necessarily finite index subgroups
makes the proof quite transparent, as we shall now recall.
In our language the statement can be expressed by saying that
the representation ring global functor  is `cyclic' in the sense that it is
generated by a single element, the class
x∈(U(1)) of the tautological 1-dimensional representation 
of the circle group U(1).subjectunitary group!U(1)
Equivalently, the morphism of global functors

    _x  :  (U(1),-) →

classified by the element x is an epimorphism. 
We recall the argument: we let i:U(1) × U(n-1)→ U(n) be the block sum embedding
and q:U(1)× U(n-1)→ U(1) the projection to the first factor.
The character formula <cit.>
for induced representations shows that the smooth transfer

    i_!(q^*(x)) ∈ (U(n))

has the same character as the tautological n-dimensional representation
of U(n). Since characters determine unitary representations of compact Lie groups,
i_!(q^*(x)) equals the class of the tautological representation τ_n of U(n). 
Any unitary representation of a compact Lie group G of dimension n
is isomorphic to α^*(τ_n) for a continuous homomorphism
α:G→ U(n); so the class of such a representation equals

    α^*(i_!(q^*(x))) ∈ (G)  .

So the global functor  is generated by the single class x=τ_1.

An interesting line of investigation, dubbed explicit Brauer induction,subjectBrauer induction!explicit
started with Snaith's paper <cit.>.
Informally speaking, an `explicit Brauer induction' is a section to the map

    (U(1),G)  → (G)

that is specified by a direct recipe, for example an explicit formula, 
and has naturality properties as the group G varies.
So such a map is an `explicit and natural' way to write a (virtual) representation
as a sum of induced representations of 1-dimensional representations.
The first explicit Brauer induction 
was Snaith's formula <cit.>;
however, Snaith's maps are not additive and not compatible with restriction to subgroups.
Later Boltje <cit.>
specified a different explicit Brauer induction
formula by purely algebraic means;
Symonds <cit.> gave a topological interpretation of Boltje's
construction. The Boltje-Symonds maps are additive and natural for
restriction along group homomorphisms; the maps are not (and in fact cannot be)
in general compatible with transfers. 
In our present language, the Boltje-Symonds maps form a natural transformation of
-abelian groups from  to (U(1),-).
We recall the construction of these maps; 
we follow Symonds' approach, as his reasoning is very much 
in the spirit of global homotopy theory.


The starting point is the formula (<ref>)
that expresses the class of the tautological U(n)-representation
in (U(n)) as a smooth induction of a specific 1-dimensional
representation of the subgroup U(1)× U(n-1), namely the one whose character
is the projection q:U(1)× U(n-1)→ U(1)
to the first factor. 
Formula (<ref>) suggest a class in (U(1),U(n)) 
as the image of the tautological U(n)-representation; 
if we also want naturality and additivity, then this fixes things completely:



There is a unique natural transformation of
-abelian groups

    b  :   → (U(1),-)

that satisfies

    b_U(n)(τ_n) =  _U(1)× U(n-1)^U(n)∘ q^*     in  (U(1),U(n))   .

Moreover:


  * The transformation b is a section to the evaluation morphism
of global power functors _x:(U(1),-)→ at the class x∈(U(1)).

  * The value of b_G at the 1-dimensional representation with
character χ:G→ U(1) is given by

    b_G[χ^*(τ_1)] =  χ^*   ∈ (U(1),G) .




Every class in (G) is a formal difference of classes of
actual representations, and every n-dimensional representation
is the restriction of τ_n along some continuous homomorphism
G→ U(n). So uniqueness is a consequence of naturality and additivity.  

Conversely, this also suggests how to define the transformation.
If V is any n-dimensional unitary G-representation,
then V is isomorphic to α^*(^n) 
for some continuous homomorphism α:G→ U(n),
unique up to conjugacy. So we set

    b_G[V]  =   b_G(α^*(τ_n))  =  α^*∘_U(1)× U(n-1)^U(n)∘ q^* ∈ (U(1),G)   .

This defines a set theoretic map

    b_G  :  ^+(G)  → (U(1),G)

from the monoid of isomorphism classes of unitary G-representations,
and these maps are automatically compatible with restriction along group homomorphisms.
The double coset formula (<ref>)
for ^U(n+m)_U(n)× U(m)∘_U(1)× U(n+m-1)^U(n+m)
implies the relation

    ^U(n+m)_U(n)× U(m) ( b_U(n+m)(τ_n+m) )  =   
    b_U(n)(τ_n)  + b_U(m)(τ_m)

in the group (U(1),U(n)× U(m)).
Hence the maps b_G are additive, and so they extend uniquely to
a group homomorphism

    b_G  :  (G)  → (U(1),G)

on the Grothendieck group, for which we use the same name. 
These homomorphisms are still compatible
with restriction along continuous group homomorphisms.

It remains to show the additional properties.
The transformation _x∘ b:→ is additive
and natural for restriction along continuous homomorphisms,
so for property (i) it suffices to show the relation _x∘ b=
in the universal examples, i.e., for the tautological representations
of the unitary groups U(n). This universal example is taken care of
by the formula (<ref>).

Property (ii) holds because for n=1 the map q is the identity of U(1).


While the above construction of the explicit Brauer map b:→(U(1),-) 
is slick, it is not yet particularly explicit. To write the class b_G[V]
as a -linear combination of transfers of 1-dimensional representations
of subgroups of G, one would now have to write 
the classifying homomorphism α:G→ U(n) for V
as the composite of an epimorphism and a subgroup inclusion and then 
expand the term α^*∘^U(n)_U(1)× U(n-1)
using the compatibility of transfers with inflation, and the
double coset formula for the restriction of a transfer.
Readers desperate for a truly explicit formula can find one in
<cit.>
or <cit.>.

subjectglobal power functor|)





§ GLOBAL MODEL STRUCTURE




In this section we construct a model structure on the category of
ultra-commu­ta­tive ring spectra with global equivalences as the weak equivalences,
see Theorem <ref>.
The strategy is the same is in the unstable situation 
in Section <ref>:
we lift the positive version of the global model structure 
to commutative monoid objects with the help of the
general lifting theorem <cit.>.

We also calculate the algebra
of natural operations on the homotopy groups of ultra-commutative ring spectra:
we show that these operations are freely generated by restrictions,
transfers and power operations,
see Theorem <ref> for the precise statement.
Then we show in Theorem <ref> that every global
power functor can be realized by an ultra-commutative ring spectrum;
the realization can moreover be chosen as a global Eilenberg-Mac Lane spectrum,
i.e., such that all equivariant homotopy groups in non-zero dimensions vanish.




The construction of the global model structure
on the category of ultra-commutative ring spectra is largely parallel
to the unstable precursor, the global model structure for ultra-commutative monoids 
in Theorem <ref>.
The analogous arguments as in the unstable case 
in Corollary <ref> show 
that the category of ultra-commutative ring spectra is complete and cocomplete
and that the forgetful functor 
to the category of orthogonal spectra creates all limits, 
all filtered colimits and those coequalizers that are reflexive in
the category of orthogonal spectra.
The category of ultra-commutative ring spectra is
tensored and cotensored over the category  of spaces,
by the same arguments in Construction <ref>
for ultra-commutative monoids.



We recall from <cit.> 
the notion of a symmetrizable cofibrationsubjectsymmetrizable cofibrationsubjectsymmetrizable acyclic cofibration
respectively symmetrizable acyclic cofibration,
see also Definition <ref>.
For a morphism i:A→ B of orthogonal spectra we let
K^n(i) denote the n-cube of orthogonal spectra whose
value at a subset S⊆{1,2,…,n} is

    K^n(i)(S)  = C_1  C_2 … C_n      with    
     C_j  = {[      A   j ∉S;      B j ∈ S. ].

All morphisms in the cube K^n(i) are smash products of identities and
copies of the morphism i:A→ B.
We let Q^n(i) denote the colimit of the punctured n-cube,
i.e., the cube K^n(i) with the terminal vertex removed, and
i^ n:Q^n(i)→ K^n(i)({1,…,n})=B^ n the canonical map.
So the morphism i^ n is an iterated  pushout product morphism.
The symmetric group Σ_n acts on Q^n(i) and B^ n 
by permuting the smash factors,
and the iterated pushout product morphism
i^ n:Q^n(i)→ B^ n is Σ_n-equivariant.
The morphism i:A→ B of orthogonal spectra 
is a symmetrizable cofibration subjectsymmetrizable cofibrationsubjectsymmetrizable acyclic cofibration
(respectively a symmetrizable acyclic cofibration) if the morphism

    i^ n/Σ_n  : Q^n(i)/Σ_n  → B^ n/Σ_n = ^n(B)

is a cofibration (respectively an acyclic cofibration) for every n≥ 1.  
Since the morphism i^ 1/Σ_1 is the original morphism i, 
every symmetrizable cofibration
is in particular a cofibration, and similarly for acyclic cofibrations.

The next theorem says that in the category of orthogonal spectra,
all cofibrations and acyclic cofibrations in the positive global model structure
are symmetrizable with respect to the monoidal structure given by the smash product.



  
  
  * Let i:A→ B be a flat cofibration of orthogonal spectra.
    Then for every n≥ 1 the morphism 
    
    i^ n/Σ_n  :  Q^n(i)/Σ_n  → B^ n/Σ_n

    is a flat cofibration. 
    In other words, all cofibrations in the global model structure
    of orthogonal spectra are symmetrizable.
  
  * Let i:A→ B be a positive flat cofibration of orthogonal spectra 
    that is also a global equivalence.
    Then for every n≥ 1 the morphism 
    
    i^ n/Σ_n  :  Q^n(i)/Σ_n  → B^ n/Σ_n

    is a global equivalence.
    In other words, all acyclic cofibrations in the positive global model structure
    of orthogonal spectra are symmetrizable.
  


(i) We recall from Theorem <ref> (iii) 
the set 

    I_ =  { G_m( ( O(m)/H× i_k )_+)  |  m,k ≥ 0, H≤ O(m)}

of generating flat cofibrations of orthogonal spectra, where i_k:∂ D^k→ D^k
is the inclusion.
The set I_ detects the acyclic fibrations 
in the strong level model structure of orthogonal spectra.
In particular, every flat cofibration is a retract of an I_-cell complex.
By <cit.> or <cit.> 
it suffices to show that the
generating flat cofibrations in I_ are symmetrizable.

For a space A, the orthogonal spectrum G_m( (O(m)/H× A)_+ ) 
is isomorphic to F_H,^m A_+;
so we show more generally that every morphism of the form

    F_G,V (i_k)_+  :   F_G,V∂ D^k_+  → F_G,V D^k_+

is a symmetrizable cofibration, where V is any representation 
of a compact Lie group G.
The symmetrized iterated pushout product

    ( F_G,V (i_k)_+ )^ n / Σ_n  :  
    Q^n(F_G,V (i_k)_+ ) /Σ_n →  (F_G,V D^k_+)^ n/Σ_n

is isomorphic to 

    F_Σ_n≀ G,V^n((i_k^ n)_+)  :  
     F_Σ_n≀ G,V^n ( Q^n (i_k)_+ )  →  F_Σ_n≀ G,V^n( (D^k)^n_+)  ,

where

    i_k^ n :  Q^n (i_k)  → (D^k)^n

is the n-fold pushout product of the inclusion i_k:∂ D^k→ D^k, 
with respect to the cartesian product of spaces.
The map i_k^ n is Σ_n-equivariant, 
and we showed in the proof of the analogous unstable result in
Theorem <ref> (i)
that i_k^ n is a cofibration of Σ_n-spaces.
Proposition <ref> (i)
then shows that i_k^ n is also a cofibration of (Σ_n≀ G)-spaces,
with respect to the action by restriction along the projection
(Σ_n≀ G)→Σ_n. 
So the morphism (<ref>) is a flat cofibration.

(ii) Theorem <ref> (iii) describes a 
set J_∪ K_ of generating acyclic cofibrations for the global model structure
on the category of orthogonal spectra.
From this we obtain a set J^+∪ K^+ 
of generating acyclic cofibration for the positive 
global model structure of Proposition <ref> 
by restricting to those morphisms in J_∪ K_ 
that are positive cofibrations, 
i.e., homeomorphisms in level 0; so explicitly, we set

    J^+  =  { G_m( ( O(m)/H× j_k)_+ ) |  m≥ 1, k ≥ 0, H≤ O(m)} ,

where j_k:D^k×{0}→ D^k× [0,1] is the inclusion, and

    K^+ =  ⋃_G,V,W :   V 0(λ_G,V,W)  ,

the set of all pushout products of sphere inclusions i_k
with the mapping cylinder inclusions of the global equivalences
λ_G,V,W: F_G,V⊕ WS^V → F_G,W.
Here (G,V,W) runs through a set of representatives
of the isomorphism classes of triples consisting of a compact Lie group G,
a G-representation V and non-zero faithful G-representation W.
By <cit.> or <cit.> 
it suffices to show that all morphisms in J^+∪ K^+
are symmetrizable acyclic cofibrations. 

We start with a morphism G_m( ( O(m)/H× j_k)_+ ) in J^+.
For every n≥ 1, the morphism

    (G_m( ( O(m)/H× j_k)_+ ))^ n/Σ_n

is a flat cofibration by part (i), and a homeomorphism in level 0
because m≥ 1. Moreover, the morphism j_k is a homotopy
equivalence of spaces, so the morphism G_m( ( O(m)/H× j_k)_+ ) 
is a homotopy equivalence of orthogonal spectra; the morphism
^n(G_m( ( O(m)/H× j_k)_+ )) is then again a homotopy equivalence 
for every n≥ 1,
by Proposition <ref> (i).
Then <cit.> shows that G_m( ( O(m)/H× j_k)_+ )
is a symmetrizable acyclic cofibration. 
This takes care of the set J^+.

Now we consider the morphisms in the set K^+.
Since G acts faithfully on the non-zero inner product space W, the action
of the wreath product Σ_n≀ G on W^n is again faithful.
So the morphism

    λ_Σ_n≀ G,V^n, W^n :  F_Σ_n≀ G, V^n⊕ W^n  S^V^n →F_Σ_n≀ G, W^n

is a global equivalence by Theorem <ref>.
The vertical morphisms in the commutative square

    @C=17mm 
    F_Σ_n≀ G, V^n⊕ W^n  S^V^n[r]^-λ_Σ_n≀ G,V^n, W^n[d]_   
    F_Σ_n≀ G, W^n[d]^
    ^n(F_G,V⊕ W S^V) [r]_-^n(λ_G,V,W)    ^n(F_G,W)

are isomorphisms; so the morphism  ^n(λ_G,V,W) is a global equivalence. 
Proposition <ref> (iii)
then shows that all morphisms in (λ_G,V,W) 
are symmetrizable acyclic cofibrations. 




Now we put all the pieces together.
We call a morphism of ultra-commutative ring spectra a
global equivalence (respectively positive global fibration)
if the underlying morphism of orthogonal spectra is a
global equivalence (respectively fibration in the positive global model structure
of Proposition <ref>).

   
  
  * The global equivalences and positive global fibrations are part of a 
    cofibrantly generated, proper, topological model structure
    on the category of ultra-commutative ring spectra,
    the global model structure.subjectglobal model structure!for ultra-commutative ring spectra
  
  * Let j:R→ S be a cofibration in the global model structure 
  of ultra-commutative ring spectra.
  
  
  * The morphism of R-modules underlying j is a cofibration
    in the global model structure of R-modules 
    of Corollary <ref> (i). 
  
  * The morphism of orthogonal spectra underlying j is an h-cofibration.
  
  * If the underlying orthogonal spectrum of R is flat, then 
    j is a flat cofibration of orthogonal spectra.
 
  


The proof is completely parallel to the proof of the analogous unstable
theorem, Theorem <ref>.

(i) The positive global model structure of orthogonal spectra
(Proposition <ref>) is monoidal 
and cofibrantly generated.
The `unit axiom' also holds: we let f:^c→ 
be any positive flat replacement
of the orthogonal sphere spectrum.
Then for every orthogonal spectrum X the induced morphism
f X:^c X→ X is a global equivalence by
Theorem <ref> (ii).
The monoid axiom holds by Proposition <ref>.
Cofibrations and acyclic cofibrations are symmetrizable by
Theorem <ref>,
so the model structure satisfies the `commutative monoid axiom'
of <cit.>.
The symmetric algebra functor  commutes with filtered colimits
by the analog of Corollary <ref> 
for ultra-commutative ring spectra.
Theorem 3.2 of <cit.> thus shows that the positive global
model structure of orthogonal spectra lifts to 
a cofibrantly generated model structure on the category of 
ultra-commutative ring spectra.
The global model structure is topological 
by Proposition <ref>;
here we take  as the set of free ultra-commutative 
ring spectra (F_H,^m) for all m≥ 1 
and all closed subgroups H of O(m),
and we take  as the set of acyclic cofibrations
(c(λ_G,V,W)) for the mapping cone inclusions c(λ_G,V,W) of
the global equivalences λ_G,V,W:F_G,V⊕ WS^W→ F_G,W,
indexed by representatives as in the definition of the set K^+.
Since weak equivalences, fibrations and pullbacks of ultra-commutative ring spectra are
are created on underlying orthogonal spectra,
right properness is inherited from the positive global model structure
of orthogonal spectra (Proposition <ref>).
We defer the proof of left properness until after the proof of part (ii).

Part (ii) is proved in exactly the same way as in the unstable case
in Theorem <ref>;
whenever the unstable proof refers to the model category of R-modules
in Corollary <ref>, the stable 
proof instead uses Corollary <ref>.
For the symmetrizability of the cofibrations,
the stable proof uses Theorem <ref> (i)
instead of Theorem <ref> (i).
We refrain from repeating the remaining details.

It remains to show that the model structure is left proper.
A pushout square of ultra-commutative ring spectra has the form

    @C=12mm
    R [d]_j [r]^-f_-≃    T[d]^j_R T
     
    S[r]_-S_R f    S_R T

where S and T are
considered as R-modules by restriction along j respectively f.
For left properness we now suppose that j is a cofibration and f 
is a global equivalence.
By part (a) of (ii), the morphism j is then a cofibration of R-modules 
in the global model structure of Corollary <ref> (i). 
Since R is cofibrant in that model structure, also S is
cofibrant as an R-module.
Proposition <ref> then shows that the functor
S_R - preserves global equivalences.
So the cobase change S_R f of f is a global equivalence.
This shows that the global model structure of ultra-commutative ring spectra 
is left proper. 




Now we can show that the restriction maps, (additive) transfer maps and
(multiplicative) power operations generate all natural operations between the 0-th
equivariant homotopy groups of ultra-commutative ring spectra. 
The strategy is the one that we have employed several times before:
the functor π_0^G from  ultra-commutative ring spectra to sets
is representable, namely by Σ^∞_+ (B_G),
the unreduced suspension spectrum of the free ultra-commutative monoid
generated by B_ G.
So we have to determine the equivariant homotopy groups 
π_0^K(Σ^∞_+ (B_G)), which just means assembling various
results already proved.


 
Let G and K be compact Lie groups.
The group of natural transformations  π_0^G →π_0^K
of set valued functors on the category of ultra-commutative ring spectra 
is a free abelian group with basis the operations

    _L^K∘ α^*∘ P^m :  π_0^G  →π_0^K

for all m≥ 0 and all (K×(Σ_m≀ G))-conjugacy classes of pairs 
(L,α) consisting of a closed subgroup L of K with finite Weyl group
and a continuous homomorphism α:L→Σ_m≀ G.


  We let W be any non-zero faithful G-representation and write
  B_G=_G,W for the global classifying space of G based on W.
  We denote by u^ucom_G the class in π_0^G( Σ^∞_+ ( B_ G))
  obtained by pushing forward the tautological class u_G,W 
  along the adjunction unit η:B_G→( B_ G) 
  and then applying the stabilization map σ^G of (<ref>).

  We apply the representability result 
  of Proposition <ref>
  to the category of ultra-commutative ring spectra and the adjoint functor pair: 
  
    @C=10mmΣ^∞_+∘ :   @<.4ex>[r]     ucom  :  U∘Ω^∙@<.4ex>[l]

  If V is any G-representation, then the restriction morphism
  ρ_G,V,W:_G,V⊕ W→_G,W is a global equivalence
  between positive flat orthogonal spaces. We showed in the proof of
  Theorem <ref> (ii)
  that the induced morphism of free ultra-commutative monoids
  (ρ_G,V,W):(_G,V⊕ W)→(_G,W) is a global equivalence.
  So the induced morphism of unreduced suspension spectra
  Σ^∞_+(ρ_G,V,W) is a global equivalence by 
  Corollary <ref>.
  In particular, the morphism of -functors
  _0(Σ^∞_+(ρ_G,V,W)) is an isomorphism. So 
  Proposition <ref> applies and shows that
  evaluation at the tautological class is a bijectionsubjectultra-commutative ring spectrum!free|seefree ultra-commutative ring spectrumsubjectfree ultra-commutative ring spectrum|(
  
    ^ucom(π_0^G,π_0^K) 
       → π_0^K(Σ^∞_+ (B_ G))  ,   τ ⟼ τ(u^ucom_G)

  to the 0-th K-equivariant homotopy group 
  of the ultra-commutative ring spectrum Σ^∞_+ (B_ G).

  The identification (<ref>)
  and Proposition <ref> (ii) 
  show that the homotopy set π_0^L((B_ G)) bijects with the set
  
    ∐_m≥ 0 (L,Σ_m≀ G)

  by sending the conjugacy class of α:L→Σ_m≀ G 
  to α^*([m](u_G))=α^*(u_Σ_m≀ G).
  Proposition <ref> 
  then implies that the group π_0^K(Σ^∞_+ (B_ G))
  is a free abelian group, and it specifies a basis consisting of the elements
  
    _L^K(σ^L(x))

  where L runs through all conjugacy classes of closed subgroups of K
  with finite Weyl group and x runs through a set of representatives 
  of the W_K L-orbits of the set π_0^L ((B_ G)).
  So together this shows that 
  π_0^K(Σ^∞_+ (B_ G)) is a free abelian group with
  basis the classes 
  
    _L^K(σ^L(α^*([m](u_G))))     =  _L^K(α^*(σ^Σ_m≀ G([m](u_G)))) 
       =  _L^K(α^*(P^m(σ^G(u_G)))) 
         =  _L^K(α^*(P^m(u^ucom_G)))

  for all (m,L,α) as in the statement of the theorem.



Our last major goal in this chapter is to show that every global power functor
is realized by an ultra-commutative ring spectrum that can moreover
be chosen to have all its equivariant homotopy groups concentrated in
dimension 0, see Theorem <ref> below.
Towards this aim we will show various auxiliary results, which may be of
independent interest.
We start with a proposition that describes the effect on the homotopy group
global power functors of `coning off' a free ultra-commutative ring
spectrum. We can only analyze this process algebraically if all spectra involved
are globally connective, for otherwise the symmetric powers ^m(A)
generate non-trivial equivariant homotopy in arbitrarily low negative dimensions.

We recall from Example <ref>
that the coproduct in the category of global power functors
is given by the box product of underlying global functors.
So if f:R→ F and g:S→ F are morphisms of global power functors,
we let f♢ g:R S→ F denote their categorical sum, i.e., the composite

    R S   f g  F F  multiplication F .

If A is an orthogonal spectrum and R an ultra-commutative ring spectrum,
then the zero morphism of orthogonal spectra from A to R
freely generates a morphism of ultra-commutative ring spectra τ_A: A→ R
that we call the `trivial morphism'.
In terms of the wedge decomposition A=⋁_m≥ 0^m(A),
this trivial morphism is the unit morphism of R on ^0(A)=, 
and trivial on all other wedge summands.
We recall that C A=A [0,1] is the cone of an orthogonal spectrum A,
and i=(- 1):A→ C A is the cone inclusion.


Let A be a positively flat orthogonal spectrum such that
the free ultra-commutative ring spectrum A is globally connective.
Let R be a globally connective ultra-commutative ring spectrum and ρ: A→ R
a morphism of ultra-commutative ring spectra.
We define T as the following pushout, 
in the category of ultra-commutative ring spectra:

    A[r]^- i[d]_ρ    (C A) [d]
     
    R [r]_ψ    T

Then T is globally connective, 
the morphism of global power functors _0(ψ):_0(R)→_0(T)
is surjective and 
a coequalizer, in the category of global power functors, of the two morphisms

    _0(R)♢_0(ρ) ,  _0(R)♢_0(τ_A)  :  _0(R)_0( A) → _0(R) .



The cone C A=A [0,1] is isomorphic to the realization 
of the simplicial orthogonal spectrum B_∙(A,A,∗),
the bar construction with respect to wedge.
Explicitly, 

    B_m(A,A,∗) =  AΔ[1]_m ,

where Δ[1] is the simplicial 1-simplex, pointed by the vertex 0;
the simplicial structure is induced by that of Δ[1].
Since Δ[1] has (m+1) non-basepoint simplices of dimension m, 
the spectrum B_m(A,A,∗) is a wedge of m+1 copies of A.

Since the free functor :→ ucom is an enriched left adjoint,
it preserves coends and cotensors with spaces. 
So (C A) is isomorphic to the realization,
internal to ultra-commutative ring spectra,
of the simplicial ultra-commutative ring spectrum (B_∙(A,A,∗)).
Under this isomorphism, the morphism i: A→(C A) identifies
A with the degeneracies of the object of 0-simplices.
As a left adjoint, the free functor preserves coproducts, 
i.e., it  takes wedges of orthogonal spectra
to smash products of ultra-commutative ring spectra.
So (B_∙(A,A,∗)) is isomorphic, 
as a simplicial ultra-commutative ring spectrum,
to B_∙^( A, A,),
the bar construction of A with respect to smash product.
subjectrealization!of simplicial ultra-commutative ring spectra

Since colimits commute among themselves and with cotensors,
we can describe the pushout T=R _ A(C A) as the realization 
of the simplicial ultra-commutative ring 
spectrum R_ A B_∙^( A, A,)
where we take pushouts in every simplicial dimension.
The smash product over A `cancels' with the 0-th smash factor
in the bar construction, so we can take T as the internal realization of 
the bar construction B_∙^(R, A,).

By the stable analog of Proposition <ref>, 
the realization internal to the category of ultra-commutative ring spectra 
can equivalently be taken in the underlying category of orthogonal spectra. 
The underlying realization
is the sequential colimit of partial realizations B^[n],
i.e., `skeleta' in the simplicial direction, defined as the coend

    B^[n] =  ∫^[m]∈_≤ n  B_m^(R, A,) Δ^m_+

of the restriction to the full subcategory _≤ n
of  with objects all [m] with m≤ n.
We refer to <cit.>
for background on realization of simplicial objects and the skeleton filtration.
The realization |B_∙^(R,  A, )| 
is the colimit of the sequence of orthogonal spectra

    R =  B^[0] → B^[1] → … → B^[n] → … .

Moreover, the n-skeleton B^[n] is obtained from the previous stage
as a pushout of orthogonal spectra:

    @C=13mm 
    L_n^Δ^n_+∪_ L_n^∂Δ^n_+
    B_n^(R, A,)∂Δ^n_+[r]^-incl[d]    
    B_n^(R, A,)Δ^n_+[d] 
    
    B^[n-1][r]    B^[n]

see (3.8) of <cit.>.
Here the orthogonal spectrum L_n^ 
is the n-th latching object in the simplicial direction, 
compare (1.5) of <cit.>,
or (<ref>).
In our situation, the latching morphism
L_n^→ B_n^(R, A,) is the morphism
  
    R i^ n :  R Q^n(i) → 
    R ( A)^ n =  B_n^(R, A,) ,

where i^ n is the iterated pushout product, with respect to
smash product of orthogonal spectra,
of the unit morphism → A. Since 
flat cofibrations are symmetrizable 
(Theorem <ref> (i)), this unit morphism
is a flat cofibration.
The pushout product property of the flat cofibrations 
(Proposition <ref>)
thus  shows that i^ n, and hence the latching morphism, is a flat cofibration.

So both horizontal morphisms 
in the pushout square (<ref>)
are h-cofibrations, and the cokernel of the inclusion B^[n-1]→ B^[n]
is isomorphic to 

    (B_n^(R, A,) / L_n^) S^n   
    R (A)^ n S^n

where A=( A)/=⋁_m≥ 1^m(A).
Since A is flat and flat cofibrations are symmetrizable,
the underlying orthogonal spectrum of A is flat.
Since A is globally connective, so is A.
Since R is also globally connective, so is
R (A)^ n, by Proposition <ref>. 
This shows that the cokernel of the inclusion B^[n-1]→ B^[n]
is globally (n-1)-connected. Since the inclusion is
also an h-cofibration,
Corollary <ref> (i)
shows that all the morphisms in the sequence (<ref>)
induce isomorphisms on the global functors _k for all k< 0
and an epimorphism on _0. Moreover, starting from B^[1] onward,
the morphisms even induce an isomorphism on _0.
Each morphism in the sequence is an h-cofibration,
and so levelwise a closed embedding.
Since equivariant homotopy groups commute with sequential colimits over closed embeddings
(Proposition <ref> (i)),
we conclude that also the morphism _k(ψ):_k(R)→_k(T) 
is an isomorphism of global functors for all k < 0
and an epimorphism for k=0.
Since R is globally connective, 
this shows in particular that the colimit T is globally connective.
Moreover, the inclusion B^[1]→ |B_∙^(R, A,)|=T 
induces an isomorphism on _0.

It remains to identify _0(ψ):_0(R)→_0(T) as a coequalizer.
The latching object L_1^ is simply the spectrum B_0^(R, A,)=R, 
and for n=1 the pushout (<ref>)
specializes to a pushout of orthogonal spectra

    @C=12mm 
     R A{0,1}_+[r]^-incl[d]_d_0+ d_1   
     R A [0,1]_+[d] 
    
    R[r]    B^[1]

The orthogonal spectrum R A{0,1}_+
is the wedge of two copies of R A.
On one copy the left vertical morphism is the simplicial face morphism
d_0:R A→ R, which is the product, with respect to
the multiplication of R, of the identity and the morphism ρ: A→ R.
On the other copy the left vertical morphism is 
the simplicial face morphism d_1:R A→ R, which is the product
of the identity of R and the trivial morphism τ_A: A→ R,
adjoint to the zero morphism from A to R.

The orthogonal spectra underlying R and A are globally connective
by hypothesis. Moreover, A is flat because A is and because
flat cofibrations are symmetrizable.
So by Proposition <ref>,
the orthogonal spectrum R A is again globally connective. 
Since the two horizontal morphisms in the pushout square (<ref>)
are h-cofibrations, the equivariant homotopy groups of the pushout
B^[1] participate in an exact Mayer-Vietoris sequence

    _0(R A)⊕_0(R A)
     → _0(R)⊕_0(R A) → _0(B^[1])  → 0  ,

where the first map is given by

    (x,y) ⟼ (_0(d_0)(x)+_0(d_1)(y),  x+y) .

Using the identification _0(B^[1])_0(T)
induced by the inclusion B^[1]→ T, we can rewrite this as an
exact sequence of global functors

    _0(R A) _0(d_0) - _0(d_1) _0(R)  _0(ψ)  _0(T)  → 0  .

Since R and A are globally connective and A is flat, the canonical morphism

    _0(R)_0( A) → _0(R A)

is an isomorphism of global power functors
by Example <ref>. 
Under this identification, the effects of the two face morphisms d_0,d_1:R A→ R
on _0 correspond to the morphisms of global power functors

    _0(R)♢_0(ρ)     respectively    _0(R)♢_0(τ_A)   .

So we may show that _0(ψ) is a coequalizer
of the two morphisms _0(d_0) and _0(d_1).
To this end we consider a morphism of global power functors
φ:_0(R)→ F
satisfying φ∘_0(d_0)=φ∘_0(d_1).
By exactness of the sequence (<ref>)
there is a unique morphism of global functors φ̅:_0(T)→ F
such that φ̅∘_0(ψ)=φ.
Since _0(ψ) is a surjective homomorphism of global power functors
and the composite φ̅∘_0(ψ) is a homomorphism
of global power functors, the morphism φ̅ is also compatible
with multiplication and power operations, hence a morphism of 
global power functors. This establishes the universal property of a coequalizer,
and hence completes the proof.



Now we have all the necessary tools to show that the 
topological construction of free ultra-commutative ring spectra
realizes the algebraic construction of free global power functors,
at least for sufficiently cofibrant and globally connective spectra.



  A free global power functorsubjectglobal power functor!freely generated by a global functor|(  
  generated by a global functor F is a pair (R,i) consisting of 
  a global power functor R and a morphism of global functors i:F→ R 
  with the following property:
  for every global power functor S and every morphism of 
  global functors j:F→ S
  there is a unique morphism of global power functors φ:R→ S
  such that φ∘ i=j.

  
In other words, (R,i) is a free global power functor for F 
if and only if it represents the functor

    → (sets) ,    S  ⟼ (F,S)  .


For every orthogonal spectrum X, the free ultra-commutative ring spectrum X
gives rise to a global power functor _0( X).
The adjunction unit η:X→ X (i.e., the inclusion of the `linear' wedge summand)
is a morphism of orthogonal spectra, and thus induces a morphism of global functors

    _0(η) :  _0(X) → _0( X) .



Let X be a positively flat globally connective orthogonal spectrum.
Then the free ultra-commutative ring spectrum X is globally connective and
the pair (_0( X),_0(η)) is a free global power functor generated
by the global functor _0(X).

 We let  be the class
of globally connective orthogonal spectra Y such that 
for some (hence any) global equivalence Y^c→ Y from a 
positively flat orthogonal spectrum the claim of the theorem holds for Y^c.
We let G be a compact Lie group and V a non-zero faithful G-representation.
Then B_G=_G,V is a global classifying space for G.
The free ultra-commutative ring spectrum
(Σ^∞_+ B_G) is isomorphic to the unreduced suspension spectrum
of the free ultra-commutative monoid (B_G); 
so (Σ^∞_+ B_G) is globally connective by 
Proposition <ref>.
By Corollary <ref> the global power functor

    _0((Σ^∞_+ B_G))

is freely generated by the class u_G^ucom in π_0^G((Σ^∞_+ B_G)).
Moreover,  the global functor _0(Σ^∞_+ B_G) is
freely generated by the stable tautological class e_G,
by Proposition <ref>.
So altogether this shows that the orthogonal spectrum Σ^∞_+ B_G
belongs to the class .

Now we establish two closure properties for the class .
We let {X_i}_i∈ I be a family of orthogonal spectra in ;
we claim that the wedge ⋁_i∈ IX_i also belongs to .
Since wedges of orthogonal spectra preserve global equivalences 
(by Corollary <ref> (i)),
we can assume that each X_i is positively flat. 
Then the wedge ⋁_i∈ IX_i is also positively flat.
The underlying orthogonal spectrum of (⋁_i∈ IX_i)
is the wedge, indexed by functions
f: I→ with finite support (i.e., f(i)=0 for almost all i∈ I)
of smash products of the finite symmetric powers:

    (  ⋁_i∈ I  X_i )   ⋁_f:I→, finite supp ⋀_f(i) 0^f(i)(X_i) .

Since flat cofibrations are symmetrizable and X_i is flat, the
symmetric power ^f(i)(X_i) is flat. 
Since (X_i) is globally connective, so is its
retract ^f(i)(X_i). So for every finitely supported function
f:I→, the morphism

    _f(i) 0 _0(^f(i)(X_i))  → _0(⋀_f(i) 0 ^f(i)(X_i) )

from the iterated box product is an isomorphism of global functors by 
Proposition <ref>. 
Since equivariant homotopy groups take wedges to sums by 
Corollary <ref> (i), 
this establishes a sequence of isomorphisms of global functors 

    _J⊂ I, J finite_j∈ J_0( (X_j))
         ⊕_f:I→, finite supp _f(i) 0 _0(^f(i)(X_i))
        ⊕_f:I→, finite supp _0(⋀_f(i) 0 ^f(i)(X_i) )
          _0( (  ⋁_i∈ I  X_i ))

We claim that the source of this isomorphism is the coproduct,
in the category of global power functors, of the global power functors
_0((X_i)).
Indeed, the coproduct of two global power functors is given by the
box product of underlying global functors, 
see Example <ref>.
As in any category, a general (possibly infinite) I-indexed coproduct 
can be obtained as the filtered colimit,
formed over the poset of finite subsets of I, of the finite coproducts.
Moreover, filtered colimits in the category of global Green functors,
and hence also of global power functors,
are created in the underlying category of global functors,
see Proposition <ref>.
This proves the claim.

Since _0((X_i)) is freely generated by the global functor _0(X_i)
and coproducts of free objects are free,
the right hand side of the isomorphism is a free global power functor
generated by the global functor ⊕_i∈ I_0(X_i ).
Equivariant homotopy groups take wedges to direct sums, so the canonical morphism

    ⊕_i∈ I _0(X_i) → _0(⋁_i∈ I X_i )

is an isomorphism of global functors.
So altogether we obtain that _0((  ⋁_i∈ I X_i))
is a free global power functor generated by the global functor
 _0(⋁_i∈ I X_i).
This concludes the proof that the class  is closed under wedges.

Now we claim that the class  is also closed under cones.
In other words, we let

    A  → B  → C  → A  S^1

be a distinguished triangle in the global stable homotopy category 
such that A and B belong to ; we must show that C also belongs to .
After replacing the distinguished triangle by an isomorphic one,
we can assume that A and B are positively flat and C=C f is
the mapping cone of a morphism f:A→ B of orthogonal spectra, i.e., the
pushout of the diagram of orthogonal spectra

    B   f  A   i   C A  ,

where the right morphism is the cone inclusion.
As a left adjoint, the free functor  preserves pushouts,
so (C f) is a pushout of the diagram of ultra-commutative ring spectra

    B    f   A    i   ( C A ) .

Since A and B belong to , both B and A
are globally connective. So Proposition <ref> applies
and shows that the morphism

    j :   B  → ( C f)

is a coequalizer, in the category of global power functors, of the two morphisms

    _0( B)♢_0( f) ,  _0( B)♢_0(τ_A)  :  _0( B)_0( A) → _0( B) .

Since A and B belong to the class , 
the global power functors _0( A) and _0( B) are freely 
generated by the global functors _0(A) respectively _0(B).
So the coequalizer property witnesses that the global power functor _0(( C f))
is freely generated by the cokernel of the morphism of global functors 
_0(f):_0(A)→_0(B).
Since A is globally connective,

    _0(A) _0(f) _0(B) → _0(C f)  →  0

is an exact sequence of global functors
by Proposition <ref>.
This concludes the proof that the pair (_0((C f)), _0(η))
is a free global power functor generated by the global functor _0(C f).
So we have also shown that the class  is closed under cones.

Altogether this shows that the class of orthogonal spectra  
is closed under wedges and cones 
and contains the suspension spectra Σ^∞_+ B_G for all
compact Lie groups G.
Proposition <ref> 
then shows that  is the class of all globally connective orthogonal spectra.
This proves the theorem.

subjectglobal power functor!freely generated by a global functor|) 




Let X be a positively flat globally (n-1)-connected orthogonal spectrum,
for some n≥ 0. Then for every m≥ 1 the orthogonal spectrum
^m(X) is globally (n-1)-connected.


We choose a cofibrant replacement q:Y→Ω^n X of Ω^n X
in the positive global model structure of 
Proposition <ref>.
Then Y is globally connective (i.e., globally (-1)-connected).
The adjoint q^♭ of q factors as the composite

    Y S^n  q S^n (Ω^n X) S^n   ϵ_X  X  .

The first morphism is a global equivalence since suspension is fully homotopical,
and the second morphism is a global equivalence
by Proposition <ref> (ii).
So the morphism q^♭ is a global equivalence.

We show that ^m(Y S^n) is globally (n-1)-connected.
We let

    ν_m =  {(x_1,…,x_m)∈^m |  x_1+…+x_m=0}

denote the reduced natural Σ_m-representation,
where Σ_m acts by permutation of coordinates.
The smash power (S^n)^ m is Σ_m-equivariantly homeomorphic to
the smash product of S^n (with trivial Σ_m-action)
and the representation sphere S^^nν_m.
This decomposition induces an isomorphism of orthogonal spectra

    ^m(Y S^n)  
     (Y^ m (S^n)^ m)/Σ_m    
    (Y^ m S^^nν_m)/Σ_m  S^n  .

The sphere S^^nν_m admits a finite Σ_m-CW-structure.
Moreover, all isotropy groups are conjugate to subgroups of the form
Σ_i_1×…×Σ_i_k with i_1+…+i_k=m.
So the orthogonal spectrum ^m(Y S^n) admits a finite filtration,
along h-cofibrations, with subquotients isomorphic to 

    (Y^ mΣ_m/(Σ_i_1×…×Σ_i_k)_+ S^k)/Σ_m S^n       
     Y^ m/(Σ_i_1×…×Σ_i_k) S^k+n
        ^i_1(Y)…^i_k(Y) S^k+n .

Since Y is positively flat and globally connective,
Theorem <ref>
says that the free ultra-commutative ring spectrum Y is 
globally connective. Since ^i_k(Y) is a retract of Y,
it is also globally connective. 
Since flat cofibrations are symmetrizable
(Theorem <ref> (i)), 
each of the orthogonal spectra ^i_j(Y) is again flat. 
So the smash product
^i_1(Y)…^i_k(Y) is globally connective by
Proposition <ref>.
Altogether this shows that each of the subquotients of ^m(Y S^n)
is globally (n-1)-connected. So the spectrum ^m(Y S^n) itself
is globally (n-1)-connected.

The free functor :→ ucom is a left Quillen functor
from the positive global model structure on orthogonal spectra
to the global model structure of ultra-commutative ring spectra.
So  preserves global equivalences 
between positively flat orthogonal spectra, hence so does its retract ^m.
So the morphism ^m(q^♭):^m(Y S^n)→^m(X) is a global equivalence,
and ^m(X) is globally (n-1)-connected.


The next theorem shows that we can `kill higher homotopy groups' in
the world of ultra-commutative ring spectra. Again, the construction
needs the hypothesis of global connectivity.

 
Let R be a globally connective ultra-commutative ring spectrum and n≥ 1.
Then there is a cofibration of ultra-commutative ring spectra
ψ:R→ T with the following properties:


  * the induced morphism of global functors _k(ψ):_k(R)→_k(T)
is bijective for k<n, and

  * the global functor _n(T) is trivial.



We choose an index set J, compact Lie groups G_j and
elements y_j ∈π_n^G_j(R), for j∈ J, 
that altogether generate the global functor _n(R).
We represent each class y_j as a morphism of
orthogonal spectra

    f_j  :  Σ^∞_+ B_ G_j  S^n → R

that sends the n-fold suspension of the
stable tautological class e_G_j to y_j; this involves an
implicit choice of non-zero faithful G_j-representation.
We form the wedge of all these morphisms 

    F  :  X  =  ⋁_j∈ J Σ^∞_+ B_ G_j  S^n   → R .

All we need to remember about F is that its source X is globally (n-1)-connected
and positively flat, and that the morphism of global functors

    _n(F) :  _n(X) → _n(R)

is surjective.
We extend F freely to a morphism of ultra-commutative ring spectra
F̃: X → R.
We let T be the pushout, in the category
of ultra-commutative ring spectra, of the diagram

    R   F̃    X     i  (C X)  ,

where C X is the cone of the orthogonal spectrum X,
and i:X→ C X is the cone inclusion.
Since X is positively flat, the cone inclusion is a positive cofibration
of orthogonal spectra, and so i is a cofibration of ultra-commutative ring
spectra. The cobase change ψ:R→ T of i is then also
a cofibration of ultra-commutative ring spectra.

We recall that the cobase change
of a free morphism i: X→(C X)
comes with a filtration by `number of factors from (C X)',
see for example <cit.>.
More precisely, the morphism ψ:R→ T is the sequential composite
of  morphisms of R-module spectra

    R  =  P_0   ψ_1  P_1   ψ_2  P_2  ψ_3  ⋯ ,

such that each step in the filtration can be obtained from the previous one
as a pushout of R-modules:

    @C=12mm 
    R Q^m(i)/Σ_m[r]^-i^ m/Σ_m[d]    
    R (C X)^ m/Σ_m[d]
    
    P_m-1[r]_-ψ_m    P_m

Since i:X→ C X is a flat cofibration, and flat cofibrations
are symmetrizable (Theorem <ref> (i)),
the upper horizontal morphism is a cofibration of R-modules,
hence so is its cobase change ψ_m:P_m-1→ P_m.
Moreover, the pushout square witnesses that the cokernel of ψ_m is isomorphic to

    Rcoker(i^ m)/Σ_m  
    R (C X/X )^ m/Σ_m   R^m(X S^1)  .

Since X is globally (n-1)-connected, its suspension X S^1
is globally n-connected, and so the m-th symmetric power
^m(X S^1) is also globally n-connected for all m≥ 1,
by Theorem <ref>.
Since R is connective and ^m(X S^1) is flat,
the smash product R^m(X S^1) is again
globally n-connected by Proposition <ref>.
In particular, the morphism  _k(ψ_m):_k(P_m-1)→_k(P_m)
is bijective for k<n and surjective for k=n.
Since all morphisms ψ_m:P_m-1→ P_m are h-cofibrations of orthogonal spectra,
they are levelwise closed embeddings, and so that canonical map

    _m _k(P_m) → _k(_m P_m)  =  _k(T)

is an isomorphism by Proposition <ref> (i).
Altogether this shows that the morphism
_k(ψ):_k(R)→_k(T)
is bijective for k<n and surjective for k=n.

The diagram

    @C=15mm_n(X) [r]^-_n(η_X)[dr]_-_n(F)   _n( X) [r]^-_n((i))[d]^_n(F̃)   _n((C X))[d]
       _n(R)[r]_-_n(ψ)   _n(T)

commutes and the composite through the upper right corner is the zero map
because C X is contractible.
Since _n(F) and _n(ψ) are surjective, we conclude that
the global functor _n(T) is trivial.



Now we can finally show that every global power
functor is realized by an ultra-commutative ring spectrum.
The analogous result in equivariant stable homotopy theory for a fixed
finite group has been obtained by Ullman <cit.>.
More is true: the next theorem effectively constructs a right adjoint functor

    H  :   → ^gl. connective(ucom)

to the functor _0 such that the adjunction counit is an isomorphism
_0(H F) F of global power functors.
In other words, for every globally connective ultra-commutative ring spectrum T,
the functor _0 restricts to a bijection

    _0 :  (ucom)(T, H F) (_0(T), F) .

However, we won't prove these more general facts.


 
Let F be a global power functor.subjectEilenberg-Mac Lane spectrum!of a global power functor 
There is an ultra-commu­ta­tive ring spectrum H F
such that _k(H F)=0 for all k 0  and an isomorphism of global power functors

    _0(H F)  F .



The t-structure on the global stable homotopy category
of Theorem <ref> 
(for the global family of all compact Lie groups) 
lets us choose an Eilen­berg-Mac Lane spectrum for the underlying global functor
of F, see Remark <ref>.
Cofibrant replacement in the positive global model structure 
of Proposition <ref> 
then provides a globally connective positive flat orthogonal spectrum X 
and an isomorphism of global functors _0(X) F.

We form the free ultra-commutative ring spectrum X.
Theorem <ref> then shows that X is globally connective 
and _0( X) is a free global power functor generated
by the global functor _0(X), with respect to
the morphism  _0(η):_0(X)→_0( X).
In particular, there is a unique morphism of global power functors

    ϵ :  _0( X) → F

such that ϵ∘_0(η) is the previous isomorphism.
The morphism ϵ is then surjective.

We choose a positively cofibrant globally connective orthogonal spectrum A
together with a morphism of orthogonal spectra ρ':A→ X
such that the image of the morphism _0(ρ'):_0(A)→_0( X)
coincides with the kernel of ϵ.
For example, we can choose compact Lie groups G_j and
elements y_j ∈π_0^G_j( X)
that altogether generate this kernel as a global functor.
Then we represent each class y_j as a morphism of
orthogonal spectra

    f_j  :  Σ^∞_+ B_ G_j  →  X

that sends the stable tautological class e_G_j to y_j,
and let ρ':A→ X be the wedge of all these morphisms.

We let ρ: A→ X be the free extension of ρ' to
a morphism of ultra-commutative ring spectra. Then we
define T as the following pushout, 
in the category of ultra-commutative ring spectra:

    A[r]^- i[d]_ρ    (C A) [d]
     X [r]_ψ    T

Since A and X are both globally connective by
Theorem <ref>, we can apply Proposition <ref>
and conclude that T is globally connective, 
the morphism of global power functors _0(ψ):_0( X)→_0(T)
is surjective, and a coequalizer, 
in the category of global power functors, of the two morphisms

    _0( X)♢_0(ρ) ,  _0( X)♢_0(τ_A)  :  _0( X)_0( A) → _0( X) .

Now _0( A) is the free global power functor generated by
the global functor _0(A), by Theorem <ref>.
So the effect of the coequalizer is to annihilate the global power ideal 
generated by the image of _0(ρ'):_0(A)→_0( X). 
This global power ideal is, by construction, the kernel of ϵ:_0( X)→ F.
So ϵ descends to an isomorphism of global power functors
between _0(T) and F.

Now we use Theorem <ref> 
to kill the higher homotopy groups of T. More precisely,
we construct a sequence of cofibrations of ultra-commutative ring spectra

    T  =  T_0  → T_1  → … → T_n  → …

by induction on n. We obtain T_n by applying Theorem <ref>
in dimension n to R=T_n-1. 
Then T_n is globally connective, _0(T_n) is isomorphic to
_0(T), and hence to F, 
and the global functors _k(T_n) are trivial for all 1≤ k≤ n.

Finally, we define H F as the colimit of the sequence (<ref>)
of ultra-commu­ta­tive ring spectra.
Each morphism in the sequence is a cofibration of ultra-commutative ring spectra,
hence an h-cofibration of underlying orthogonal spectra
(see Theorem <ref> (ii)),
and so levelwise a closed embedding.
Since equivariant homotopy groups commute with sequential colimits over closed embeddings
(see Proposition <ref> (i)),
the colimit has the desired properties of an
ultra-commutative Eilenberg-Mac Lane spectrum for F.

subjectfree ultra-commutative ring spectrum|)









CHAPTER: GLOBAL THOM AND K-THEORY SPECTRA




The final chapter of this book is devoted to an in-depth study 
of interesting examples of ultra-commutative ring spectra, in particular
global Thom spectra and ultra-commutative global models for various flavors
of equivariant K-theory spectra.
In Section <ref> we discuss global refinements of
the classical Thom spectrum M O, as well as Thom spectra for the other families
of classical Lie groups, such as M S O and M U. 
We carefully analyze two different global forms 
of the Thom spectrum M O that represents unoriented bordism,
namely the ultra-commutative Thom ring spectrum , 
and an  E_∞-orthogonal ring spectrum .
Both Thom spectra are the homogeneous degree 0 summands in certain
-graded periodic extensions  respectively .
The Thom spectrum  is the natural target for the Thom-Pontryagin map
from geometric equivariant bordism.
The ultra-commutative ring spectrum  is 
the -analog of tom Dieck's equivariant homotopical bordism.
The equivariant homology theory represented by  is a localization
of the one represented by , formed by inverting certain `inverse Thom classes',
see Corollary <ref>.
The Thom spectrum  comes with a rank filtration
that we use to show that  is globally connective,
and to calculate the global functor _0(), see Theorem <ref>.

Section <ref>
is devoted to equivariant bordism and its relation to
the equivariant homology theories represented by the global Thom spectra
of Section <ref>.
We recall various facts about equivariant bordism groups of smooth
compact G-manifolds in some detail, highlighting the global perspective.
The main result of this section is Theorem <ref> 
which says that when G is isomorphic to a product of
a finite group and a torus, then the Thom-Pontryagin map
is an isomorphism from G-equivariant bordism to 
G-equivariant -homology.
This result is usually credited to Wasserman,
because it can be derived from his equivariant transversality
theorem <cit.>.
Theorem <ref> gives a localized version
of this result: the Thom-Pontryagin map
is an isomorphism from stable equivariant bordism to 
[1/τ]-theory, without any restriction on the compact Lie group.
Given that [1/τ]-theory is isomorphic to -theory
(by Corollary <ref>),
this is equivalent to a result of Bröcker and Hook <cit.>
that identifies stable equivariant bordism with equivariant -homology.


Section <ref> 
discusses the ultra-commutative ring spectrum ,
the connective global K-theory spectrum, 
see Construction <ref>.
The degree zero equivariant cohomology theory represented by 
tries hard to be equivariant K-theory (see Theorem <ref> for the
precise statement),
and there is a ring homomorphism (G)→π_0^G()
from the complex representation ring that is 
an isomorphism whenever G is finite, and almost a morphism
of global power functors, compare Theorem <ref>.
We also introduce the Bott class in the group π_2^e() 
(see Construction <ref>) and the more general
equivariant Bott classes associated to  G-^c-representations
(see Construction <ref>).


The final Section <ref> reviews , the
periodic global K-theory spectrum, see Construction <ref>.
This is an ultra-commutative ring spectrum whose G-homotopy type realizes 
G-equivariant periodic K-theory, in the sense that
the equivariant cohomology represented by  on finite G-CW-complexes is
isomorphic to equivariant K-theory, see Corollary <ref>.
Moreover, _0() is isomorphic, as a global power functor, 
to the complex representation ring functor ,
by Theorem <ref>.
Periodic global K-theory receives a morphism of ultra-commutative ring spectra
j:→ from connective global K-theory; 
the image of the Bott class under j becomes a unit
(Theorem <ref>),
justifying the adjective `periodic' for .
Our final example of an ultra-commutative ring spectrum is ,
the global connective K-theory
(see Construction <ref>),
a certain global refinement 
of Greenlees' equivariant connective K-theory <cit.>.







§ GLOBAL THOM SPECTRA



subjectglobal Thom spectrum|(

In this section we discuss two different global forms 
of the Thom spectrum M O that represents unoriented bordism,
namely the ultra-commutative Thom ring spectrum , 
and a variation  that is only E_∞-commutative. 
Both Thom spectra are the homogeneous degree 0 summands in certain
-graded periodic extensions  respectively .
All four orthogonal spectra are Thom spectra over
certain orthogonal spaces defined in Section <ref>.
The partners are easy to identify from the notation: 
the relevant orthogonal spaces either have
a 𝐁 or 𝐛 in their name,
and in the corresponding Thom spectrum this letter is replaced by 
an 𝐌 respectively 𝐦.
 

While the spectrum  has less structure than 
(it is `only' E_∞, not ultra-commutative),
it is closely related to geometry. Indeed, the Thom spectrum 
is the natural target for the Thom-Pontryagin map
from geometric equivariant bordism.
The ultra-commutative ring spectrum  is 
the -analog of equivariant homotopical bordism, 
due to tom Dieck <cit.>;
the unoriented version  is studied in detail in <cit.>.
The following commutative diagram gives a schematic overview of the relevant
equivariant homology theories:

    @C=10mm𝒩_*^G [r]^-Θ^G[d]    _*^G [d] [r]    _*^G [d] 
    𝔑_*^G:S[r]_-Θ^G^   _*^G  [r]    _*^G

The two theories in the middle column are the equivariant homology
theories represented by the orthogonal Thom spectra  and .
The vertical transformations are isomorphisms for G=e, but not in general.
In fact, when G has more than one element, then π_*^G()
has non-trivial elements in negative degrees, while  is globally connective.
The vertical transformations are localizations, i.e., they become
isomorphisms after inverting all inverse Thom classes,
compare Corollary <ref>.
The two theories  and  in the last column are the periodic versions
of  and ; each is a wedge of all suspensions and desuspensions
of the non-periodic version.

In the left column, 𝒩_*^G is geometrically defined equivariant bordism,
and 𝔑_*^G:S is stable equivariant bordism,
a certain localization of 𝒩_*^G.
So the two theories in the left column are not represented by
orthogonal spectra, but they are defined from bordism classes of G-manifolds; 
we will recall these geometric theories in Section <ref>.
The transformations labeled Θ^G are the equivariant Thom-Pontryagin construction
and its `stabilization'. The upper Thom-Pontryagin
map Θ^G:𝒩_*^G→π_*^G() is an isomorphism whenever G is isomorphic
to a product of a finite group and a torus; 
this result seems to have been folklore
at some point, and we give a proof in Theorem <ref>. 
The upper Thom-Pontryagin map is not an isomorphism for more general
compact Lie groups; in fact the geometric bordism theory 𝒩_*^G
cannot in general be represented by an orthogonal G-spectrum because
the Wirthmüller map fails to be an isomorphism for those closed
subgroups H of G that act non-trivially on the tangent space T_e H (G/H),
compare Remark <ref>.
The stabilized Thom-Pontryagin
map Θ^G:𝔑_*^G:S→π_*^G() is an isomorphism
in complete generality, by a theorem of Bröcker 
and Hook <cit.>; we derive this fact in a different way,
see Remark <ref>.

The Thom spectrum  comes with a rank filtration, the subquotients
of which we identify in Theorem <ref>.
We use the rank filtration to show that  is globally connective,
and we calculate the global functor _0() in Theorem <ref>.



We start with the ultra-commutative ring spectrum ,symbol - Thom spectrum over the additive Grassmannian
the Thom spectrum over the additive Grassmannian ,
discussed in Example <ref>.subjectadditive Grassmannian
We recall that the value of  at an inner product space V is

    (V) =  ∐_n≥ 0  Gr_n(V)  ,

the disjoint union of all Grassmannians in V.
Over the space (V) sits a tautological euclidean vector bundle 
(of non-constant rank) whose total space consists of pairs 
(x,U)∈ V×(V) such that x∈ U.
We define (V) as the Thom space of this tautological vector bundle, i.e., 
the one-point compactification of the total space.
The structure maps are given by

    (V,W) (V)  → (W) ,(w,φ)   (x,U) 
      ⟼  ( w +φ(x), φ^⊥⊕φ(U) ) ,

where φ^⊥=W-φ(V) is the orthogonal complement 
of the image of φ:V→ W.
Multiplication maps
are defined by direct sum, i.e., 

    μ_V,W :  (V) (W)     →  (V⊕ W)   
    
    (x,U) (x',U')     ⟼ ( (x,x'),  U⊕ U') .

Unit maps are defined by

    η(V) :  S^V → (V)  ,     v ⟼ (v,V) .

The multiplication maps are binatural, associative, commutative and unital,
and all this structure makes  into an ultra-commutative ring spectrum.
The orthogonal spectrum  is graded, with k-th homogeneous summand given by

    ^[k](V) =   T h(G r_|V|+k(V))  .

So  is concentrated in non-positive gradings, i.e.,
^[k] is trivial for k >0.
The unit morphism η:→ is an isomorphism onto the summand ^[0].

Now we let V be a representation of a compact Lie group G.
We define the inverse Thom classsubjectinverse Thom class!in  

    τ_G,V ∈ _0^G(S^V)

as the class represented by the G-map 
  
    t_G,V :   S^V     → T h(G r(V)) S^V  =  (V) S^V  ,  
    v ⟼  (0, {0}) (-v)    .

If V has dimension m, then the class τ_G,V 
has internal degree -m, i.e., it lies in the homogeneous summand ^[-m].
The justification for the name `inverse Thom class' is that in
the theory , the image of the class τ_G,V
becomes invertible, and its inverse is the Thom class of V
(considered as G-vector bundle over a point). 
We explain this in more detail in Theorem <ref> below.
So while the theory  is not globally orientable,
and does not have Thom isomorphisms for equivariant bundles,
informally speaking the inverses of the prospective Thom classes 
are already present in . 


[ as a wedge of semifree spectra]
We recall from Construction <ref>
that the semifree orthogonal spectrum generated
by the tautological O(m)-representation ν_m on ^m is 
given bysubjectsemifree orthogonal spectrum

    F_O(m),ν_m =  (ν_m,-)/O(m) .

We claim that the spectrum  is isomorphic
to the wedge of these semifree orthogonal spectra. Indeed, the maps

    (ν_m,V)/O(m) → ^[-m](V)  ,  
    (v,φ)· O(m) ⟼ (v, φ^⊥ )

define an isomorphism of orthogonal spectra from F_O(m),ν_m
to the homogeneous summand ^[-m]. 
This isomorphism takes the equivariant homotopy class
a_O(m),ν_m in π_0^G(F_O(m),ν_m S^ν_m) defined in (<ref>)
to the inverse Thom class τ_O(m),ν_m of the tautological O(m)-representation,
by direct inspection of the definitions.


Now we suppose that φ:V→ W is an isomorphism of
orthogonal G-representations.
Then φ compactifies to a G-equivariant homeomorphism 
S^φ:S^V→ S^W and hence induces an isomorphism 

    (S^φ)_*  :  _0^G(S^V)  → _0^G(S^W)  .

The following properties of the inverse Thom classes τ_G,V
are all straightforward from the definition.
The power operations that show up in part (v) 
are defined in (<ref>); 
the norm maps of part (vi) are introduced in Remark <ref>.


The inverse Thom classes τ_G,V have the following properties.  


  * For every isomorphism φ:V→ W of orthogonal G-representations,
the induced isomorphism (S^φ)_* 
takes the class τ_G,V to the class τ_G,W.

  * The class τ_G,0 of the trivial 0-dimensional G-representation
is the multiplicative unit 1 in _0^G(S^0)=π_0^G().

  * For every continuous homomorphism α:K→ G 
of compact Lie groups the relation

    α^*(τ_G,V)  =  τ_K,α^*V

holds in  _0^K(S^α^*(V)). 

  * For all orthogonal G-representations V and W
the relation

    τ_G,V·τ_G,W =  τ_G,V⊕ W

holds in  _0^G(S^V⊕ W).

  * For all orthogonal G-representations V and all k≥ 1
the relation

    P^k(τ_G,V) =  τ_Σ_k≀ G,V^k

holds in  _0^Σ_k≀ G(S^V^k).

  * For every closed subgroup H of G of finite index 
and all orthogonal H-representations W the relation

    N_H^G(τ_H,W) =  τ_G,_H^G W

holds in  _0^G(S^_H^G W).




The next proposition shows that multiplication by the inverse Thom class
τ_G,V is realized by a certain morphism
of orthogonal G-spectra j^V_:→^V. 
The value at an inner product space U is the map

    j_^V(U) :  (U)    → (U⊕ V) =  (^V)(U)
    
    (x,L)     ⟼ ((x,0), L⊕ 0)   .

If V has dimension m, then j^V_ is homogeneous of degree -m
in terms of the internal grading of , i.e.,
j^V_ takes the wedge summand ^[k] to the summand ^V^[k-m].  
The _*-isomorphism λ^V_X:X S^V→^V X 
was defined in (<ref>).

 
Let V be a representation of a compact Lie group G.
The composite

    π_k^G( A) -·τ_G,V π_k^G( A S^V) 
     (λ^V_ A)_* π_k^G(^V  A)

coincides with the effect of the morphism j_^V A: A→^V A.


In (<ref>) 
we defined an isomorphism ψ^V_X:π_k^G(^V X)→π_k^G(X S^V),
natural for morphisms of orthogonal G-spectra X, essentially given by
smashing with the identity of S^V.
We also defined ε_V:π_k^G(X S^V)→π_k^G(X S^V)
as the effect of the antipodal map of S^V.
Now we observe that  multiplication by the class τ_G,V
factors as the composite

    π_k^G( A)    (j^V_ A)_* π_k^G(^V  A) 
        ψ^V_ A  π_k^G( A S^V)  ε_V  π_k^G( A S^V)  .

Besides the definitions,
this uses that the map j^V_(U) S^V equals the composite

    (U) S^V (U) t_G,V (U)   (V) S^V
       μ_U,V S^- (U⊕ V) S^V ,

where t_G,V is the defining representative of the class τ_G,V.
The claim then follows from the fact,
established in Proposition <ref> (i),
that (λ^V_ A)_* is inverse to ε_V∘ψ^V_ A.



We define two ultra-commutative ring spectra  
and . The latter is a periodic version of the former, 
and the former is the homogeneous degree 0 summand with respect
to a natural -grading of the latter.
Non-equivariantly,  is a version of the unoriented Thom spectrum M O,
and it is a global refinement of
equivariant homotopical bordism, due to tom Dieck <cit.>;
tom Dieck first considered the unitary version in <cit.>,
and the paper <cit.> by Bröcker and Hook
studies the orthogonal version.

The spectrum  is a Thom spectrum over the orthogonal space 
discussed in Example <ref>.
We recall that the value of  at an inner product space V is

    (V) =  ∐_n≥ 0  Gr_n(V^2)  ,

the disjoint union of all Grassmannians in V^2.
Over the space (V) sits a tautological euclidean vector bundle 
(again of non-constant rank) with total space consisting of pairs 
(x,U)∈ V^2×(V) such that x∈ U.
We define (V)symbol - periodic global Thom spectrum
as the Thom space of this tautological vector bundle, i.e., the
one-point compactification of the total space.
The structure maps are given by

    (V,W) (V)     →    (W)
    
    (w,φ)   (x,U)       ⟼  ((w,0)+(φ)(x), (φ)(U)) .

Multiplication maps

    μ_V,W :  (V) (W)  → (V⊕ W)

are defined by sending (x,U) (x',U') to 
(κ_V,W(x,x'),κ_V,W(U⊕ U')) 
where κ_V,W:V^2⊕ W^2 (V⊕ W)^2 is the preferred isometry 
defined by

    κ_V,W((v,v'),(w,w')) =   ((v,w),(v',w'))  .

Unit maps are defined by

    S^V → (V)  ,     v ⟼ ((v,0), V⊕ 0) .

The multiplication maps are binatural, associative, commutative and unital,
and all this structure makes  into an ultra-commutative ring spectrum.

The orthogonal space  is -graded, with k-th homogeneous summand

    ^[k](V) =   Gr_|V|+k(V^2)  .

The spectrum  inherits a -grading,
where the summand ^[k](V) of degree k 
is the Thom space of the tautological (|V|+k)-plane bundle over ^[k](V);
then (V) is the one-point union of the
Thom spaces  ^[k](V) for -|V|≤ k≤ |V|, and thus

    =  ⋁_k∈ ^[k]

as orthogonal spectra.

We define =^[0] as the homogeneous degree zero wedge summand
of ; this is then an ultra-commutative ring spectrum in its own right.
Explicitly, (V) is the Thom space of the tautological |V|-plane
bundle over G r_|V|(V^2).symbol - global Thom spectrum





Certain variations of the construction of  and  are possible, and
have been used at other places in the literature.
Indeed, if U is any euclidean vector space, finite or infinite dimensional, 
and u∈ S(U) a unit vector,
we obtain an ultra-commutative ring spectrum _U,u in exactly the same way
as above, with value at V given by 
the Thom space of the tautological vector bundle over Gr_|V|(V U).
The chosen vector u enters in the definition of the unit and structure maps.
For U=^2 and u=(1,0), the construction specializes to  as above.

If the dimension of U is at least 2, then we always get the same global homotopy type.
Indeed: any linear isometric embedding ψ:U→ U' such that ψ(u)=u'
induces a morphism of ultra-commutative ring spectra ψ_*:_U,u→_U',u'.
If the dimension of U is at least 2, this morphism is a global equivalence. 



Since  is an ultra-commutative ring spectrum, 
the equivariant homotopy groups _0() form a global power functor.
The global power functor _0() is an interesting algebraic structure,
but a complete algebraic description does not seem to be known.
Since 2=0 in π_0^e(),
the global power functor _0() takes values in _2-vector spaces.
In _0(), the stronger relation _e^C_2(1)=0 holds,
compare Theorem <ref> below.

The orthogonal spectrum underlying  comes with a -grading, 
i.e., a wedge decomposition (<ref>)
into summands ^[k]. The geometric splitting induces a direct sum decomposition
of π_0^G() for every compact Lie group G 
and makes it into a commutative -graded ring.
The m-th power operation takes the summand ^[k] to the summand ^[m k].




We move on to explain the periodicity of the ultra-commutative ring spectrum . 
We let t∈π_-1(^[-1])
be the class represented by the point

    (0,{0})  ∈ T h(G r_0(^2))  =  ^[-1]()  .

We let σ∈π_1(^[1])
be the class represented by the map

    S^2  → T h(G r_2(^2))  =  ^[1]()  ,  
    x ⟼ (x,^2) .

As the next proposition shows,  is periodic in the sense
that t is a unit in the graded ring π_*^e(), with inverse σ.

The orthogonal spectrum  has an even stronger kind of
`R O(G)-graded' periodicity.
We define the inverse Thom classsubjectinverse Thom class!in  

    τ_G,V ∈ _0^G(S^V)

as the class represented by the G-map 
  
    t_G,V :   S^V     → T h(G r(V^2)) S^V  =  (V) S^V 
    
    v     ⟼  ( (0,0), 0⊕ 0) (-v)    .

Here we abuse notation by denoting the inverse Thom class in -theory
by the same symbol as the  inverse Thom class in -theory
defined in (<ref>). The justification
for this abuse is that the homomorphism c:→
introduced in (<ref>) below takes one inverse Thom class
to the other.
If V has dimension m, then the class τ_G,V 
has internal degree -m, i.e., it lies in the homogeneous summand ^[-m].
The justification for the name `inverse Thom class' is that
it is inverse to the Thom class σ_G,V in ^0_G(S^V), 
defined in (<ref>) below.

The periodicity class t of (<ref>) is essentially
the inverse Thom class of the 1-dimensional representation of the trivial group.
More precisely, t S^1 = τ_e,, i.e., the suspension isomorphism

    - S^1  :  π_-1^e()    π_0^e( S^1) =  _0^e(S^1)

takes the periodicity class t to the inverse Thom class τ_e,.
Indeed, the suspension of the defining representative (<ref>)
for t differs from the  defining representative
for τ_e, by the inversion map -:S^1→ S^1.
So t S^1=-τ_e,; however, since 2=0 in π_0^e(),
this yields the claim.

The next proposition shows that multiplication by any inverse Thom class
is invertible in -theory. Moreover, multiplication by τ_G,V
is realized, in a certain precise way,
by a periodicity morphism of orthogonal G-spectra j_^V:→^V:
the value at an inner product space U is the map

    j_^V(U) :  (U)    → (U⊕ V) =  (^V)(U)
    
    (x,L)     ⟼   (i(x),i(L))

induced by the linear isometric embedding
i:U⊕ U→ U⊕ V⊕ U⊕ V with i(u,u')=(u,0,u',0).
The morphism j^V_ is even a homomorphism of left -module spectra.
If V has dimension m, then j^V_ is homogeneous of degree -m in terms
of the -grading of , i.e., it restricts to a morphism
of orthogonal G-spectra

    j_^V :  ^[k+m] → ^V^[k]

where k is any integer.
In the special case V= with trivial G-action, the map

    j  =  j_^ :   →

is a morphism of orthogonal spectra (with trivial G-action).
As usual we denote by p_G:G→ e the unique group homomorphism
to the trivial group, and p_G^* is the associated inflation homomorphism.




  
  
  * The relation t·σ =1 holds in π_0^e().
    For every compact Lie group G, every based G-space A
    and all k∈, the maps 
    
    _k+1^G(A)    · p_G^*(t) _k^G(A) 
           and  _k^G(A)     · p_G^*(σ) _k+1^G(A)

    are mutually inverse isomorphisms.
  
  * For every representation V of a compact Lie group G, the morphism
    
    j^V_ :   → ^V

    is a _*-isomorphism of orthogonal G-spectra.
    In particular, the morphism j=j_^ :→
    is a global equivalence.
  
  * For every based G-space A, the composite
    
    π_0^G( A) -·τ_G,V π_0^G( A S^V) 
         (λ^V_ A)_* π_0^G(^V  A)

    coincides with the effect of the morphism 
    j^V_ A: A→^V A.
    In particular, exterior multiplication by the inverse Thom class τ_G,V 
    is invertible in equivariant -homology.
  


(i) The class t·σ is represented by the composite

    S^2  x↦ (0,{0}) (x,^2) ^[-1]()^[1]()
     μ_, ^[0](⊕)

where the first map is the smash product of the defining representatives
for t and σ. Expanding the definition of μ_.
identifies this composite as the map

    S^2  → ^[0](⊕)  ,   
    x ⟼ ( (⊕τ_,⊕)(0,0,x), 0⊕⊕ 0⊕ ) .

This differs from the representative of the unit 1∈π_0^G()
by the action of the linear isometry

    ^4  → ^4  ,    (a,b,c,d) ⟼ 
    (b,d,c,a)  .

This isometry has determinant 1, so we conclude that t·σ=1
in π_0^e().

(ii) We show first that j_^V induces an isomorphism on 0-th 
G-equivariant homotopy groups. To this end we define a map

    Φ :  π_0^G(^V ) → π_0^G()

in the opposite direction as follows.
We define a G-map 

    s_G,V :  S^V⊕ V → T h(G r(V⊕ V))  =  (V) 
      by  s_G,V(v,w) = ( (v,w), V⊕ V)   .

If f:S^U→(U⊕ V)=(^V)(U) 
represents a class in π_0^G(^V ),
then we define Φ[f] as the class of the composite

    S^U⊕ V⊕ V f s_G,V(U⊕ V)(V)   μ_U⊕ V,V(U⊕ V⊕ V)   .

This recipe is compatible with stabilization, so Φ is indeed well-defined.

The composite

    S^V⊕ V  s_G,V  (V)  j_^V(V) (V⊕ V)

is G-equivariantly homotopic to the unit 
map η(V⊕ V):S^V⊕ V→(V⊕ V).
So 

    (j_^V)_*(Φ[f])    =  [j_^V(U⊕ V⊕ V)∘μ_U⊕ V,V∘(f s_G,V)] 
       =  [μ_U⊕ V,V⊕ V∘ ((U⊕ V) j_^V(V))∘ (f s_G,V)] 
       =  [μ_U⊕ V,V⊕ V∘ (f ( j_^V(V)∘ s_G,V))] 
       =  [μ_U⊕ V,V⊕ V∘ (fη(V⊕ V))]  =  [f] .

The second equation is the fact that j_^V is a homomorphism of left -modules.
This proves that (j_^V)_*∘Φ is the identity.

On the other hand, the composite

    (U) S^V⊕ V j_^V(U) s_G,V (U⊕ V)(V)  μ_U⊕ V,V (U⊕ V⊕ V)

agrees with the opposite structure map σ^_U,V⊕ V
of the spectrum .
So if φ:S^U→(U) represents a class in π_0^G(), then

    Φ( (j_^V)_*[φ])    =  [μ_U⊕ V,V∘(j_^V(U)∘φ) s_G,V] 
       =  [μ_U⊕ V,V∘(j_^V(U) s_G,V)∘ (φ S^V⊕ V)] 
       =  [σ^_U,V⊕ V∘(φ S^V⊕ V)] =  [φ] .

This proves that Φ∘ (j_^V)_* is the identity,
and thus completes the proof that π_0^G(j_^V) is an isomorphism.
Since j_^V is a homomorphism of left -modules,
its effect on homotopy groups is π_*^G()-linear.
In particular, it commutes with the action of the element p_G^*(t),
where t∈π_-1^e() is the periodicity element defined 
in (<ref>). Since p_G^*(t) is invertible by part (i)
and π_0^G(j_^V) is an isomorphism, the map π_k^G(j_^V) is 
then an isomorphism for every integer k.
Applying the above to a closed subgroup H of G and the underlying H-representation
of V shows that j_^V induces isomorphisms of equivariant stable homotopy groups
for all closed subgroups of G. So j_^V is a _*-isomorphism.

(iii) The proof of the first claim proceeds in the same way as its analog for
 in Proposition <ref>.
Since the morphisms j^V_ and λ^V_ A
are both _*-isomorphisms of orthogonal G-spectra,
they induce isomorphisms on π_0^G.
So exterior multiplication by the inverse Thom class τ_G,V 
is invertible.


The orthogonal spectra  and  admit `shift morphisms'
j_^V: →^V and j_^V : →^V
defined in (<ref>) respectively (<ref>).
However, the morphism j^V_  is not a _*-isomorphism,
whereas the morphism j^V_ is a _*-isomorphism,
by Theorem <ref> (ii).
This is a reflection of the fact that the inverse Thom classes τ_G,V
are not invertible in equivariant -homology,
whereas their -counterparts are.





[Thom classes for representations]
The Thom spectrum  comes with distinguished Thom classes for representations.
We let V be a representation of a compact Lie group G.
We consider the G-map

    s_G,V :  S^V⊕ V → T h(G r(V⊕ V))  =  (V)  ,
       (v,w) ⟼ ( (v,w), V⊕ V)   .

If V has dimension m, then s_G,V is a homeomorphism onto the 
homogeneous summand ^[m](V).
The adjoint S^V→_*(S^V,(V)) of s_G,V represents the Thom class
symbolσ_G,V - Thom class in ^0_G(S^V)subjectThom class!in 

    σ_G,V ∈ ^0_G(S^V)  =  π_0^G(_*(S^V, ))

in the  G-equivariant -cohomology of S^V.


The following theorem is a special case of a Thom isomorphism 
in the equivariant cohomology theory represented by .
It also makes precise in which way the inverse Thom class τ_G,V
is inverse to the Thom class σ_G,V.
This relation between τ_G,V and σ_G,V
is the ultimate justification for naming τ_G,V the `inverse Thom class'.

subjectThom isomorphism!for 
Let V be a representation of a compact Lie group G.
Then the composite

    ^0_G(S^V) =  π_0^G(Ω^V )  ·τ_G,Vπ_0^G((Ω^V) S^V) (ϵ^V_)_*π_0^G()

is inverse to multiplication by σ_G,V,
where ϵ^V_:(Ω^V) S^V→ is the evaluation morphism.
In particular, ^0_G(S^V) is a free module 
of rank 1 over the ring π_0^G(),
and the Thom class σ_G,V is a generator.


We consider a G-map f:S^U→Ω^V (U)
that represents a class in ^0_G(S^V).
Then ϵ^V_([f]·τ_G,V) is represented by the following composite:

    S^U⊕ V f t_G,V    (Ω^V (U))(V) S^V 
    evaluate    (U)(V)   μ^_U,V  (U ⊕ V)  ,

where t_G,V:S^V→(V) S^V 
is the defining representative for the class τ_G,V
from (<ref>). 
If we let f=s_G,V^♯:S^V→Ω^V (V) be adjoint to
the defining representative for σ_G,V, then the composite comes out as the map

    S^V⊕ V → (V ⊕ V)  ,   
    (v,w) ⟼ ( (v,0,-w,0),V⊕ 0⊕ V⊕ 0) .

This composite is equivariantly homotopic to the map

    (v,w) ⟼ ( (v,w,0,0) ,  V⊕ V⊕ 0⊕ 0)

which represents the multiplicative unit. So we have shown that

    ϵ^V_(σ_G,V·τ_G,V) = 1  .

All maps in sight are left π_0^G()-linear, so we deduce that

    ϵ^V_(x·σ_G,V·τ_G,V) = 
    x·ϵ^V_(σ_G,V·τ_G,V) = x

for every class x∈π_0^G().
On the other hand, the composite

    π_0^G(Ω^V )  ·τ_G,Vπ_0^G((Ω^V) S^V) (λ^V_Ω^V)_*π_0^G(^V Ω^V)

is the effect of the morphism Ω^V j^V_:Ω^V→^VΩ^V, 
by the same reasoning as in Theorem <ref> (iii). 
Since j^V_ is a _*-isomorphism, so is Ω^V j^V_ by 
Proposition <ref> (ii).
On the other hand, λ^V_Ω^V is a  _*-isomorphism 
by Proposition <ref> (ii).
So multiplication by the class τ_G,V is an isomorphism.
The morphism ϵ^V_ is a _*-isomorphism,
also by Proposition <ref> (ii).
So the composite (ϵ^V_)_*∘(-·τ_G,V)
is bijective. Since it is also left inverse to multiplication by σ_G,V,
this proves the first claim.




[Thom classes for equivariant vector bundles]
The Thom spectrum  comes with a distinguished orientation, 
given by Thom classes for equivariant vector bundles.
These Thom classes generalize the classes σ_G,V 
defined in (<ref>),
when we view a G-representation as a G-vector bundle over a one-point
G-space.

We recall the definition of the Thom classes.
Given a compact Lie group G, an orthogonal G-spectrum X and a
compact based G-space B, we define the G-equivariant X-cohomology group
of B as

    X^0_G(B)  =  π_0^G(_*(B, X) )  .

We let ξ:E→ B be a G-equivariant vector bundle.
The bundle has a classifying G-map ψ:B→ G r(V)
for some G-representation V, i.e., such that ξ is isomorphic
to the pullback of the tautological G-vector bundle over the Grassmannian.
We let ψ̅:E→ V be a map that covers ψ, i.e., ψ̅ is
fiberwise linear and satisfies ψ̅(e)∈ψ(ξ(e)). 
We define a based G-map

    S^V  T h(ξ) → (V) = T h(G r(V^2))  by 
    v e  ⟼ ((v,ψ̅(e)), V⊕ψ(ξ(e))) .

We denote the equivariant cohomology class represented by the adjoint 
S^V→_*(T h(ξ), (V)) of this map by

    σ_G(ξ) ∈ _G^0(T h(ξ))

and refer to it as the 
Thom classsubjectThom class!in !of a G-vector bundle 
of the G-vector bundle ξ.
If the bundle ξ has constant rank m,
then the image of the map lies in the wedge summand ^[m].
It is straightforward to see that the Thom classes just defined
are natural for pullback of bundles, compatible with
restriction along continuous homomorphisms, and 
the Thom class of an exterior product of bundles is
the exterior product of the Thom classes.

The Thom diagonal of the G-vector bundle ξ:E→ B is the map

    Δ :  
    T h(ξ) → B_+ T h(ξ)  ,    e ⟼ ξ(e) e  .

This diagonal induces an action map of equivariant cohomology groups

    ^0_G(B_+) ×^0_G(T h(ξ)) 
      ×  ^0_G(B_+ T h(ξ))   Δ^* ^0_G(T h(ξ))  .

Since the diagonal is coassociative and counital, the action map
makes the group ^0_G(T h(ξ)) into a left module over the
commutative ring ^0_G(B_+).
The Thom isomorphism then says that whenever B admits the structure
of a finite G-CW-complex, then ^0_G(T h(ξ)) is a free module 
of rank 1 over ^0_G(B_+), with the Thom class σ_G(ξ)
as a generator.
Theorem <ref> is the special case when the base
consists of a single point.
We refrain from giving the proof of the general Thom isomorphism 
in equivariant -theory.


[Euler classes]
We let G be a compact Lie group and V an m-dimensional G-representation.
As usual, the Thom classes σ_G,V give rise to
Euler classessymbole(V) - Euler class in π_0^G()subjectEuler class!in 
by `restriction to the zero section', i.e.,

    e(V) =  i^*(σ_G,V) ∈ ^0_G(S^0) =  π_0^G()  .

Here i:S^0→ S^V is the inclusion
of the fixed points 0 and ∞, and
i^*:^0_G(S^V)→^0_G(S^0) the induced map on equivariant cohomology groups.
The Euler class is thus represented by the based G-map

    S^V  → (V)  ,   v ⟼ ( (v,0),  V⊕ V)  .

Since the Thom class lives in the homogeneous summand ^[m],
so does the Euler class.
If V has nonzero G-fixed points, then the inclusion i:S^0→ S^V
is G-equivariantly null-homotopic, so e(V)=0 whenever V^G 0.



[Shifted Thom and Euler classes in ]subjectshifted inverse Thom class!in  
The author thinks that the periodic theory  is the most
natural home for the Thom classes, the Euler classes and the inverse Thom classes, 
but the more traditional place
to host them is the degree 0 wedge summand =^(0].
Indeed, for an m-dimensional G-representation V,
the Thom class σ_G,V and the Euler class e(V) lie 
in the homogeneous summand ^[m],
and we can use the periodicity of  to move the classes into ,
at the expense of shifting their degrees by m.
In other words, by multiplying by a suitable power of the
periodicity class t∈π_-1^e(^[-1]), we define

    σ̅_G,V =  p_G^*(t^m)·σ_G,V∈ ^m_G(S^V) 
       and  e̅(V) =  p_G^*(t^m)· e(V)  ∈ π_-m^G()  ,

where p_G:G→ e is the unique group homomorphism.
The Thom isomorphism theorem for  then says that ^*_G(S^V) 
is a free graded module of rank 1 over the graded ring π_*^G(),
and the shifted Thom class σ̅_G,V is a generator.
This version of the Thom isomorphism follows directly 
from Theorem <ref> because 
 is globally equivalent to the wedge of all suspensions and desuspensions
of . 
More precisely, the maps 

    ⊕_n∈ π_n^G() → π_0^G()  
       and  ⊕_n∈ ^-n_G(S^V) → ^0_G(S^V)  ,

given on the n-th summand by multiplication by p_G^*(t^n),
are isomorphisms; moreover the latter isomorphism 
takes the shifted Thom class σ̅_G,V
to the original Thom class σ_G,V.

Similarly, we can use the periodicity of  to move the inverse Thom class 
τ_G,V∈_0^G(S^V) into ,
at the expense of shifting it from degree 0 to homological degree m.
The class τ_G,V lies in the homogeneous summand ^[-m],
so by multiplying by a suitable power of the
periodicity class σ∈π_1^e(^[1]) we define

    τ̅_G,V =  p_G^*(σ^m)·τ_G,V∈ _m^G(S^V)  .

Theorem <ref> (iii) then implies that
for every based G-space A, the map

    -·τ̅_G,V :  π_*^G( A) → π_*+m^G( A S^V)

is an isomorphism.





Our next task is to show that the Thom spectrum  is
a localization of the Thom spectrum , 
obtained by formally inverting all inverse Thom classes.
This result can be viewed as a `thomification' of the fact
that the morphism of ultra-commutative monoids i:→
induces a group completion of abelian monoids
[A,i]^G : [A,]^G→[A,]^G
for every compact Lie group G and every G-space A, 
compare Proposition <ref>.

The ultra-commutative ring spectra  and  are connected by a homomorphism

    a  :   →

whose value at an inner product space V is

    a(V) :  (V) → (V)  ,    
    (x,L) ⟼ ((x,0), L⊕ 0) .

For varying V, these maps form a morphism of 
-graded ultra-commutative ring spectra.
The morphism (<ref>)
induces natural transformations of equivariant homology theories

    (a A)_*  :  _*^G(A) → _*^G(A)

for all compact Lie groups G and all based G-spaces A.
We observe that 

    (a S^V)_*(τ_G,V) =  τ_G,V ,

i.e., the morphism a takes the -inverse Thom class
to the -inverse Thom class with the same name.
This relation is immediate from the explicit
representatives of the inverse Thom classes in (<ref>)
respectively (<ref>).







We define a localized version of equivariant -homology by

    ^G_k(A)[1/τ] =  _V∈ s(_G) _k^G(A S^V)  ;

for V⊂ W, the structure map in the colimit system is the multiplication

    _k^G(A S^V)  -·τ_G,W-V _k^G(A S^V S^W-V)   _k^G(A S^W)   .

In equivariant -theory, the inverse Thom classes become
invertible by Theorem <ref> (iii).
So for a G-representation V we can consider the map

    _k^G(A S^V)   (a A S^V)_*  ^G_k(A S^V)  (-·τ_G,V)^-1 ^G_k(A) .

By the multiplicativity of a,
these maps are compatible as V varies over the poset s(_G),
so they assemble into a natural transformation

    a^♯ :  _k^G( A)[1/τ]  = _V∈ s(_G) _k^G(A S^V)  → ^G_k(A) .




For every compact Lie group G, every based G-space A 
and every integer k the map

    a^♯ :  _k^G(A)[1/τ]  → _k^G(A)

is an isomorphism.


To simplify the exposition we prove the claim for k=0 only, 
the argument for a general integer being essentially the same.
Alternatively, we can observe that source and target of a^♯
are periodic, so it suffices to establish bijectivity in a single dimension.
We show two separate statements that amount to the injectivity
respectively surjectivity of the map a^♯.

(a) 
We show that for every class x in the kernel 
of the map (a A)_*:_0^G(A) →^G_0(A),
there is a G-representation V such that x·τ_G,V=0.
Indeed, we can represent any such class x by a based G-map
f:S^V→(V) A, for some G-representation V,
such that the composite

    S^V   f  (V) A a(V) A (V) A

is equivariantly null-homotopic. 
In (<ref>) we defined a morphism of orthogonal G-spectra
j^V_:→^V. 
We observe that (V)=(V⊕ V) and a(V)=j^V_(V).
So we can apply Proposition <ref> and conclude that

    (λ^V_ A)_*(x·τ_G,V)    =  
     (j_^V A)_*(x) 
       =  [ (j_^V(V) A)∘ f] =  [ (a(V) A)∘ f] =  0  .

Since λ^V_ A: A S^V→^V A is
a _*-isomorphism (by Proposition <ref> (ii)),
this implies the desired relation x·τ_G,V=0.

(b) 
We show that for every class y in _0^G(A),
there is a G-representation V and a class x in _0^G(A S^V)
such that y·τ_G,V= (a A S^V)_*(x).
To this end we represent y by a based G-map
f:S^V→(V) A. Because (V) = (V^2)=(^V )(V),
the map f also defines a class in π_0^G(^V A).
Since λ^V_ A: A S^V→^V A is
a _*-isomorphism by Proposition <ref> (ii),
there is a unique class x∈_0^G(A S^V) such that

    (λ^V_ A)_*(x) =  [f] .

On the other hand, the map a(V^2):(V^2)→(V^2)
is equal to the map j^V_(V):(V)→(V^2).
So

    (λ^V_ A)_*(  (a A S^V)_*(x))    =   
      (^V a A)_*((λ^V_ A)_*(x))
       =   (^V a A)_*[f] =   [(a(V^2) A)∘  f]
       =   [(j^V_(V) A)∘  f] =  (j^V_ A)_* (y) 
       =  (λ^V_ A)_* (y·τ_G,V)  .

The sixth equation is Theorem <ref> (iii),
the others are either definitions or naturality properties.
Since λ^V_ A is a _*-isomorphism,
we can conclude that 

    (a A S^V)_*(x) =   y·τ_G,V .





[The global Thom spectra  and ]
We define two E_∞-orthogonal ring spectra symbol - global Thom spectrum 
and ,symbol - periodic global Thom spectrum 
the Thom spectra over the orthogonal spaces  and 
defined in Examples <ref>
respectively <ref>.
The spectrum  is a periodic version of ,
and conversely  is the homogeneous degree 0 summand with respect
to a certain -grading of .
Non-equivariantly,  is another version of the unoriented Thom spectrum M O.
The equivariant homology theory represented by  is the natural target
of the equivariant Thom-Pontryagin map from equivariant bordism, 
and that map is trying hard to be an isomorphism, see Theorem <ref>  below.


We recall that the value of  at an inner product space V is

    (V) =  ∐_n≥ 0  G r_n(V⊕^∞)  ,

the disjoint union of all the Grassmannians in V⊕^∞.
The map (φ):(V)→(W) induced by a 
linear isometric embedding φ:V→ W is defined as

    (φ)(L)  =  
    (φ⊕^∞)(L)  +  ((W-φ(V))⊕ 0)  .

In other words: we apply the linear isometric 
embedding φ⊕^∞:V⊕^∞→ W⊕^∞ 
to the subspace L and add the orthogonal complement of the image of φ
(sitting in the first summand of W⊕^∞).

Over the space (V) sits a tautological euclidean vector bundle 
(again of non-constant rank);
the total space of this bundle consist of pairs 
(x,U)∈ (V⊕^∞)×(V)  such that x∈ U.
We define (V) as the Thom space of this tautological vector bundle.
The structure maps are given by

    (V,W)(V)     →  (W)
    
    (w,φ) (x,U)      ⟼ ((w,0)+(φ)(x), (φ)(U)) .

As we explained in Remark <ref>,
the orthogonal spaces  and  have natural E_∞-structures.
Correspondingly, the orthogonal spectra  and 
have natural E_∞-structures,
by which we mean an action of the linear isometries operad. 
This multiplication is, however, not ultra-commutative.
Multiplication maps

    μ_V,W :  ((^∞)^2,^∞)_+ (V) (W)  → (V⊕ W)

are defined by sending ψ (x,U) (x',U') 
to (ψ_♯(x,x'),ψ_♯(U⊕ U')),
where ψ_♯ is the linear isometric embedding

    V⊕^∞⊕ W⊕^∞ → 
    V⊕ W⊕^∞  ,  ψ_♯(v,y,w,z) =  (v,w,ψ(y,z)) .

Unit maps are defined by

    S^V → (V)  ,     v ⟼ ((v,0),  V⊕ 0) .

All this structure makes  into an E_∞-orthogonal ring spectrum.

The orthogonal spectrum  is -graded, where
the summand ^[k](V) of degree k is defined as 
the Thom space of the tautological
(|V|+k)-plane bundle over ^[k](V)=G r_|V|+k(V⊕^∞);
then (V) is the one-point union of the
Thom spaces  ^[k](V) for |V|+k≥ 0.
So we have a wedge decomposition

    =  ⋁_k∈ ^[k]

as orthogonal spectra.
We define =^[0] as the homogeneous wedge summand of degree 0.



In the rest of this section we will also use products
on the equivariant homology theories represented by the Thom spectra 
and .
As we just explained,
the spectrum  comes with an E_∞-multiplication
which is, however, neither strictly associative, nor strictly commutative.
So we briefly explain how to define these products.
We choose a linear isometric embedding ψ:^∞⊕^∞→^∞.
We define continuous maps

    ψ_V,W :  (V) (W)     →  (V⊕ W)       by
    ψ_V,W((x,U),(x',U'))     =  (ψ_♯(x,x'),ψ_♯(U⊕ U'))  ,

where ψ_♯ was defined in (<ref>).
These maps form a bimorphism, which corresponds to a morphism of orthogonal spectra

    ψ_⋆ :   →

by the universal property of the smash product.
Since the space ((^∞)^2,^∞)
of linear isometric embeddings is contractible,
the morphism ψ_⋆ is independent up to homotopy of the choice of ψ.
Even though the multiplication map ψ_⋆ is neither associative
nor commutative, the contractibility of the space ((^∞)^3,^∞) 
implies that the square

    @C=18mm[r]^-ψ_⋆[d]_ψ_⋆   [d]^-ψ_⋆
    [r]_-ψ_⋆

commutes up to homotopy,
and the contractibility of the space ((^∞)^2,^∞)
implies that the composite

    τ_,   ψ_⋆

is homotopic to ψ_⋆.
So whenever we pass to induced maps on equivariant homotopy groups,
an E_∞-multiplication is as good as a strictly
associative and commutative multiplication. 
However, an E_∞-multiplication does not entitle us to power operations.

Given a compact Lie group G and based G-spaces A and B,
we define a multiplication

    · :  ^G_k(A)×_l^G(B)  → _k+l^G(A B)

as the composite

    π_k^G( A)×π_l^G( B)     [(<ref>)] ×  π_k+l^G( A B) 
       (twist)_* π_k+l^G( A B) 
       (ψ_⋆ A B)_* π_k+l^G( A B)  .


We move on to explain the periodicity property of . 
As the theories  and , the theory 
also has its own inverse Thom classes and shift morphisms.
We define the inverse Thom classsubjectinverse Thom class!in  

    τ_G,V ∈ _0^G(S^V)

as the class represented by the G-map 

    S^V  → T h(G r(V⊕^∞)) S^V  =  (V) S^V  ,  
    v ⟼  ( (0,0), 0⊕ 0) (-v)    .

Here we abuse notation one more time and also denote the inverse Thom class 
in -theory
by the same symbol as its counterparts in -theory
and -theory defined in (<ref>)
respectively (<ref>).
The justification for this abuse is that the inverse Thom classes
match up under certain homomorphisms relating ,  and .
As usual, if V has dimension m, then the class τ_G,V 
lies in the homogeneous summand ^[-m].
A shift morphism of orthogonal G-spectra j_^V:→^V 
is defined as for  and : the value at an inner product space U is the map

    j_^V(U) :  (U)    → (U⊕ V) =  (^V)(U)
    
    (x,L)     ⟼   (i(x),i(L))

induced by the linear isometric embedding
i:U⊕^∞→ U⊕ V⊕^∞ with i(u,x)=(u,0,x).
If V has dimension m, then j^V_ is homogeneous of degree -m.
In the special case V= with trivial G-action, the map

    j  =  j_^ :   →

is a morphism of orthogonal spectra (with trivial G-action).

On the level of homotopy groups, the periodicity is realized by
multiplication with a periodicity element t∈π_-1^e(^[-1])
represented by the point

    (0,{0}) ∈ T h(G r_0(⊕^∞)) =  ^[-1]()  .





  
  
  * The morphism j=j_^ :→
    is a homotopy equivalence of orthogonal spectra, hence a global equivalence.
    For every compact Lie group G and every based G-space A, the induced map
    
    (j A)_* :  π_*^G( A)  → π_*^G( A)

    is an isomorphism.
  
  * For every compact Lie group G, every G-representation V and every
    based G-space A, the composite
    
    π_0^G( A) -·τ_G,V π_0^G( A S^V) 
         (λ^V_ A)_* π_0^G(^V  A)

    coincides with the effect of the morphism 
    j^V_ A: A→^V A.
    In particular, exterior multiplication by the inverse Thom class τ_G, 
    of the trivial 1-dimensional G-representation
    is invertible in equivariant -homology.
  
  * For every compact Lie group G, every based G-space A
    and every integer k the multiplication map
    
    _k+1^G(A)    · p_G^*(t)  _k^G(A)

    is an isomorphism. In particular, the class t∈π_-1^e()
    is a unit in the graded homotopy ring π_*^e().
  


(i) The morphism j is based on the 
linear isometric embedding i:^∞→⊕^∞ 
defined by i(x)=(0,x). This linear isometric embedding is homotopic, 
through linear isometric embeddings,
to the linear isometry ^∞⊕^∞ 
sending (x_1,x_2,x_3,…) to (x_1,(x_2,x_3,…)).
Such a homotopy induces a homotopy from the morphism j to an
isomorphism between  and . So j is homotopic to an isomorphism,
hence a homotopy equivalence.

The first statement in (ii) is proved by the same argument as the analogous statement 
for  in Proposition <ref>.
For V=, the trivial 1-dimensional G-representation,
the morphisms j^_ A and λ_ A
are both global equivalences by part (i) 
respectively Proposition <ref> (i).
So they induce isomorphisms on π_0^G.
Hence exterior multiplication by the class τ_G, is invertible.

(iii) The composite

    _k+1^G(A) -· p_G^*(t)_k^G(A) []- S^1 _k+1^G(A S^1)

differs from multiplication by the inverse Thom class τ_G,
by the effect of the involution A S^- of A S^1.
Multiplication by τ_G, is an isomorphism by part (ii),
and the suspension isomorphism is bijective by
Proposition <ref>.
So multiplication by p_G^*(t) is bijective as well.



In the earlier Theorem <ref>
we showed that equivariant -homology is obtained from 
equivariant -homology by inverting all inverse Thom classes.
Now we add  to this picture, which turns out to be
an intermediate localization. 
As we will now explain, -theory is obtained from
-theory by inverting the inverse Thom classes of all trivial representation.
Then -theory is obtained from
-theory by inverting the remaining inverse Thom classes,
i.e., the ones of non-trivial representations.
Informally speaking, the first localization turns -theory 
into a theory that is periodic in the -graded sense;
the second localization then turns -theory 
into a theory that is periodic in the R O(G)-graded sense.
Schematically:

    @C=17mm_*^G(A)   @=>[r]^-invert τ_G,     _*^G(A)
      @=>[r]^-invert all τ_G,V     _*^G(A)


The ring spectra  and  are connected by a homomorphism

    b  :   →

whose value at an inner product space V is

    b(V) :  (V) → (V)  ,    
    (x,L) ⟼ ((x,0), L⊕ 0) .

For varying V, these maps form a morphism of 
-graded orthogonal E_∞-ring spectra.
The morphism b:→
induces natural transformations of equivariant homology theories

    (b A)_*  :  _*^G(A) → _*^G(A)

for all compact Lie groups G and all based G-spaces A.
We observe that 

    (b S^V)_*(τ_G,V) =  τ_G,V ,

i.e., the morphism b takes the -inverse Thom class
to the -inverse Thom class with the same name.
This relation is again immediate from the explicit
representatives of the inverse Thom classes in (<ref>)
respectively (<ref>).

We define a localized version of equivariant -homology by

    ^G_k(A)[τ_G,^-1] =  _n≥ 0 _k^G(A S^n)  ,

the colimit of the sequence

    _k^G(A)  -·τ_G, _k^G(A S^1)  -·τ_G,_k^G(A S^2)  -·τ_G, …

along multiplication by the inverse Thom class τ_G,∈_0^G(S^1).
In equivariant -theory, the class τ_G,
becomes invertible by Theorem <ref> (ii).
So we can consider the maps

    _k^G(A S^n)   (b A S^n)_*  ^G_k(A S^n)  (-·τ_G,^n)^-1 ^G_k(A) .

By the multiplicativity of b, these maps are compatible,
so they assemble into a natural transformation
b^♯ : _k^G(A)[τ_G,^-1]→^G_k(A).



For every compact Lie group G, every based G-space A and every integer k
the map 

    b^♯ :  _k^G(A)[τ_G,^-1]  → _k^G(A)

is an isomorphism.


The `standard' linear isometric embedding

    ^n  → ^∞ ,    
    (x_1,…,x_n) ⟼ (x_1,…,x_n,0,0,…)

induces a continuous map

    ψ^n(V) :  (^n )(V) =  T h(G r(V⊕^n))
     → T h(G r(V⊕^∞)) =  (V) .

As V varies, these maps form a morphism of orthogonal spectra
ψ^n:^n→. Since ψ^n= ψ^n+1∘(^n j_), 
the morphisms ψ^n are compatible with the sequence of morphisms

    j_    j_ ^2  ^2 j_ ⋯  .

Moreover, the morphisms ψ^n express  as the colimit of this sequence.
So A is the colimit 
of the sequence of orthogonal G-spectra ^n A.
The map j_(V) is an h-cofibration of based O(V)-spaces.
So if G acts on V by linear isometries,
then j_(V) A is an h-cofibration of based G-spaces, hence a closed embedding
by Proposition <ref>.
The morphism j_ A and all its shifts are thus levelwise closed embeddings.

Proposition <ref> (i)
shows that equivariant homotopy groups commute with sequential colimits
over closed embeddings; so the canonical map

    _n≥ 0 π_k^G(^n A) → π_k^G( A)

is an isomorphism.
The diagram

    @C=12mmπ_k^G( A S^n-1)[r]^-·τ_G,[d]_λ^n-1_ A^   π_k^G( A S^n)[d]^λ^n_ A_[r]^-π_k^G(b A S^n)   π_k^G( A S^n)[d]_^(-·τ_G,^n)^-1
    π_k^G(^n-1 A)[r]_-π_k^G(^n-1 j_ A)   π_k^G(^n A)[r]_-π_k^G(ψ^n A)   π_k^G( A)

commutes by Proposition <ref> and because
(^n-1j_)∘λ^n-1_ = λ^n-1_∘ (j_ S^n-1).
The three vertical maps are isomorphisms.
So π_k^G() is also a colimit of the sequence of
multiplication maps by τ_G., with respect to the maps
that define b^♯.



There is one localization result left: it remains to exhibit 
-theory as the localization of -theory, 
by inverting the inverse Thom classes of arbitrary representations.
This is in fact a formal consequence of 
Theorem <ref>
and Theorem <ref> which exhibit both
_*^G(A) and _*^G(A) as localizations of _*^G(A),
the former being a more drastic localization than the latter.

We spell this out in more detail.
We define a localized version of equivariant -homology by

    ^G_0(A)[1/τ] =  _V∈ s(_G) _0^G(A S^V)  ;

for V⊂ W, the structure map in the colimit system
is the multiplication

    _0^G(A S^V)  -·τ_G,W-V _0^G(A S^V S^W-V)   _0^G(A S^W)   .

In (<ref>) we introduced a morphism 
of ultra-commutative ring spectra a:→.
In (<ref>) we introduced the morphism
of E_∞-ring spectra b:→.
These morphisms induce multiplicative natural transformations

    (a A)_*     :  _*^G(A) → _*^G(A)     respectively
    
     (b A)_*     :  _*^G(A) → _*^G(A)

for all compact Lie groups G and all based G-spaces A.
Moreover, the morphisms match up the inverse Thom classes
in the sense that

    (a S^V)_*(τ_G,V) =  τ_G,V     and    
     (b S^V)_*(τ_G,V) =  τ_G,V ;

indeed, this is the justification for our abuse of notation of
using the same name for the inverse Thom classes in ,  and .

Theorem <ref> says that the map (b A)_*
becomes an isomorphism after inverting the inverse Thom class τ_G,
of the trivial 1-dimensional G-represen­tation.
So it also becomes an becomes after inverting the inverse Thom classes
of all representations, i.e., the induced transformation

    (b A)_*[1/τ]  :  _*^G(A)[1/τ] → _*^G(A)[1/τ]

is an isomorphism.
On the other hand, the transformation (a A)_* induces an
isomorphism a^♯ from _*^G(A)[1/τ] to _*^G(A), by
Theorem <ref>.
So combining these two theorems yields:


For every compact Lie group G, every based G-space A 
and every integer k the map

    a^♯∘ ((b A)_*[1/τ])^-1 :  _k^G(A)[1/τ]  → _k^G(A)

is an isomorphism.


While the authors thinks that the periodic theories  and  give the
most convenient formulation of the localization result, 
the more traditional formulation is in terms of the degree 0 summands
=^[0] and =^[0]. So we also take the
time to reformulate Corollary <ref>
in terms of  and .
Since the inverse Thom classes do not lie in the degree zero summands,
we instead use the shifted inverse Thom classessubjectshifted inverse Thom class!in subjectinverse Thom class!shifted|seeshifted inverse Thom class  

    τ̅_G,V =  p_G^*(σ^m)·τ_G,V∈ _m^G(S^V)

introduced in (<ref>), 
and its -analog

    τ̅_G,V =  p_G^*(σ^m)·τ_G,V∈ _m^G(S^V)   .

In the -case, the periodicity class σ∈π_1^e(^[1])
was defined in (<ref>), and it is inverse to the class t. 
In the -case, the periodicity class σ∈π_1^e(^[1])
was not yet defined, and we take it to be the inverse
of the  periodicity class t from (<ref>).
Both theories  and  are periodic (in the -graded sense),
i.e., the maps

    ⊕_m∈ _m^G(A) → _0^G(A)     and  ⊕_m∈ _m^G(A) → _0^G(A)

that multiply by the appropriate powers of the periodicity classes are isomorphisms.

We define a localized version of equivariant -homology by

    ^G_k(A)[1/τ̅] =  _V∈ s(_G) _k+|V|^G(A S^V)  ;

for V⊂ W, the structure map in the colimit system is the multiplication

    _k+|V|^G(A S^V)  -·τ̅_G,W-V _k+|W|^G(A S^V S^W-V)  _k+|W|^G(A S^W)   .

The periodicity of  is inherited by the localized theory, 
i.e., the upper horizontal map in the following commutative square

    ⊕_m∈_m^G(A)[1/τ̅][r] 
    [d]_a^♯∘ ((b A)_*[1/τ̅])^-1   _0^G(A)[1/τ] [d]^a^♯∘ ((b A)_*[1/τ])^-1
    ⊕_m∈_m^G(A)[r]    _0^G(A)

is an isomorphism.
The lower horizontal map is an isomorphism by the periodicity of , and
the right vertical map is an isomorphism by Corollary <ref>.
So the left vertical map is an isomorphism. Since the left map
is homogeneous with respect to the -grading, it is an
isomorphism in every degree. So we conclude:


For every compact Lie group G, every based G-space A 
and every integer m the map

    a^♯∘ ( (b A)_*[1/τ̅])^-1 :  _m^G(A)[1/τ̅]  → _m^G(A)

is an isomorphism.




Now we investigate the global homotopy type of the Thom spectrum 
in more detail; 
one reason for wanting to understand  better is the close connection
to equivariant bordism, compare Theorem <ref>  below. 
Our main tool is the following `rank filtration';
there is an analog of the rank filtration for , 
but we refrain from making it explicit.


[Rank filtration of ]subjectrank filtration! of 
The orthogonal spectrum  is a global Thom spectrum over
an orthogonal space ;   
in Proposition <ref>
we identified  as a certain global homotopy colimit
of the global classifying spaces B_ O(m). More precisely,
the filtration of ^∞ by the subspaces ^m induces a
filtration of  by orthogonal subspaces _(m),
and _(m) receives a global equivalence from B_ O(m).
We now define and study the corresponding orthogonal Thom spectrum
_(m) over _(m), which turns out to be an m-fold suspension
of the orthogonal spectrum M_ T(m), the global refinement
of the spectrum traditionally denoted M T (m).


Construction <ref> defines 
the semifree orthogonal spectrum F_G,V
generated by a G-representation V. We are interested in the
tautological O(m)-representation ν_m with underlying inner product space ^m,
and to simplify the notation we abbreviate the corresponding semifree spectrum to 

    F_m = F_O(m),ν_m .

The shift functor ^m=^^m 
by the inner product space ^m was defined in Construction <ref>.
We set

    _(m) =  ^m F_m ,

the m-th shift of F_m.symbol_(m) - truncated global Thom spectrum  
Unpacking this definition reveals the value of _(m)
at an inner product space V as the space

    _(m)(V) =  (ν_m,V⊕^m)/O(m) .

To justify the notation _(m) we clarify the connection
to the orthogonal space _(m) defined in (<ref>).
The value of _(m) at V is

    _(m)(V) =  G r_|V|(V⊕^m)  ,

the Grassmannian of |V|-planes in V⊕^m.
Over the space _(m)(V) sits a tautological euclidean |V|-plane bundle, 
with total space consisting of pairs 
(x,U)∈ (V⊕^m)×_(m)(V)  such that x∈ U.  
Passage to orthogonal complements provides a homeomorphism:

    _(m)(V) =  (ν_m,V⊕^m)/O(m)      T h(G r_|V|(V⊕^m))  
    
    [x, φ]     ⟼ (x,φ(ν_m)^⊥ )

In this sense, _(m)(V) `is' the Thom space over _(m)(V).
Just as the orthogonal spaces _(m) form an exhaustive filtration
of , the orthogonal spectra _(m) form an exhaustive filtration
of the Thom spectrum , compare Proposition <ref> below.


As we explained in Remark <ref>,
the semifree orthogonal spectrum F_m=F_O(m),ν_m
is a global refinement of the Thom spectrum M T(m) of the
negative of the tautological m-plane bundle over G r_m(^∞).
Since shift and suspension are globally equivalent
(by Proposition <ref> (i)),
_(m) is globally equivalent to the
m-fold suspension of the orthogonal spectrum M_ T (m)=F_O(m),ν_m 

    _(m) =  ^m F_m ≃_ F_m  S^m =   M_ T(m)  S^m .


We define a morphism of orthogonal spectra

    i  :  F_m →  F_m+1 ;

the value at an inner product space V is the closed embedding

    i(V)  :  F_m(V) = (ν_m,V)/O(m)
         -⊕(ν_m+1,V⊕)/O(m+1) = ( F_m+1)(V) 
    
    [x,φ]        ⟼     [(x,0),φ⊕]

where we identify
ν_m⊕ with ν_m+1 by 
sending ((x_1,…,x_m),y) to (x_1,…,x_m,y).
In fact, there are very few morphisms of orthogonal spectra from F_m to F_m+1:
by the representing property of the semifree spectrum F_m, such
morphisms biject with O(m)-fixed points
of ( F_m+1)(ν_m)=(ν_m+1,ν_m⊕)/O(m+1);
this space only has two elements, and the morphism i
corresponds to the non-basepoint element.
We define 

    j^m =  ^m i  : _(m) = ^m F_m  → ^m( F_m+1) =  ^m+1F_m+1 =  _(m+1) .

We define a morphism

    ψ^m  :  _(m) =  ^m F_m  →

at an inner product space V as the map

    ψ^m(V)  :  (ν_m,V⊕^m)/O(m)    → T h(G r_|V|(V⊕^∞))
    
    [x,φ]      ⟼   ( i(x), i(φ^⊥)  ) ,

where φ^⊥=(V⊕^m)-φ(ν_m)
is the orthogonal complement of the image of φ and
 i:V⊕^m → V⊕^∞ is the `standard' embedding given by

    i(v,x_1,…,x_m) =  (v,x_1,…,x_m,0,0,…)  .

The rank filtration expresses the orthogonal spectrum  
as the colimit of the sequence of closed embeddings

    _(0) j^0 _(1) j^1 … → _(m) j^m … .

The following proposition is straightforward from the definitions.
The periodic analog was already used in the proof of 
Theorem <ref>,
since it expresses  as a sequential colimit of the spectra ^m.
We omit the proof.

 
For every m≥ 0 the morphism 
j^m: _(m)→_(m+1) is levelwise a closed embedding. 
These morphisms satisfy ψ^m+1∘ j^m= ψ^m.  
With respect to the morphisms ψ^m:_(m)→,
the orthogonal spectrum  is a colimit of the sequence (<ref>).



Since colimits along sequences of closed embeddings
are invariant under global equivalences 
(Proposition <ref> (v)),
Proposition <ref> says that  is a homotopy colimit
of the sequence (<ref>).
The underlying non-equivariant statement, i.e., that M O is
a homotopy colimit of the spectra Σ^m M T(m),
can for example be found in <cit.>. 
The identification of  as a homotopy colimit of semifree orthogonal spectra
now allows an algebraic description of ,E, 
the group of morphisms in the global stable homotopy category ,
into any orthogonal spectrum E.

We define a distinguished class

    τ_m ∈ π_m^O(m)(_(m) S^ν_m)

in the m-th O(m)-equivariant _(m)-homology group of S^ν_m
as the class represented by the based O(m)-map

    S^ν_m⊕^m    → (ν_m,ν_m⊕^m)/O(m) S^ν_m =  _(m)(ν_m) S^ν_m
    
    (v,x)    ⟼     [ (0,x),i]  (-v)  ,

where i:ν_m→ν_m⊕^m is the embedding as the first summand.


  Let m≥ 0 be a natural number.
  
  
  * The pair (_(m),τ_m) represents the functor
    
    → (sets) ,    E  ⟼ E_m^O(m)(S^ν_m)
         =  π_m^O(m)(E S^ν_m)  .

  
  * The morphism j^m:_(m)→_(m+1) satisfies the relation
    
    (j^m S^ν_m)_*(τ_m)  S^1  =  ^O(m+1)_O(m)(τ_m+1)

    in the group  π_m+1^O(m)(_(m+1) S^ν_m S^1).
  
  * The morphism ψ^m:_(m)→ sends τ_m
    to the shifted inverse Thom class τ̅_O(m),ν_m
    defined in (<ref>).subjectshifted inverse Thom class!in  
 


(i) 
In (<ref>) we defined a distinguished equivariant homotopy class

    a_m  =  a_O(m),ν_m ∈ π_0^O(m)(F_m S^ν_m)   .
 
Expanding the definitions of a_m and of the morphism

    λ^m_F_m S^ν_m : F_m S^ν_m S^m → ^m F_m S^ν_m =  _(m) S^ν_m

shows that

    τ_m =  (λ^m_F_m S^ν_m)_*(a_m S^m)  .

By Theorem <ref>,
the pair (F_m,a_m) represents the functor

    →(sets) ,    E  ⟼ E_0^O(m)(S^ν_m) = π_0^O(m)(E S^ν_m) .

We claim that the following composite

    _(m), E    =  ^m F_m, E λ^m_F_m,E F_m S^m, E
       adjunction F_m,Ω^m E eval at a_m π_0^O(m)((Ω^m E) S^ν_m) 
       assembly_* π_0^O(m)(Ω^m (E S^ν_m))  α^m  π_m^O(m)(E S^ν_m)

coincides with evaluation at the class τ_m. 
Here the fourth map is induced by the assembly morphism

    (Ω^m E) S^ν_m → Ω^m( E S^ν_m)  ,

and

    α^m : π_0^O(m)(Ω^m Y)  π_m^O(m)(Y)

is the analog of the loop isomorphism (<ref>). 
Since all the maps are natural in E, it suffices to check this claim
for the identity of the universal example E=_(m)=^m F_m.
Since the square

    @C=15mm 
    E S^ν_m[r]^-η[d]_λ̃^m_E S^ν_m   Ω^m(E S^ν_m S^m) [d]^Ω^m(λ^m_E S^ν_m)
    
    (Ω^m ^m E) S^ν_m[r]_assembly   Ω^m (^m E S^ν_m)

commutes, we obtain

    α^m(assembly_*((λ̃^m_F_m S^ν_m)_*(a_m)))     =  α^m( Ω^m(λ^m_F_m S^ν_m)_*(η_*(a_m))) 
       =   (λ^m_F_m S^ν_m)_*(α^m(η_*(a_m))) 
       =   (λ^m_F_m S^ν_m)_*( a_m S^m) =  τ_m .

This verifies the relation in the universal example.
Since all the individual maps in the above composite are bijective,
so is the composite, which proves the representability property
of the pair (_(m),τ_m).

(ii)
The class (j^m S^ν_m)_*(τ_m)  S^1 is represented by the composite:

    S^ν_m⊕^m+1    t_m S^1 (ν_m,ν_m⊕^m)/O(m) S^ν_m S^1 
       j^m(ν_m⊕^m) S^ν_m S^1 (ν_m+1,ν_m⊕^m+1)/O(m+1) S^ν_m S^1
    
     (v,x,s)    ⟼ [ (0,x),i] (-v) s
     ⟼ [ (0,x,0),i⊕ ] (-v) s   .

If we stabilize this representative along the linear isometric embedding

    j :  ν_m⊕^m+1 → ^O(m+1)_O(m)(ν_m+1)⊕^m+1 ,   
     (v,x,s) ⟼ (v,0,x,s)

we obtain another representative, namely

    S^ν_m⊕⊕^m+1    → (ν_m+1,^O(m+1)_O(m)(ν_m+1)⊕^m+1)/O(m+1) S^ν_m S^1
    
     (v,u,x,s)    ⟼   [ (0,u,x,0),i⊕ ]  (-v) s  .

Here i⊕:ν_m+1→^O(m+1)_O(m)(ν_m+1)⊕^m+1
sends (v,u) to (v,0,0,u).

On the other hand, 
 ^O(m+1)_O(m) (τ_m+1) is represented by the underlying
O(m)-map of the O(m+1)-map

    S^ν_m+1⊕^m+1     t_m+1  (ν_m+1,ν_m+1⊕^m+1)/O(m+1) S^ν_m+1
    
     (v,u,x,s)    ⟼   [ (0,0,x,s),i' ] (-v, -u) ,

where i':ν_m+1→^O(m+1)_O(m)(ν_m+1)⊕^m+1
sends (v,u) to (v,u,0,0).
The two representatives differ by conjugation with the O(m)-equivariant
linear isometry

    ν_m⊕⊕^m⊕ → ν_m⊕⊕^m⊕ ,   
    (v,u,x,s) ⟼ (v,-s,x,u) ,

so they represent the same class in 
the group  π_m+1^O(m)(_(m+1) S^ν_m S^1).

(iii) Substituting the definitions of τ_m and of the morphism ψ^m
shows that (ψ^m S^ν_m)_*(τ_m) is represented by the map

    S^ν_m⊕^m → (ν_m) S^ν_m ,  
    (v,x) ⟼   [ (0,i(x)), 0⊕ i(^m)]  (-v)  ,

where i:^m→^∞ is the `standard' embedding as the leading m coordinates.
The periodicity class σ is represented by the map

    s :  S^1  → T h(G r(^∞))  =  (0) ,   
    x ⟼ ((x,0,0,…),⊕ 0) .

The multiplication of  is an E_∞-multiplication,
so as explained in (<ref>),
multiplication of classes in -theory
involves choices of linear isometric embeddings.
We choose a linear isometric embedding

    ψ :  (^∞)^m  → ^∞

that satisfies 

    ψ( (x_1,0,0,…),…,(x_m,0,0,…)) =  
    (x_1,…,x_m,0,0,…)  .

If we base the multiplication of  on such a choice ψ,
then the product of the defining representative for τ_O(m),ν_m
and m factors of the map s above is precisely the 
previous representative for the class (ψ^m S^ν_m)_*(τ_m).
This proves the relation (ψ^m S^ν_m)_*(τ_m)=τ_O(m),ν_m· p_O(m)^*(σ^m) in_m^O(m)(S^ν_m).





The fact that  is the global homotopy colimit of the sequence of
orthogonal spectra _(m) (see Proposition <ref>)
has the following consequence. 

 
For every orthogonal spectrum E the following sequence is short exact:

    0  → lim_m^1   E_m+1^O(m)(S^ν_m)
     → , E → lim_m    E_m^O(m)(S^ν_m) → 0

Here  the right map is evaluation at the shifted inverse Thom
classes τ̅_O(m),ν_m and 
the inverse limit and derived limit are formed along the maps

    E_m+1^O(m+1)(S^ν_m+1) ^O(m+1)_O(m) E_m+1^O(m)(S^ν_m S^1) (- S^1)^-1 E_m^O(m)(S^ν_m)  .



Since  is the sequential homotopy colimit, 
in the triangulated global stable homotopy category, 
of the sequence of orthogonal spectra (<ref>), 
the Milnor exact sequence takes the form:

    0  → lim_m^1 _(m) S^1,E → , E → lim_m   _(m),E → 0

By  Proposition <ref>
the pair (_(m),τ_m) then represents the functor

    →(sets) ,    E  ⟼ E_m^O(m)(S^ν_m) ,

and the morphism j^m:_(m)→_(m+1) has the correct behavior
on the universal classes.




Now that we recognized  as the global homotopy colimit of
the rank filtration (<ref>), we study how
one filtration term is obtained from the previous one.
The answer given by Theorem <ref> below
takes the form of a distinguished triangle
in the global stable homotopy category, witnessing that
the mapping cone of j^m:_(m-1)→_(m)
`is' the m-fold suspension of the suspension spectrum
of the global classifying space  B_O(m).

We define a morphism of orthogonal spectra

    T_m = T_O(m)^O(m+1) :  Σ^∞_+ B_O(m+1) → F_m

as the adjoint of the O(m+1)-equivariant map

    r  :   S^ν_m+1     → (ν_m,ν_m+1)/O(m) 
     =  F_m(ν_m+1) 
    
     r(A·(0,…,0,t))    = A·( (0,…,0, (t^2-1)/t),incl)· O(m) ,

where A∈ O(m+1) and t∈[0,∞].
Theorem <ref> (ii) shows that 
the morphism T_m represents the dimension shifting transfer
from O(m) to O(m+1), in the sense of the relation

    (T_m)_*(e_O(m+1)) =  _O(m)^O(m+1)(a_m)

between the tautological classes.
From T_m we define another morphism of orthogonal spectra

    ∂ =  λ_F_m^m∘ (T_m S^m)
      =     (^m T_m)∘λ_Σ^∞_+ B_ O(m+1)^m
     : 
       Σ^∞_+ B_O(m+1) → ^m F_m  =  _(m) .

The space

    ((Σ^∞_+ _O(m+1),ν_m+1)(ν_m+1))^O(m+1) =  (S^ν_m+1(ν_m+1,ν_m+1)/O(m+1) )^O(m+1)

has two points, the basepoint and 0_ν_m+1· O(m+1).
So there is a unique non-trivial morphism of orthogonal spectra 

    a : F_m+1 → Σ^∞_+ _O(m+1),ν_m+1=Σ^∞_+ B_O(m+1)  .

For every orthogonal spectrum E
the morphism λ_E^m+1:E S^m+1→^m+1 E 
is a global equivalence
(by an iteration of Proposition <ref> (i)),
so it becomes invertible in the global stable homotopy category.
We can thus define a morphism in  as

    q  =  (λ_Σ^∞_+ B_ O(m+1)^m+1)^-1∘ (^m+1 a)
     :  _(m+1) → Σ^∞_+ B_O(m+1) S^m+1 .



subjectglobal classifying space!of O(n)  
The sequence

    Σ^∞_+ B_O(m+1) S^m  ∂ _(m) j^m _(m+1)  q  Σ^∞_+ B_O(m+1) S^m+1

is a distinguished triangle in the global stable homotopy category.
The behavior of the first morphism on the stable tautological class is
given by

    ∂_*( e_O(m+1) S^m)  =  _O(m)^O(m+1)(τ_m) .



The tautological O(m+1)-representation ν_m+1 is faithful,
and the action of O(m+1) on the unit sphere S(ν_m+1) is transitive.
The stabilizer group of the unit vector (0,…,0,1)
identifies with the group O(m), and the orthogonal complement of
this vector becomes the tautological representation ν_m. 
We can thus apply Theorem <ref>
and obtain a distinguished triangle:

    F_m+1  a  Σ^∞_+ B_O(m+1)  T_m  F_m -λ_F_m+1^-1∘ i    F_m+1 S^1

By Example <ref>
shifting preserves distinguished triangles; 
so the following sequence is also distinguished:

    ^m F_m+1 ^m a ^m Σ^∞_+ B_O(m+1) ^m T_m ^m F_m -^m(λ_F_m+1^-1∘ i)  ^m F_m+1 S^1

The rotation of this triangle is the lower sequence in the following diagram:

    @C=10mmΣ^∞_+ B_O(m+1) S^m[r]^-∂[d]_λ^m_Σ^∞_+ B_O(m+1)   _(m)[r]^-j^m@=[d]   _(m+1)[d]^^m(λ_F_m+1^-1)[r]^-q   Σ^∞_+ B_O(m+1)  S^m+1[d]^λ^m_Σ^∞_+ B_O(m+1) S^1
    ^m Σ^∞_+ B_O(m+1)[r]_-^m T_m   ^m F_m [r]_-^m(λ_F_m+1^-1∘ i)   ^m F_m+1 S^1   [r]_-^m a S^1   ^m Σ^∞_+ B_O(m+1)  S^1

The left and middle squares commute by definition of the morphisms ∂
respectively j^m; the right square commutes by the relation

    (λ^m_Σ^∞_+ B_O(m+1) S^1)∘ q     =  
    (λ^m_Σ^∞_+ B_O(m+1) S^1)∘ (λ_Σ^∞_+ B_ O(m+1)^m+1)^-1∘ (^m+1 a)
       =  
    (^m λ_Σ^∞_+ B_ O(m+1))^-1∘ (^m+1a) 
       =  ^m(λ_Σ^∞_+ B_ O(m+1)^-1∘( a)) =  ^m((a S^1)∘λ_F_m+1^-1)  .

So the upper sequence is a distinguished triangle.

The final relation is a consequence of various other previously established relations:

    ∂_*( e_O(m+1) S^m )     =  
    (λ_F_m^m)_*( (T_m S^m )_*(e_O(m+1) S^m ) ) 
       =  (λ_F_m^m)_*( (T_m)_*(e_O(m+1)) S^m )  
     
    _(<ref>)   =  (λ_F_m^m)_*(_O(m)^O(m+1)(a_m) S^m )  
       =  (λ_F_m^m)_*(_O(m)^O(m+1)((F_mτ_ν_m,^m)_*(a_m S^m )))  
       =  _O(m)^O(m+1)((λ_F_m^m S^ν_m)_*((F_mτ_ν_m,^m)_*(a_m S^m )))  
       =  _O(m)^O(m+1)((λ^m_F_m S^ν_m)_*(a_m S^m))
     =  _O(m)^O(m+1)(τ_m) .

The second and fifth equations are naturality.
The fourth equation is the compatibility of transfer and suspension isomorphism,
see Proposition <ref>.



For calculations of equivariant homotopy groups of 
we also need to understand the composite:

    Σ^∞_+ B_ O(m+1)  S^m
      ∂  _(m)  q  Σ^∞_+ B_ O(m)  S^m

We start from the calculation

    (a∘ T_m)_*(e_O(m+1))    =  
      a_*( _O(m)^O(m+1)(a_m ) ) = _O(m)^O(m+1)(  a_*(a_m ) )
       =  _O(m)^O(m+1)(  a· e_O(m)) ) = _O(m)^O(m+1)(e_O(m)) .

On the other hand,

    q∘∂    =  
    (λ_Σ^∞_+ B_ O(m)^m)^-1∘ (^m a)∘λ_F_m^m∘ (T_m S^m)  
       =  (λ_Σ^∞_+ B_ O(m)^m)^-1∘λ_Σ^∞_+ B_O(m)^m∘ (a S^m)∘ (T_m S^m)  =   (a∘ T_m) S^m ,

by definition.
Combining these two facts gives

    ( q∘∂)_*(e_O(m+1) S^m)     =  
    ((a∘ T_m) S^m)_*(e_O(m+1) S^m) 
       =  
    (a∘ T_m)_*(e_O(m+1)) S^m =  _O(m)^O(m+1)(e_O(m)) S^m .

In other words, the composite q∘∂
represents the degree zero transfer _O(m)^O(m+1).

Now we can easily show that  is globally connective
and describe the global functor _0().
We denote by _e^O(1) the global subfunctor of the Burnside ring functor
 generated by _e^O(1)∈(O(1)).


 
The orthogonal spectrum  is globally connective
and the action of the Burnside ring global functor on the unit element
1∈π_0^e() induces an isomorphism of global functors

    /_e^O(1)  _0()  .



The suspension spectrum Σ^∞_+ B_O(m+1) is globally connective
(Proposition <ref>);
so the distinguished triangle of Theorem <ref>
implies that the morphism j^m:_(m)→_(m+1)
induces an isomorphism of global functors 

    _k(_(m))  _k(_(m+1))

for k≤ m-1 and an exact sequence of global functors

    (O(m+1),-) →  _m(_(m)) → _m(_(m+1)) → 0  .

Here we used the isomorphisms

    (O(m+1),-)     []τ↦τ(e_O(m+1)) _0(Σ^∞_+ B_O(m+1)) 
       [] - S^m  _m(Σ^∞_+ B_O(m+1) S^m)   .

Since _(0) is isomorphic to the global sphere spectrum,
which is globally connective, we conclude inductively that _(m)
is globally connective for all m≥ 0, and that the inclusion
_(1)→_(m) induces an isomorphism on _0 for all m≥ 1.
Since  is a colimit of the sequence of closed embeddings
j^m:_(m)→_(m+1), the map

    _m _k(_(m)) → _k()

induced by the morphisms ψ^m:_(m)→
is an isomorphism of global functors for every integer k.
So  is globally connective and the morphism ψ^1:_(1)→
induces an isomorphism

    _0(_(1))  _0() .

The unit morphism identifies _(0) with the global sphere spectrum,
so  the action on the class 1∈_(0)
is an isomorphism of global functors _0(_(0)).
For m=1, the exact sequence thus becomes an exact sequence of global functors

    (O(1),-) (_e^O(1),-) (e,-) → _0(_(1)) → 0  ;

here we used (<ref>)
to identify the first morphism as the one induced by the transfer _e^O(1).
This proves the claim about _0().




Theorem <ref> gives a nice compact description of
the 0-th equivariant homotopy groups of the Thom spectrum ,
but we may still ask for a more explicit calculation of the
group π_0^G() for an individual compact Lie group G.
Given the presentation of _0() as the quotient of
 by the global subfunctor generated by _e^O(1),
this is a purely algebraic exercise.
Since the element _e^O(1)(1) is trivial in the group π_0^O(1)(), also

    2  =  ^O(1)_e(^O(1)_e(1)) =  0

in π_0^e();
thus all equivariant homotopy groups π_*^G() are _2-vector spaces.
The next proposition pins down an _2-basis 
of π_0^G() in terms of the subgroup structure of G.
For a closed subgroup H of G we use the familiar notation

    t_H^G  =  _H^G(p_H^*(1))  ∈ π_0^G()  ,

where p_H:H→ e is the unique homomorphism.


 
  For every compact Lie group G, an _2-basis of π_0^G()
  is given by the classes t_H^G, indexed by conjugacy classes
  of those closed subgroups H of G whose Weyl group is finite and of odd order.


  We abbreviate C=O(1).
  The group (G) is free abelian with basis the classes t_H^G 
  for all conjugacy classes of subgroups H with finite Weyl group.
  So the claim follows if we can show that the value _e^C(G) 
  of the global functor _e^C at G is the subgroup of (G) 
  generated by 2·(G) and the classes t_H^G 
  for those closed subgroups H whose Weyl group is finite of even order.

  By Theorem <ref>, 
  the group (C,G) is freely generated by the elements
  _L^G∘α^* indexed by (G× C)-conjugacy classes
  of pairs (L,α) where L is a closed subgroup of G with finite Weyl group
  and α:L→ C is a continuous group homomorphism. 
  So _e^C(G) is generated as an abelian group by the elements
  _L^G∘α^*∘_e^C.

  A homomorphism to C is either trivial or surjective,
  and the generating elements come in two flavors.
  If α is the trivial homomorphism, then
  
    _L^G∘α^*∘_e^C  =  _L^G∘ p_L^*∘^C_e∘_e^C  =  
      2·_L^G∘ p_L^* =  2· t_L^G .

  These elements generate the subgroup 2·(G).
  If α is surjective with kernel H, then
  
    _L^G∘α^*∘_e^C  =  _L^G∘_H^L∘ p_H^*  =  _H^G∘ p_H^*  =   t_H^G .

  The group H that arises in this way is normal of index 2 in L.
  So L is contained in the normalizer N_G H, and 2 divides [N_G H:H]. 
  So the order of the Weyl group W_G H is even.
  Conversely, if H  is a subgroup of G with finite Weyl group of even order,
  then we can choose a subgroup C≤ W_G H of order 2.
  The preimage L of C under the projection N_G H→ W_G H
  then contains H as an index 2 subgroup. By the above, the class t_H^G 
  is then one of the generating elements of _e^C(G).





[Geometric fixed points of  and ]
subjectgeometric fixed points!of 
subjectgeometric fixed points!of 
We give a description of the geometric fixed points of  and .
The geometric fixed points of other global Thom spectra such as
_(m),  or  can be worked out in a similar fashion, 
but we leave that to the interested reader.
We let V be a representation of a compact Lie group G.
We write V^⊥=V-V^G for the orthogonal complement of the G-fixed subspace.
A point ((v,x),U)∈(V)=T h(G r(V⊕^∞)) 
is G-fixed if and only if the subspace U
of V⊕^∞ is G-invariant and the vector v∈ V is G-fixed.
The first condition guarantees that U=U^G⊕ (U∩ V^⊥),
and U^G is a subspace of V^G⊕^∞.
So we obtain a homeomorphism

    (V^G)(V^⊥)^G        (V)^G 
    
    (x, U) W     ⟼ ( x ,  U⊕ (V^⊥- W))  .

Under this identification, the structure map

    (σ_V,W)^G  :  S^V^G(W)^G  → (V⊕ W)^G

becomes the  smash product of the structure map 

    σ_V^G,W^G :  S^V^G(W^G)  → (V^G⊕ W^G)

with the map

    (i)^G  :  (W^⊥)^G  → (V^⊥⊕ W^⊥)^G

induced by the embedding W^⊥→ V^⊥⊕ W^⊥
as the second summand.
So in the colimit over V∈ s(_G) this gives an isomorphism

    Φ_*^G()   _*((_G^⊥)^G_+)

to the non-equivariant -homology groups 
of the G-fixed point space of (_G^⊥),
the disjoint union of all Grassmannians in _G^⊥=_G-(_G)^G.

The formula (<ref>) is a compact way 
to express the geometric fixed points of , 
but it can be decomposed and rewritten further, thereby making it more explicit.
The orthogonal spectrum  is a -indexed wedge of homogeneous
summands ^[k]. The space (_G^⊥), and hence also
its G-fixed points, is the disjoint union indexed by the dimension of
the subspaces, i.e.,

    (_G^⊥)^G  =  ∐_j≥ 0 (G r_j(_G^⊥))^G 
     =  ∐_j≥ 0  G r_j^G,⊥ .

The two decompositions induce a direct sum decomposition of the right
hand side of the isomorphism (<ref>) as

    _*((_G^⊥)^G_+)    =  ⊕_k∈⊕_j≥ 0 ^[k]_*( (G r_j^G,⊥)_+)
       ⊕_k∈, j≥ 0 _*-k( (G r_j^G,⊥)_+) .

The second isomorphism uses the periodicity of 
to identify ^[k] with S^k.

The condition (U⊕ (V^⊥-W))=(V)
is equivalent to (U)=(V^G)+(W).
So the homeomorphism (<ref>) identifies (V)^G
with the wedge of the spaces ^[j](V^G) G r_j(V^⊥) for j≥ 0. 
The composite of the isomorphism (<ref>)
and the isomorphism (<ref>)
thus takes the wedge summand of Φ^G_*()
corresponding to =^[0] to the sum of the terms with k=j.
So the isomorphisms restrict to an isomorphism

    Φ^G_*()  ⊕_j≥ 0 _*-j( (G r_j^G,⊥)_+) .

The space G r_j^G,⊥ can be decomposed further:
every G-invariant subspace of _G^⊥ is the direct sum of
its isotypical components, indexed by the non-trivial irreducible G-representations. 
The irreducibles come in three flavors (real, complex or quaternionic),
and so the space G r_j^G,⊥ is a disjoint union
of products of classifying spaces of the groups O(n), U(n), and S p(n)
for various n.

The reader may want to compare the previous description of the geometric
G-fixed points of  with Proposition <ref> (i),
which identifies the G-fixed points of the orthogonal space  as

    (_G)^G ≃ ∐_j≥ 0    B O× G r_j^G,⊥ .

This illustrates the general phenomenon that `geometric G-fixed points of
a global Thom spectrum are the Thom spectrum over the G-fixed points'.






The Thom spectrum  has an oriented analog.
For m≥ 0 we define an orthogonal spectrum _(m) by

    _(m) =  ^m F_S O(m),ν_m ,

the m-th shift of the semifree orthogonal spectrum generated
by the S O(m)-representation ν_m.
In much the same way in Construction <ref>, 
_(m)(V) `is' (by passage to orthogonal complements)
the Thom space over the tautological
bundle over the oriented Grassmannian G r^+_|V|(V⊕^m)
of oriented |V|-planes in V⊕^m. 

We define a morphism

    i  :  F_S O(m),ν_m →  F_S O(m+1),ν_m+1

at an inner product space V as the closed embedding

    i(V)  : (ν_m,V)/ S O(m)
        - ⊕ (ν_m+1,V⊕)/S O(m+1)  
    
    [x,φ]        ⟼     [(x,0),φ⊕] .

Then we set

    j^m =  ^m i  : _(m) = ^m F_S O(m),ν_m → ^m+1 F_S O(m+1),ν_m+1 =  _(m+1)

and we define  as the colimit of the sequence of closed embeddings

    _(0)  j^0   _(1)  j^1  … → _(m)  j^m  … .

The orthogonal spectrum  supports 
inverse Thom classessubjectinverse Thom classsubjectThom class!inverse|seeinverse Thom class
for oriented representations,
i.e., oriented inner product spaces V 
equipped with an orientation preserving isometric action of a compact Lie group G.





 
Now we mention the unitary analogs  and  
of the Thom spectra  and .
Beside the complexification, there is an extra twist to the
unitary definitions, because we need to `loop by imaginary spheres'
to really get orthogonal spectra.
The unitary Thom spectra have periodic versions  respectively , 
but we will not go into any details about those.symbol - periodic unitary global Thom spectrum


We define an orthogonal spectrum  in analogy
with .symbol - unitary global Thom spectrum
Non-equivariantly,  is the unitary Thom spectrum M U. 
The spectrum  is essentially a Thom spectrum over the orthogonal space ,
the complex analog of the E_∞-orthogonal monoid space 
discussed in Example <ref>.
As before we denote by V_=_ V the complexification 
of a real inner product space V. 
The value of  at V is

    (V) =   Gr_|V|^(V_⊕^∞)  ,

the Grassmannian of -linear subspaces of V_⊕^∞
of the same dimension as V.
The map (φ):(V)→(W) induced by a 
linear isometric embedding φ:V→ W is defined as

    (φ)(L)  =  
    (φ_⊕^∞)(L)  +  ((W_-φ_(V_))⊕ 0)  .

In other words: we apply the linear isometric 
embedding φ_⊕^∞:V_⊕^∞→ W_⊕^∞ 
to the subspace L and add the orthogonal complement of the image of φ_
(sitting in the first summand of W_⊕^∞).

We let i V denote the imaginary subspace of V_, i.e., the
-subspace consisting of the elements i v for v∈ V.
Over the space (V) sits a tautological hermitian vector bundle 
with total space consisting of the pairs 
(x,U)∈ (V_⊕^∞)×(V)  such that x∈ U.
We define (V) as 

    (V)  =  _*(S^i V, T h( (V))) ,

the space of based maps from the imaginary sphere S^i V
to the Thom space of this tautological vector bundle.
The structure map of the spectrum  starts from the 
(O(V)× O(W))-equivariant map

    S^V_ T h(( W))    →   T h( (V⊕ W)) 
    
     v   (x, U)        ⟼  
    ( (v,x), U⊕ (0⊕ W_⊕ 0)) .

The structure map 

    σ_V,W :  S^V(W)  → (V⊕ W)

is then adjoint to the composite

    S^V_*(S^i W, T h((W)))    S^i(V⊕ W)
         S^V__*(S^i W, T h( (W))) S^i W
    S^V_    S^V_ T h(( W)) → T h( (V⊕ W))  .

Looping by S^i V is essential for obtaining an orthogonal spectrum;
without it, we would end up with a structure one may call a `unitary spectrum'.


Most of our results about  have analogues for .
The orthogonal spectrum  comes with an E_∞-structure,
which is, however,  not ultra-commuta­tive.
There are unitary versions of the shifted inverse Thom classes 
subjectshifted inverse Thom class!in 

    τ̅^U_G,V ∈ _2 n^G(S^V)  ,

in the G-equivariant -homology groups of S^V, defined
for unitary representations V of a compact Lie group G.
Here n is the complex dimension of V, so that 2 n is its real dimension.
The orthogonal spectrum  is the union of a
sequence of orthogonal subspectra

    _(0) ⊂ _(1) ⊂ … ⊂ _(m) ⊂ …

and  is also the global homotopy colimit of this sequence.
The unit morphism is a global equivalence ≃_(0)
and _(m) is globally equivalent to the 2 m-th 
suspension of the semifree orthogonal spectrum generated by
the tautological unitary representation ν_m^U of U(m) on ^m:

    _(m) ≃ F_U(m),ν_m^U S^2 m

There are distinguished triangles
in the global stable homotopy category:subjectglobal classifying space!of U(n)  

    Σ^∞_+ B_ U(m+1)  S^2 m + 1 → _(m) → _(m+1) → Σ^∞_+ B_ U(m+1)  S^2 m +2

and the first map is classified by the U(m+1)-equivariant homotopy class

    _U(m)^U(m+1)(τ_U(m),ν_m^U S^1)
         in  π_2 m +1^U(m+1)(_(m)).

This uses that the tangent U(m)-representation 
of U(m+1)/U(m) is isomorphic to ν_m^U⊕.
So loosely speaking, _(m+1) is obtained from _(m)
by coning off this transfer class.
A consequence is then that all _(m) and  are globally connective.
Since the first unitary relation _e^U(1)(1 S^1)=0
in π_1^U(1)() lives in a positive dimension, 
the description of the global functor _0() 
is easier than the corresponding
calculation of _0() in Theorem <ref>.
Indeed, the action of the Burnside ring global functor on the element
1∈π_0^e() induces an isomorphism of global functors

    _0()  .

Moreover, there is an exact sequence of global functors

    (U(1),-) → _1() → _1() → 0  ,

where the first morphism is the action on the class
_e^U(1)(1 S^1) in π_1^U(1)().

Corollary <ref> has a unitary analog that
describes morphisms in the global stable homotopy category from :
for every orthogonal spectrum E the sequence 

    0  → lim_m^1   E_2 m+1^U(m)(S^ν_m^U)
     → , E → lim_m    E_2 m^U(m)(S^ν_m^U) → 0

is short exact.
The right map is evaluation at the unitary shifted inverse Thom classes,
and the inverse and derived limits are formed along the maps

    E_2 m+2^U(m+1)(S^ν_m+1^U) ^U(m+1)_U(m) E_2 m+2^U(m)(S^ν_m^U S^2) (- S^2)^-1 E_2 m^U(m)(S^ν_m^U)  .

We leave it to the interested reader to formulate the analogous properties
of the rank filtrations for the special unitary and symplectic
global Thom spectra 𝐦𝐒𝐔 and 𝐦𝐒𝐩.








subjectunitary global bordism
We define another unitary global Thom spectrum ,symbol - unitary global Thom spectrum 
an ultra-commutative ring spectrum and the unitary analog of .
Non-equi­variantly,  is another version of the complex bordism spectrum MU;
for a compact Lie group G, the orthogonal G-spectrum _G 
is a model for tom Dieck's homotopical equivariant bordism <cit.>.
Closely related, strictly commutative ring spectrum models
for these homotopy types have been discussed in various places, 
for example <cit.>, <cit.>,
<cit.> or <cit.>.

For an inner product space V we consider the complex Grassmannian 

    (V) =   Gr_|V|^(V_^2)  .

Over the space (V) sits a tautological hermitian vector bundle and we set

    (V) =  _*(S^i V, T h((V)))  ,

the i V-loop space of the Thom space of this tautological vector bundle.
The structure maps are defined in a similar way as for ,
and a commutative multiplication is given by

    μ_V,W :  (V) (W)  → (V⊕ W)  ,   
    f g  ⟼ f· g  .

Here f:S^i V→ T h((V)), g:S^i W→ T h((W)),
and f· g denotes the composite

    S^i V S^i W f g    T h((V)) T h((W))
       (x,U) (x',U')↦ (κ_V,W(x,x'),κ_V,W(U⊕ U')) 
    T h((V⊕ W))  ,

where κ_V,W is the preferred isometry from
V_^2⊕ W_^2 (V⊕ W)_^2 sending (v,v',w,w') to (v,w,v',w').
Unit maps are defined by

    S^V → (V)  ,     v ⟼ 
    [ v'↦ ( (v+v',0),   V_⊕ 0 ) ] .

These multiplication maps unital, associative and commutative,
and make  into an ultra-commutative ring spectrum.


The global Thom spectrum  comes with distinguished Thom classes
symbolσ^U_G,V - unitary Thom class in _V^G(S^2 n) 

    σ^U_G,V ∈ ^2 n_G(S^V)  ,

for unitary representations V of compact Lie groups G,
where n is the complex dimension of V.
As in the orthogonal situation in Theorem <ref>
the Thom class σ_G,V^U
is inverse to the image of the inverse Thom class τ_G,V^U,
and σ_G,V^U restricts to a unitary Euler class
in π_-2 n^G().subjectEuler class!in 
The proof of Corollary <ref> generalizes to the
unitary situation and proves that the G-equivariant homology
represented by  is the localization 
at the unitary inverse Thom classes of the theory represented by .

Some general facts about the equivariant homotopy groups of 
are known for abelian compact Lie groups A.
In that case, π_*^A() is a free module on even dimensional generators
over the non-equivariant homotopy ring π_*^e();
this calculation was announced by Löffler in <cit.>,
and a proof by Comezaña can be found in <cit.>.
Since the graded ring π_*^e() is concentrated in even degrees
and the periodic version  is a wedge of even suspensions of ,
the groups π_*^A() are concentrated in even degrees.
Since these groups are also 2-periodic, for abelian compact Lie groups
all the information is concentrated in the ring π_0^A(). 
The non-equivariant homotopy groups π_0^e()
are a polynomial ring in countably many generators.
For cyclic groups of prime order, Kriz <cit.>
has described π_0^C_p() as a pullback of two explicit
ring homomorphisms. 
For the cyclic group of order 2, Strickland <cit.> 
has turned this into an explicit
presentation of π_0^C_2() as an algebra over π_0^e(). 

subjectglobal Thom spectrum|)




§ EQUIVARIANT BORDISM


subjectequivariant bordism|(

In this section we recall equivariant bordism groups and their
relationship to the equivariant homology groups defined by the global Thom spectrum 
introduced in Example <ref>. 
The main result is Theorem <ref> 
which says that when G is isomorphic to a product of
a finite group and a torus, then the Thom-Pontryagin map
is an isomorphism from G-equivariant bordism to 
G-equivariant -homology.
Theorem <ref> is usually credited to Wasserman
because it can be derived from his equivariant transversality
theorem <cit.>; as far as I know, the only place
where the translation is spelled out
in detail is the unpublished part of Costenoble's thesis,
see <cit.>. 
Wasserman's theorem is based on equivariant differential topology;
for finite groups, tom Dieck <cit.> gives
a different proof of Theorem <ref>
based on the geometric and homotopy theoretic isotropy separation sequences.
We generalize tom Dieck's proof, translated into our present language,
with an emphasis on global aspects.

In Theorem <ref> we also present a localized version
of this result: the Thom-Pontryagin map
is an isomorphism from stable equivariant bordism to 
[1/τ]-theory, without any restriction on the compact Lie group.
Given that [1/τ]-theory is isomorphic to -theory
(by Corollary <ref>),
this is equivalent to a result of Bröcker and Hook <cit.>
that identifies stable equivariant bordism with equivariant -homology.






The serious study of equivariant bordism groups was initiated
by the work <cit.> of Conner and Floyd.
We recall equivariant bordism as a homology theory for G-spaces X,
where G is a compact Lie group. 
A singular G-manifoldsubjectsingular G-manifold
over X is a pair (M, h) consisting of a closed smooth G-manifold M
and a continuous G-map h: M→ X.
Two singular G-manifolds (M, h) and (M', h') are bordant
if there is a triple (B, H,ψ) consisting of a compact smooth G-manifold B,
a continuous G-map H:B→ X and an equivariant diffeomorphism

    ψ :  M ∪ M'   ∂ B

such that (H∘ψ)|_M=h  and to (H∘ψ)|_M'=h'.

Bordism of singular G-manifolds over X is an equivalence relation.
Reflexivity and symmetry are straightforward; transitivity is established by 
gluing two bordisms along a common piece of the boundary.
To get a smooth structure on the glued bordism that is compatible 
with the G-action one needs smooth equivariant collars; 
the existence of such collars is guaranteed 
by <cit.>.


We denote by 𝒩_n^G(X)symbol𝒩_n^G(X) - equivariant geometric bordism
the set of bordism classes of n-dimensional singular G-manifolds over X.
This set becomes an abelian group under disjoint union. Every element x of
𝒩_n^G(X) satisfies 2 x=0.
The groups 𝒩_n^G(X) are covariantly functorial
in continuous G-maps, by postcomposition.






  Let G be a compact Lie group. 
  
  
  * Let φ,φ':X→ Y be equivariantly homotopic continuous G-maps.
    Then φ_*=φ'_* as homomorphisms
    from 𝒩_n^G(X) to 𝒩_n^G(Y).
  
  * For every G-weak equivalence
    φ:X→ Y the induced homomorphism φ_*:
    𝒩_n^G(X)→𝒩_n^G(Y) is an isomorphism.
  
  * Let {X_i}_i∈ I be a family of G-spaces. 
    Then the canonical map
    
    ⊕_i∈ I 𝒩_n^G(X_i)  → 𝒩_n^G(∐_i∈ I  X_i)

    is an isomorphism.



(i) We let H:X× [0,1]→ Y be an equivariant homotopy from φ
to φ' and (M,h) a singular G-manifold over X.
Then (M×[0,1], H∘ (h×[0,1]),ψ) is a bordism 
from (M,φ h) to (M,φ' h), 
where ψ:M∪ M→∂(M× [0,1])
identifies one copy of M with M×{0}
and the other copy with M×{1}.
So φ_*[M,h]=[M,φ∘ h]=[M,φ'∘ h]=φ'_*[M,h].

(ii) For surjectivity of φ_* we consider any singular G-manifold
(M,g) over Y. Illman's theorem <cit.>
shows that M admits the structure of a G-CW-complex.
Since φ is a G-weak equivalence there exists
a continuous G-map h:M→ X such that φ h 
is equivariantly homotopic to g. 
Part (i) then shows that φ_*[M,h]=(φ h)_*[M,_M]=g_*[M,_M]=[M,g].

The argument for injectivity is similar. 
We consider a singular G-manifold (M,h) over X
that represents an element in the kernel of φ_*. 
There is then a null-bordism (B,H,ψ) of (M,φ h).
By Illman's theorem <cit.> and the discussion immediately
preceding it there is a G-CW-structure on B for which
the boundary is an equivariant subcomplex. So the map 
ψ:M→ B that identifies M with the boundary of B is a G-cofibration.
Since φ is a G-weak equivalence, there exists
a continuous G-map H':B→ X such that H'∘ψ =h.
The triple (B,H',ψ) thus witnesses that [M,h]=0.
Since φ_* is a group homomorphism, it is injective. 

Property (iii) holds because compact manifolds only have finitely 
many connected components, so all
continuous reference maps from singular manifolds or bordisms have image
in a finite union. 



Now we state the key exactness property of equivariant bordism 
in the form of a Mayer-Vietoris sequence. 
The definition of the boundary map needs the existence of G-invariant
separating functions as provided by the following lemma.

 
Let M be a compact smooth G-manifold, C and C'
two disjoint, closed, G-invariant subsets of M, and

    s  :  ∂ M  →

a smooth G-invariant map such that

    C∩∂ M ⊆ s^-1(0)      and    
     C'∩∂ M ⊆ s^-1(1)  .

Then there exists a smooth G-invariant extension 
r:M→ of s such that C⊆ r^-1(0) and
C'⊆ r^-1(1).


Since M is compact, hence normal, the Tietze extension theorem provides
a continuous map r_0:M→ that extends s and satisfies
C⊆ r_0^-1(0) and C'⊆ r_0^-1(1).
We average r_0 to make it G-invariant, i.e., we define r_1:M→ by 

    r_1(x) =  ∫_G r_0(g· x)   d g .

The integral is taken with respect to the normalized invariant measure
(Haar measure) on G. The new map r_1 is again continuous,
compare <cit.>.
Since r_0 is already G-invariant on C∪ C'∪∂ M,
the new map r_1 coincides with r_0 on this subset.
In particular, r_1 is smooth on C∪ C'∪∂ M.
By <cit.> we can then find 
a smooth G-invariant map r:M→ that coincides with r_1 
on C∪ C'∪∂ M; this is the desired separating function.


[Boundary map in equivariant bordism]
We define a boundary homomorphism for a Mayer-Vietoris sequence.
We let X be a G-space and A,B⊂ X open G-invariant subsets
with X=A∪ B. Then a homomorphism

    ∂ :  𝒩_n^G(X) → 𝒩_n-1^G(A∩ B)

is defined as follows.


We let (M,h) be a singular G-manifold that represents
a class in 𝒩_n^G(X). The sets h^-1(X-A) and h^-1(X-B) are 
G-invariant, disjoint closed subsets of M;
we let r:M→ be a G-invariant smooth separating function
as provided by Lemma <ref>, i.e., 
such that h^-1(X-A)⊆ r^-1(0) and
h^-1(X-B)⊆ r^-1(1).
We let t∈ (0,1) be any regular value of r. Then

    M_t  =   r^-1(t)

is a smooth closed G-submanifold of M of dimension n-1 (possibly empty),
and h_t=h|_M_t lands in A∩ B;
so (M_t, h_t) is a singular G-manifold over A∩ B.
  





In the situation above, the bordism class [M_t,h_t] 
is independent of the choice of regular value t,
of the choice of separating function and of the representative
for the given class in 𝒩_n^G(X).
The resulting map

    ∂ :  𝒩_n^G(X) → 𝒩_n-1^G(A∩ B)   ,   
    [M,h] ⟼ [M_t, h_t]

is a group homomorphism.  


We let t<t' be two regular values in (0,1) of the separating function r.
Then

    (r^-1[t,t'],  h|_r^-1[t,t'], incl )

is a bordism from (r^-1(t),h|_r^-1(t)) to (r^-1(t'),h|_r^-1(t')),
so the bordism class does not depend on the regular value.

Now we let (M,h) and (N,g) be two singular G-manifolds over X
in the same bordism class, and we let (B,H,ψ) be a G-bordism
from (M,h) to (N,g).
We let r:M→ and r̅:N→ be two G-invariant separating
functions. Lemma <ref> lets us extend this data 
to a smooth G-invariant separating function

    Ψ :  B  →

such that Ψ∘ψ|_M=r, Ψ∘ψ|_N=r̅,

    H^-1(X-A)⊆Ψ^-1(0)    and  
     H^-1(X-B)⊆Ψ^-1(1)  .

We choose a simultaneous regular value t∈(0,1) for Ψ, r and r̅. Then 

    (Ψ^-1(t),  H|_Ψ^-1(t), ψ|_r^-1(t)∪r̅^-1(t))

is a bordism from 
(r^-1(t),h|_r^-1(t)) to (r̅^-1(t),g|_r̅^-1(t)).
This shows at the same time that the bordism class is independent of
the choice of separating function and of the choice of representing
singular G-manifold.
Additivity of the resulting boundary map is then clear: a separating function
for a disjoint union can be taken as the union of separating functions
for each summand.



Now we formulate the property that makes equivariant bordism 
a homology theory for G-spaces.

 
  Let G be a compact Lie group, X a G-space 
  and A,B⊂ X open G-invariant subsets
  with X=A∪ B. Let i^A:A∩ B→ A, i^B:A∩ B→ B,
  j^A:A→ X and j^B:B→ X denote the inclusions.
  Then the following sequence of abelian groups is exact:
  
    …→𝒩_n^G(A∩ B)  (i^A_*,i^B_*) 𝒩_n^G(A) ⊕𝒩_n^G(B)  ()0ptj^A_*-j^B_* 𝒩_n^G(X)   ∂  𝒩_n-1^G(A∩ B) →…




While Proposition <ref> has been frequently used,
there does not seem to be any published account with a complete proof
in the generality of compact Lie groups;
the argument is somewhat involved, but it proceeds along the lines of
the non-equivariant argument as given for example in
<cit.>.
As input one needs that certain basic tools from differential topology 
generalize to the G-equivariant context, such as for example the
existence of equivariant collars and bicollars, and that 
non-equivariant smoothing of corners is compatible with G-actions.
We honor the tradition in this area and refrain from giving further details.



We define the reduced bordism groupsubjectequivariant bordism!reduced
of a based G-space X as

    𝒩_n^G(X)  =  ( 𝒩_n^G(∗)  →𝒩_n^G(X) ) ,

the cokernel of the homomorphism induced by the basepoint inclusion.
If (M, h) is a singular G-manifold over X, 
then we use the notation
M,h for the class it represents in the reduced bordism group
𝒩^G_n(X).
The unique map u:X→∗ is retraction to the basepoint inclusion, so the map

    (proj, u_*) :  𝒩_n^G(X)  → 𝒩_n^G(X)⊕𝒩_n^G(∗)

is an isomorphism. 
On the other hand, if we add a disjoint G-fixed
basepoint to an unbased G-space Y, then the composite 

    𝒩_n^G(Y)  incl_* 𝒩_n^G(Y_+)  proj 𝒩_n^G(Y_+)

is an isomorphism.


We consider a continuous G-map f:X→ Y and let

    C f =  C X ∪_f Y  =  ( X×[0,1]∪_f Y ) / X×{0}

denote its unreduced mapping cone. The two open sets

    A  =  X× [0,1) / X×{0}     and     
    B  =  X× (0,1]∪_f Y

are G-invariant and together cover the mapping cone.
The intersection A∩ B is homeomorphic to X×(0,1),
so the open covering has an associated boundary homomorphism

    ∂ :  𝒩_n^G(C f)  → 𝒩_n-1^G(X× (0,1))

as in Construction <ref>.
We take the cone point as the basepoint of C f; this is contained
in the subset A, so the map ι:𝒩_*(∗)→𝒩_*(C f)
induced by the basepoint inclusion factors through 
j^A_*:𝒩_n^G(A)→𝒩_n^G(C f),
and the composite ∂∘ι is trivial by exactness
of the excision sequence. The boundary map thus factors over 
the reduced bordism group. We define a `reduced boundary map' ∂̅
as the compositesubjectconnecting homomorphism!in equivariant bordism

    𝒩_n^G(C f)  ∂  𝒩_n-1^G(X×(0,1))  proj_* 𝒩_n-1^G(X)  .





 
  Let G be a compact Lie group and f:X→ Y a continuous G-map. 
  Then the following sequence of abelian groups is exact:
  
    … → 𝒩_n^G(X)   f_*  𝒩_n^G(Y)    i_*  𝒩_n^G(C f)   ∂̅  𝒩_n-1^G(X) → …



  We use the open covering of the mapping cone C f 
  as in the definition of the boundary map.
  In the diagram
  
    @C=15mm 
        X  [r]^-f [d]_x↦ (x,1/2)    Y [r]^-i  [d]_-i      C f @=[d]  
    
        A∩ B  [r]_-incl    B [r]_-incl    C f

  the right square commutes and the left square commutes up to equivariant homotopy.
  Moreover, all vertical maps are equivariant homotopy equivalences, so
  they induce isomorphisms in equivariant bordism,
  by Proposition <ref>.
  So the resulting diagram of bordism groups commutes:
  
    𝒩_n^G(X) [r]^-f_*[d]_   𝒩_n^G(Y) [r]^-i_*[d]_^-i_*   𝒩_n^G(C f) @=[d]
    𝒩_n^G(A∩ B) [r]    𝒩_n^G(B) [r]    𝒩_n^G(C f)

  Moreover, all vertical maps in this diagram are isomorphisms,
  so we can substitute 𝒩_n^G(X) and 𝒩_n^G(Y)
  into the long exact excision sequence 
  of Proposition <ref>.
  Since A is equivariantly contractible to the cone point,
  we can also replace the corresponding summand by the coefficient group,
  and the result is an exact sequence
  
    …→𝒩_n^G(X)   (f_*,u_*)  𝒩_n^G(Y) ⊕𝒩_n^G(∗)  → 𝒩_n^G(C f)   ∂  𝒩_n-1^G(X) →…

  The sequence then remains exact if we divide out the summand
  𝒩_n^G(∗) and replace the absolute bordism group of C f
  by the reduced group 𝒩_n^G(C f).




If f:A→ B is an h-cofibration of G-spaces, then 
the projection q:C f→ B/A
from the mapping cone to the quotient is a based equivariant homotopy equivalence.
So we can substitute 𝒩_*^G(B/A)
into the long exact mapping cone sequence of Proposition <ref> 
and obtain a long exact sequence of abelian groups:
    
    … → 𝒩_n^G(A)   f_*  𝒩_n^G(B)    q_*  𝒩_n^G(B/A)  → 𝒩_n-1^G(A) → …
  



The next proposition says, loosely speaking, that 
in a reduced bordism group the part of
a singular G-manifold that sits over the basepoint can be ignored. 

 
Let G be a compact Lie group
and h:N→ X a singular G-manifold over a based G-space X. 
Let V be a G-representation and M a closed smooth G-manifold
such that (M)+(V)=(N).
Let j:M× D(V)→ N be a smooth G-equivariant embedding.
Suppose that h sends N-j(M×D(V)) to the basepoint.
Then 

    N,h =  M× S(⊕ V), f   in  𝒩_n^G(X),

where f:M× S(⊕ V)→ X is defined by

    f(m, (x,v)) = 
    h(j(m,v))     if x≤ 0, and
      ∗    if x≥ 0.



We define a G-space by

    B  =  ( N×[-1,0]   ∪  N× [0,1] ) / ∼ ,

where the equivalence relation identifies the two copies
of (n,0) for every n∈ N- j(M×D(V)).
The group G acts by g·[n,t]=[g n,t].
The space B is a topological (n+1)-manifold whose boundary
consists of three disjoint parts that we now parametrize.
There are two obvious embeddings

    ψ ,  ψ'  :  N  → B      by    ψ(n)=[n,-1]      and    ψ'(n)=[n,1]

as the two endpoints in the direction of the internal [-1,1].
We define another embedding 

    i  :  M× S(⊕ V)    → B      by
    
     i(m,(x,v))    =     [(j(m,v), 0]^left    for x≤ 0, and
        [(j(m,v), 0]^right    for x≥ 0.

Here the superscripts `left' and `right' indicate whether we refer to the point
[n,0] as the image of (n,0) in N× [-1,0] or in N× [0,1].
The manifold boundary of B is then the disjoint union of the images
of ψ, ψ' and i.

The topological manifold B admits a smooth structure for which the given G-action
is smooth and such that the embeddings ψ, ψ' and i are smooth;
the construction involves `smoothing of corners' 
(also called `straightening of angles') near the image of j(M× S(V))× 0
and is explained for example
in Construction 15.10.3 of tom Dieck's textbook <cit.>.
Tom Dieck has no group actions around, 
but we also have to ensure that the given G-action on B
is smooth. This can be arranged by insisting that the collars used 
in <cit.>
are G-equivariant collars, which is possible for example
by <cit.>.

Now we define a continuous G-map
H:B → X by H(n,t)=h(n) on N×[-1,0]
and as the constant map to the base point of X on N×[0,1].
Then

    H∘ψ =  h      and     H∘ i  =  f  ,

so the bordism (B,H,ψ+ ψ'+ i) witnesses the relation

    [N,h]  =  [N,H∘ψ']  +   [M× S(⊕ V),f]

in the unreduced bordism group 𝒩_n^G(X). 
Since H∘ψ' is constant to the basepoint of X,
the class [N,H∘ψ'] vanishes in the reduced bordism group 
𝒩_n^G(X); this proves the claim. 






The equivariant bordism groups come with natural products, given by the biadditive maps

    × :  𝒩_m^G(X) × 𝒩_n^G(Y)  → 𝒩_m+n^G(X× Y) ,   
    [M,h]× [N,g] =  [M× N, h× g] .

These products are suitably associative, commutative and unital.
The product pairing descends to a pairing on reduced bordism groups
if the G-spaces X and Y are based.
Indeed, the composite

    𝒩_m^G(X) 𝒩_n^G(Y)      ×  𝒩_m+n^G(X× Y)   q_*  𝒩_m+n^G(X Y)  proj𝒩_m+n^G(X Y)

annihilates the image of 𝒩_m^G(∗)𝒩_n^G(Y)
and the image of 𝒩_m^G(X)𝒩_n^G(∗),
where q:X× Y→ X Y is the quotient map;
so the composite factors uniquely over a homomorphism

    :  𝒩_m^G(X) ⊗ 𝒩_n^G(Y) 
     → 𝒩_m+n^G(X Y) .



We will frequently use certain distinguished bordism classes
associated to representations.
We let V be an m-dimensional representation of a compact Lie group G.
Stereographic projection is a G-equivariant 
homeomorphismsubjectstereographic projection

    Π_V :  S(⊕ V)     S^V 
     ,    (x,v)  ⟼ v/1-x

from the unit sphere of ⊕ V to the one-point compactification S^V.
We define a reduced G-bordism class over S^V bysymbold_G,V - equivariant bordism class of a representation

    d_G,V =   S(⊕ V),Π_V   ∈ 𝒩_m^G(S^V) .

The following multiplicativity property is expected, but not entirely obvious.



Let V and W be two representations of a compact Lie group G.
Then the relation

    d_G,V d_G,W = d_G,V⊕ W

holds in 𝒩_m+n^G(S^V⊕ W).

 
We define a `distorted' version 
τ_V : S(⊕ V) → S^V
of the stereographic projection as the composite

    S(⊕ V)   Π_V  S^V   J   S^V ,

where the second map J is given by

    J(v) =  v/1-|v|   for |v| < 1, and
     ∞   for |v|≥ 1.

The map J is equivariantly based homotopic to the identity of S^V,
so the pair (S(⊕ V),τ_V) 
is another representative for the bordism class d_G,V.
The map

    j :  V⊕ W     → S(⊕ V)× S(⊕ W)
    
    j(v,w)    =  ( |v|^2-1/|v|^2+1, 2 v/|v|^2+1,
      |w|^2-1/|w|^2+1,2 w/|w|^2+1)

is a smooth G-equivariant embedding.
The map j is the product of the inverse stereographic projections 
Π_V^-1×Π_W^-1:S^V× S^W→ S(⊕ V)× S(⊕ W),
restricted to V⊕ W.

If a quadruple (x,v,y,w) is in the image j(D(V⊕ W)) of the unit disc, 
then in particular x≤ 0 and y≤ 0.
Equivalently, the points (x,v,y,w) of S(⊕ V)× S(⊕ W)
that are in the complement of j(D(V⊕ W)) have x > 0 or y > 0.
So the map

    q∘(τ_V×τ_W) : S(⊕ V)× S(⊕ W) → S^V⊕ W

sends the complement of j(D(V⊕ W)) to the basepoint at infinity,
where q:S^V× S^W→ S^V⊕ W is the projection.
Proposition <ref> thus shows that

    q_*(d_G,V    d_G,W)  =  
    q_*(S(⊕ V),τ_VS(⊕ W),τ_W) 
       =   S(⊕ V)× S(⊕ W), q∘(τ_V×τ_W) =   S(⊕ V⊕ W), f  ,

where
f:S(⊕ V⊕ W)→ S^V⊕ W is defined by

    f(x,v,w) = 
    q(τ_V×τ_W(j(v,w)))     if x≤ 0, and
        ∞    if x≥ 0.

We claim that for all (x,v,w)∈ S(⊕ V⊕ W) the relation

    f(x,v,w)  Π_V⊕ W(-x, - v, -w)

holds in S^V⊕ W.
Assuming this claim, we can finish the proof as follows:
since the G-map Π_V⊕ W^-1∘ f
never takes a point to its antipode,
the linear homotopy between  Π_V⊕ W^-1∘ f and the
identity in the ambient vector space ⊕ V⊕ W can be
normalized to land in the unit sphere. 
This yields an equivariant based homotopy between Π_V⊕ W^-1∘ f 
and the identity of S(⊕ V⊕ W).
Hence f is equivariantly based homotopic 
to the stereographic projection Π_V⊕ W,
and so (S(⊕ V⊕ W),f) represents the bordism class d_G,V⊕ W.

It remains to prove the claim.
The only point of S(⊕ V⊕ W) 
that Π_V⊕ W sends to the point at infinity is (1,0,0).
Since f(-1,0,0)=(0,0), the claim is true for all
(x,v,w) such that f(x,v,w)=∞.
It remains to consider those tuples for which f(x,v,w)∞,
which means that  x < 0 and 
|v|< 1 and |w| < 1.
On such points the map f is given by

    f(x,v,w)     =  q ( τ_V(|v|^2-1/|v|^2+1,2 v/|v|^2+1), τ_W(|w|^2-1/|w|^2+1,2 w/|w|^2+1))
       =  q(J(v),J(w)) ,

whereas

    Π_V⊕ W(-x,-v,-w)  =  ( -v/1+x,  -w/1+x)  .

If these two expressions were the same, then

    v/1-|v| =  -v/1+x     and    w/1-|w| =  -w/1+x .

For v 0 this implies

    |v|-1 =   1+x  ,

which is impossible since |v|<1 and x≥ -1.
If w 0 we obtain the same kind of contradiction.
The final case is (x,v,w)=(-1,0,0), in which case

    f(1-,0,0)  =  (0,0)   ∞ =  Π_V⊕ W(1,0,0) .




We shall now recall that if G acts trivially on V, 
then exterior multiplication by d_G,V is an isomorphism,
the suspension isomorphism in reduced equivariant bordism.
In general, however, the class d_G,V is not invertible
and exterior multiplication by d_G,V need not be an isomorphism.
The theory obtained by formally inverting all
the classes d_G,V is stable equivariant bordism,
to which we return in Remark <ref> below.
The projection p:C f→ X S^1 from the reduced mapping to the suspension
was defined in (<ref>).


If G acts trivially on V and X is a cofibrant based G-space,
then the  exterior product map

    -  d_G,V :  𝒩_n^G(X)  → 𝒩_n+|V|^G(X S^V)

is an isomorphism.  
For every continuous G-map f:X→ Y between based G-spaces
the connecting homomorphism in the mapping cone sequence equals the composite

    𝒩_n^G(C f)  p_*  𝒩_n^G(X S^1) (- d_G,)^-1𝒩_n-1^G(X) .



We start with the special case V=.
We apply Proposition <ref>
to the map f:X→∗ to a one-point G-space.
The cone of this map is

    X^♢ =  X× [0,1] / ∼ ,

the unreduced suspension of X, where X×{0} and X×{1} 
are collapsed to one point each.
Since X has a G-fixed point, 
the map f_*:𝒩_*^G(X)→𝒩_*^G(∗)
is a split epimorphism. So the long exact sequence provided by 
Proposition <ref> reduces to a short exact sequence:

    0   → 𝒩_n+1^G(X^♢)   ∂̅  𝒩_n^G(X)  f_*  𝒩_n^G(∗)   → 0

Since X is cofibrant in the based sense, the projection 

    ψ :  X^♢ → X S^1  ,   
    q[x,s] =  x2s-1/s(1-s)

that collapses {x_0}× [0,1] is an equivariant homotopy equivalence.

Then  the composite

    𝒩_n+1^G(X S^1)  [] ψ_*^-1  𝒩_n+1^G(X^♢)   ∂̅  𝒩_n^G(X)  proj 𝒩_n^G(X)

is an isomorphism.
We claim that the relation

    proj(∂̅(ψ_*^-1(x d_G,)))  =  x

holds for all classes x∈𝒩_n^G(X).
Since proj∘∂̅∘ψ_*^-1 
is an isomorphism, so is smash product with the class d_G,.

This relation, in turn, is a consequence of the geometric origin
of the class d_G,, the product in bordism  and the boundary map.
In more detail, we suppose that x=M,h for a singular
G-manifold (M,h) over X.
We define a continuous map H:M× S(⊕)→ X^♢ by

    H(m,(x,y))  =  [h(m), (y+1)/2]     for x ≤ 0, and 
       [x_0, (y+1)/2]     for x ≥ 0.

Then the following square commutes up to G-equivariant homotopy:

    M× S(⊕) [r]^-H [d]_h×Π_   
    X^♢[d]^ψ
    
    X× S^1 [r]_-q     X S^1

Hence

    ψ_* M× S(⊕), H  =   M× S(⊕), q∘(h×Π_)  =  M,h d_G, ,

and thus

    ∂̅(ψ_*^-1(M,h d_G,))  = ∂̅ M× S(⊕), H  .

To calculate this geometric boundary we use the smooth separating function

    r :  M× S(⊕)  → [0,1]  ,   
    r(m,(x,y)) =(y+1)/2 .

Then 1/2 is a regular value of this separating function, and 
the preimage over this regular value is 
r^-1(1/2)=M×{(1,0), (-1,0)}, two disjoint copies of M.
The function H takes the copy M×{(1,0)} to the basepoint
of X, so this copy does not contribute to the reduced bordism group.
The restriction of H to the other copy  M×{(-1,0)} 
is the original map h, so we obtain

    ∂̅[M× S(⊕), H]   =  M×{(1,0),(-1,0)}, H|_M×{(1,0),(-1,0)} ≡ M,h

in the reduced bordism group of X.

The general case now follows easily. Since the claim is true for V=,
it also holds for V=^n by the associativity of the smash product pairing
and the classes d_G,, compare Proposition <ref>.
If G acts trivially on V, then it is equivariantly isomorphic to ^n
for some n.



The bordism theories 𝒩_*^G for different compact Lie groups
are related by geometrically defined restriction and induction maps.
Every continuous group homomorphism α:K→ G is automatically smooth
(see for example <cit.>),
and thus induces a restriction homomorphism

    α^*  :  𝒩_n^G(X) → 𝒩_n^K(α^*(X))  ,   
     [M,h] ⟼ [α^*(M),α^*(h)]

by restricting all actions along α.
Restriction maps preserve the distinguished classes (<ref>)
in the sense that α^*(d_G,V)=d_K,α^*(V).
The product in equivariant bordism is compatible with restriction maps in the sense that

    α^*(x× y)  =  α^*(x)×α^*(y) .



For every closed subgroup H of G and every closed smooth H-manifold M
of dimension n, the induced space G×_H M
has a preferred smooth structure making it 
a smooth closed G-manifold of dimension d+n
with d=(G/H)=(G)-(H).
Indeed, since the diagonal H-action on G× M is smooth and free,
there is a unique smooth structure on G×_H M
such that the projection G× M→ G×_H M
is a submersion, see for example <cit.>,
and the G-action is smooth with respect to this smooth structure.
We can also apply G×_H- to bordisms, so this
gives a well-defined induction homomorphism

    G×_H-  :  𝒩_n^H(Y)  → 𝒩_d+n^G(G×_H Y) ,    
    [M,h] ⟼ [G×_H M, G×_H h] .

For Y=∗ we can compose the induction map with the effect of
the projection G×_H∗→∗ and arrive at an induction homomorphism
on coefficient groups

    _H^G  :  𝒩_n^H(∗)  → 𝒩_d+n^G(∗) ,   
    [M] ⟼ [G×_H M] .

The induction map _H^G is compatible with inflations,
in the sense of the formula

    α^*∘_H^G  =  _L^K ∘ (α|_L)^* :  𝒩_n^H(∗)  → 𝒩_d+n^K(∗)

for a continuous epimorphism α:K→ G, where L=α^-1(H).
If H has finite index in G, then the induction _H^G preserves
the dimension, and then it satisfies the double coset formula.
So for fixed n≥ 0 the coefficient groups 𝒩_n^G(∗)
almost form a global functor; the only missing structure are the
transfer maps for closed inclusions that are not of finite index.

Multiplication, restriction and induction satisfy a reciprocity relation.
We let H be a closed subgroup of G, 
X an H-space and Y a G-space.
Then the following diagram commutes:

    @C=13mm@R=6mm𝒩_m^H(X)×𝒩_n^G(Y)
    [d]_×^G_H[r]^-(G×_H-)×   𝒩_d+m^G(G×_H X)×𝒩_n^G(Y)
    [d]^×
    𝒩_m^H(X)×𝒩_n^H(^G_H(Y)) [d]_×   𝒩_d+m+n^G((G×_H X) × Y) [d]^χ_*
    𝒩_m+n^H(X×^G_H(Y)) [r]_-G×_H-   𝒩_d+m+n^G(G⋉_H( X×^G_H(Y)))

Here 

    χ :  
    (G×_H X) × Y  → 
    G⋉_H( X×^G_H(Y) )  ,    
    ([g,x],y) ⟼ [g, (x, g^-1 y)]

is the G-equivariant shearing isomorphism.
The proof of the commutativity is straightforward from the definitions,
using the shearing diffeomorphism for equivariant manifolds.
If we specialize to X=Y=∗ and postcompose with the projection
to the one-point G-space, we obtain the reciprocity formula 

    _H^G(x×^G_H(y))  =  _H^G(x)× y

for classes x∈𝒩_m^H(∗) and y∈𝒩_n^G(∗)
in the coefficient groups.



The next proposition shows that the distinguished bordism classes d_G,V 
measure the failure of the Wirthmüller isomorphism in equivariant bordism,
see Remark <ref> below.
We consider a closed subgroup H of a compact Lie group G 
and continue to write

    L =  T_e H(G/H)

for the tangent H-representation.
For an H-space Y the H-equivariant continuous  map

    l_Y  :  G×_H Y  → Y_+ S^L

was defined in Construction <ref>.



 
For every closed subgroup H of a compact Lie group G and every
H-space Y, the composite

    𝒩_n^H(Y)   G×_H- 𝒩_n+d^G(G×_H Y)   ^G_H 𝒩_n+d^H(G×_H Y)   (l_Y)_* 𝒩_n+d^H(Y_+ S^L)

is exterior multiplication by the class d_H,L∈𝒩_d^H(S^L),
where d=(G/H).


We let (M,h) be a singular H-manifold that
represents a class in 𝒩_n^H(Y).
The G×_H h:G×_H M→ G×_H Y represents the class G×_H[M,h].
As in the construction of the collapse map l_Y we choose 
a slice around 1∈ G orthogonal to H, i.e., a smooth embedding
s: D(L) →  G satisfying s(0)=1 and

    s(h· l)  =  h · s(l) · h^-1

for all (h,l)∈ H× D(L) and such that the differential at 0 of the composite

    D(L)  G   G/H

is the identity of L.
The slice property implies that the map 

    s̅ :  D(L)× H  → G  ,    (l,h) ⟼ s(l)· h

is a tubular neighborhood of H inside G. Moreover, this 
embedding is equivariant for the action of H^2, acting on source
and target by

    (h_1,h_2)· (l,h) =  (h_1 l,  h_1 h h_2^-1) 
       respectively   
    (h_1,h_2)· g =   h_1 g h_2^-1 .

The map 

    l_H^G  :  G_+  → S^L H_+

was defined as the H^2-equivariant collapse map 
with respect to the tubular neighborhood s̅.
So explicitly,

    l_H^G(g)  =  
     ( l / (1-|l|) ) h     if g=s(l)· h with (l,h)∈ D(L)× H, and 
      ∞    if g is not in the image of s̅.

We obtain a smooth H-equivariant embedding

    j : M× D(L)  → G×_H M      by    
    j(m,l)  =  [ s(l), m]  ,

where H acts diagonally on the left.
The composite 

    G×_H M   G×_H h  G×_H Y  l_H^G_H Y_+   
    (S^L  H_+)_H Y_+    Y_+ S^L

sends the complement of j(M× D(L)) to the basepoint at infinity.
Proposition <ref> (for (H,L) instead of (G,V))
shows that then

    ((l_Y)_*∘^G_H∘ (G×_H-))[M,h]     =  G×_H M, l_Y∘ (G×_H h)
       =  M× S(⊕ L), f

in the reduced bordism group of Y_+ S^L,
where f:M× S(⊕ L)→ Y_+ S^L is defined by

    f(m,(x,l)) = 
    ( l_Y∘(G×_H h))(j(m,l))     if x≤ 0, and
        ∞    if x≥ 0.

So f equals the composite

    M× S(⊕ L) h×Ψ  Y× S^L  
    Y_+ S^L

where 

    Ψ(x,l) =  l/(1-|l|)

for x≤ 0, and Ψ(x,l)=∞ for x≥ 0.
The map Ψ:S(⊕ L)→ S^L is homotopic, in the equivariant based sense,
to the stereographic projection Π_L.
We can thus conclude that

    ((l_Y)_*∘^G_H∘ (G×_H-))   [M,h]  =  M× S(⊕ L), f
       =   [M,h]S(⊕ L),Ψ =  [M,h] d_H,L .





[Failure of the Wirthmüller isomorphism
in equivariant bordism]
subjectWirthmüller isomorphism!in equivariant bordism
subjectWirthmüller map!in equivariant bordism
As we now explain, there is no general Wirthmüller isomorphism
in equivariant bordism, i.e., the Wirthmüller map 

    _H^G =  (l_Y)_*∘^G_H  :  𝒩_*^G(G×_H Y) → 𝒩_*^H(Y_+ S^L)

is not in general bijective for closed subgroups H of G. 
Here Y is an H-space and l_Y is the H-equivariant collapse map
defined in Construction <ref>.
This immediately shows that there cannot be a natural isomorphism,
compatible with restriction to subgroups,
between equivariant bordism and the equivariant homology theory represented by
any orthogonal G-spectrum. Indeed, Proposition <ref>
says that the composite

    _H^G∘(G×_H -) :  𝒩_n^H(Y)  → 𝒩_n+d^H(Y_+ S^L)

is exterior multiplication by the class d_H,L∈𝒩_d^H(S^L).
Since the induction map G×_H - is an isomorphism,
the Wirthmüller map is bijective if any only if the class d_H,L
acts invertibly on 𝒩_*^H(Y).
In this sense, the distinguished bordism class d_H,L 
precisely measures the failure of the Wirthmüller isomorphism in equivariant bordism.
One can show that the class d_H,L is not 
invertible as soon as H acts non-trivially on L,
i.e., in this situation the class d_H,L 
does not generate 𝒩_*^H(S^L)
as an 𝒩_*^H-module.

If G is a product of a finite group and a torus, then H acts
trivially on L for every closed subgroup H of G, and hence
the Wirthmüller isomorphism does hold for such compact Lie groups G.
And in fact, for this class of groups, the Thom-Pontryagin construction
provides an natural isomorphism between equivariant bordism and
the equivariant homology theory represented by the Thom spectrum ,
compare Theorem <ref>  below.




Our argument to compare the geometric bordism theory with the equivariant homotopy
groups of the Thom spectrum  is based on the isotropy separation sequence.
We will now identify the `geometric fixed point term'
in the isotropy separation sequence for equivariant bordism.


As before we let E be a universal space for the family of proper closed subgroups
of G. So E is a cofibrant G-space without G-fixed points, 
and (E)^H is contractible for every proper closed subgroup of G.subjectuniversal space!for the family of proper subgroups
We let Ẽ denote the unreduced suspension of E.
Then (Ẽ)^G=S^0 consists of the two cone points,
and (Ẽ)^H is contractible for every proper subgroup of G.

We recall the `geometric fixed point' homomorphism,subjectgeometric fixed point homomorphism!in equivariant bordism
see (<ref>) below, that identifies the
reduced bordism group 𝒩^G_n(Ẽ)
in terms of non-equivariant bordism groups.
Loosely speaking, the class Φ_geomM,h:M→Ẽ remembers the
bordism class of the part of the fixed point manifold
that lies over the fixed point 0,
together with the normal data of the embedding into M.
As before, we use the abbreviation

    G r_j^G,⊥ =  ( G r_j(_G^⊥) )^G

for the space of j-dimensional G-invariant subspaces of _G^⊥=_G-(_G)^G.
We first define a fixed point map 

    Φ̅ :  𝒩_n^G(Ẽ) → ⊕_j≥ 0 𝒩_n-j (G r_j^G,⊥ )

into the unreduced, non-equivariant bordism group as follows. 
We let (M,h:M→Ẽ) be an
n-dimensional singular G-manifold.
Since G acts smoothly, M^G is a disjoint union of smooth submanifolds
of varying dimensions, compare <cit.>.
Since Ẽ has exactly two fixed points, and both are isolated,
h must map every path component of the fixed point manifold M^G
to either 0 or ∞. We denote by

    M_0^G  =  M^G ∩ h^-1(0)

the union of those components of M^G that lie over the point 0.
We let M^(j) be the union of all (n-j)-dimensional components of M^G_0.

The Mostow-Palais embedding theorem <cit.>
provides a smooth G-equivariant embedding i: M→ V, 
for some G-representation V; we can assume that V is in fact
a subrepresentation of the complete universe _G. 
Then for every fixed point x∈ M^(j), 

    (d i)(T_x (M^(j)))  =   ( (d i)(T_x M) )^G  ,

i.e., the tangent space inside M^(j) `is' the G-fixed part
of the tangent space in M. 
So we can define a continuous map

    ν_j  :  M^(j) → ( G r_j(V^⊥))^G incl 
     G r_j^G,⊥

by sending a fixed point x∈ M^(j) to the orthogonal complement
of (d i)(T_x (M^(j))) inside (d i)(T_x M).
By its very construction, the map ν_j classifies 
the normal bundle of the inclusion M^(j)→ M.
The geometric fixed point map is then given by

    Φ̅[M,h]  =  ∑_j=0^n  [M^(j), ν_j] .

Since the image of the map i_*:𝒩_n^G(∗)→𝒩_n^G(Ẽ)
is concentrated over the basepoint ∞, it is annihilated by
the map Φ̅. So Φ̅ factors
over the reduced bordism group of Ẽ as a homomorphism

    Φ_geom :   𝒩_n^G(Ẽ) → ⊕_j≥ 0 𝒩_n-j (G r_j^G,⊥ ) .

The fact that the geometric fixed point map (<ref>)
is an isomorphism is ubiquitous in calculations of equivariant bordism groups,
and it goes back to Conner and Floyd <cit.>.
In the classical literature, the reduced equivariant bordism groups 
of Ẽ usually appear in a different guise, namely as
the groups 𝒩_*^G[,] of bordism classes of
smooth compact G-manifolds with boundary, but where there
are no fixed points on the boundary, 
compare Remark <ref> below.





For every compact Lie group G and every n≥ 0
the geometric fixed point map (<ref>)
is an isomorphism.


The proof is by explicit geometric constructions.
We start with surjectivity.
We consider a non-equivariant closed smooth (n-j)-manifold N
and a continuous map f:N→ G r_j^G,⊥.
The space G r_j^G,⊥ is the filtered colimit of its
closed subspaces (G r_j(V))^G for V∈ s(_G^⊥).
Since N is compact, the image of f lands in (G r_j(V))^G 
for some finite-dimensional G-representation V with V^G=0.
The space (G r_j(V))^G is a smooth manifold, 
and by smooth approximation 
we can assume without loss of generality that f is a smooth map.
We define a closed smooth n-dimensional manifold by

    M =  { (n,x,v)∈ N × S(⊕ V)  |  
    v∈ f(n) } .

Another way to say this is that M is a double of
the unit disc bundle of the pullback 
of the tautological j-plane bundle along f:N→ (G r_j(V))^G. 
The group G acts smoothly on M by
g· (n,x,v) = (n, x, g v).

At this point it will be convenient to use a specific model 
for the space Ẽ, namely
the unit sphere in ⊕_G^⊥, 
compare Example <ref>.
We can then define a continuous G-map

    h  :  M   → Ẽ = S(⊕_G^⊥)      by     
     h(n,x,v) =  (x,v)  .

So the pair (M,h) represents a bordism class 

    [M,h]   ∈ 𝒩_n^G(Ẽ) .

Since V^G=0, the G-fixed points of M
are a disjoint union of two copies of N embedded as
N× (-1,0) respectively N× (1,0), 
and h maps one copy to each of the two fixed points of Ẽ.
The normal bundle of the copy of N over the non-base point
is the bundle classified by the original map f, so we obtain

    Φ_geom[ M,h] =  [N,f] .

This shows that every class in 𝒩_n-j(G r_j^G,⊥)
is in the image of the geometric fixed point map, and so
the map (<ref>) is surjective.

Now we consider a reduced equivariant bordism class M,h over Ẽ
in the kernel of the fixed point map (<ref>).
Being in the kernel of Φ_geom means that 
for every 0≤ j≤ n there is a non-equivariant null-bordism B_j with
∂ B_j= M^(j) and a continuous map F_j:B→ G r_j^G,⊥
whose restriction to M^(j) classifies the normal bundle
of M^(j) inside M.
As in the first part of the proof we can compress F_j
to a G-map B→ (G r_j(V))^G for some finite-dimensional G-subrepresentation
V of _G^⊥ and replace this factorization by a homotopic smooth
map F_j:B→ (G r_j(V))^G.
We use this data to `cut out' the fixed points M_0^G from M by replacing a
tubular neighborhood by the sphere bundles of the maps F_j; this produces
a new singular G-manifold over Ẽ, bordant to (M,h),
that has no more fixed points over 0. 

The construction is done separately
and disjointly over each of the components M^(j) of the fixed points M^G_0.
To simplify the exposition we restrict to the special case where 
the fixed points M^G_0 are of constant dimension n-j
(i.e., all other components M^(i) for i j are empty).
Then we proceed as follows.
We let ν denote the normal bundle of M^(j) inside M.
The equivariant tubular neighborhood theorem provides
a smooth G-equivariant embedding

    ψ :  D(ν)  → M

of the unit disc bundle of ν whose composite with 
the zero section s:M^(j)→ D(ν) is the inclusion,
see for example <cit.>.
In particular, the image of ψ is a closed G-invariant 
tubular neighborhood of M^(j).
By shrinking the neighborhood, if necessary, we can make its image
disjoint from all other components of M^G except M^(j).
The bundle arising by pulling back the tautological bundle 
along F:B→ (G r_j(V))^G has its own disc bundle with total space

    D(F)  =  { (b,v)∈ B× V  |  v∈ F(b),  |v|≤ 1 } .

The sphere bundle S(F) is then a smooth compact G-manifold with boundary

    ∂(S(F)) =  S(F|_∂ B)  =  S(ν) .

Now we form the G-manifold

    M̅ =  ( M- ψ(D(ν))∪_S(ν) S(F) ,

where the gluing uses the restriction ψ: S(ν) → M
of the tubular neighborhood to the sphere bundle.
A smooth structure on M̅ is provided by choices of G-equivariant
collars of ψ(S(ν)) inside M- ψ(D(ν))
and of S(ν) inside S(F).

The boundary of the disc bundle of F decomposes as

    ∂(D(F))  =  S(F) ∪_S(F|_∂ B) D(F|_∂ B) 
     =  S(F) ∪_S(ν) D(ν)  .

By equivariant smoothing of corners the disc bundle D(F)
can be given a smooth structure such that the given G-action is smooth
and that the embeddings of D(ν) and S(F) into D(F) are smooth.
We define a bordism as the G-space

    W =  ( M× [0,1])∪_D(ν) D(F)

where the gluing is along (ψ,1):D(ν)→ M× [0,1].
The space W is a topological (n+1)-manifold whose boundary
is the union of two disjoint parts that we now parametrize.
An obvious embedding is given by

    ψ :  M  → W  ,  ψ(m)=[m,0]  .

A second embedding 

    i  :  M̅ =  ( M-ψ(D(ν)))∪_S(ν) S(F)  → W

identifies M-ψ(D(ν))
with M-ψ(D(ν))× 1
and includes the sphere bundle S(F) into the disc bundle D(F).
The boundary ∂ W is then the disjoint union of the images of ψ and i.

The topological manifold W admits a smooth structure for which the given G-action
is smooth and such that the embeddings ψ and i are smooth;
in the non-equivariant version, this is explained
in Construction 15.10.3 of <cit.>.
To ensure that the given G-action on W
is smooth, we must insist that the collars involved in the construction
are G-equivariant collars, which is possible for example
by <cit.>.

Now we have to arrange the equivariant reference maps to Ẽ.
In fact, this extra data goes along for the ride, as the only homotopical information
it encodes is a decomposition of the G-fixed points into
two disjoint subspaces, the preimages of the two fixed points of Ẽ. 
In more detail, we let W be a compact smooth G-manifold, possibly with boundary.
Then W admits the structure of a finite G-CW-complex
by Illman's triangulation theorem <cit.>. 
By the proof of Proposition <ref>, 
the fixed point map

    (-)^G  :  ^G(W,Ẽ) → (W^G,{0,∞})

is a weak equivalence and Serre fibration.
A continuous map to the discrete space {0,∞}
is equivalent to a decomposition into two disjoint open subsets.
So altogether we conclude that for every disjoint union decomposition W^G=A∪ B
there is a G-map h:W→Ẽ with h(A)={0}
and h(B)={∞}, and any two such maps are equivariantly homotopic.
We use this property three times, namely for the G-manifolds W,
M and M̅.

The G-fixed points of W are a disjoint union of
M_∞^G× [0,1] and (M^(j)× [0,1])∪_M^(j)× 1 B.
So there is a G-map H: W→Ẽ
such that H( M_∞^G× [0,1])={∞}
and H(( M^(j)× [0,1])∪_M^(j)× 1 B )={0}.
The triple (W,H,ψ+i) is then a bordism that witnesses the relation
M,H ψ=M̅,H i in the group
𝒩_n^G(Ẽ).
The map H i:M̅→Ẽ sends all of M̅^G to the fixed point ∞,
so H i is equivariantly homotopic to the constant map 
with value ∞. By homotopy invariance, 
the class M̅,H i thus vanishes in the reduced bordism group.
On the other hand, the G-map Hψ agrees with the original map h
on the fixed points M^G. So Hψ is equivariantly homotopic to h,
and homotopy invariance yields

    M,h =  M,H ψ =  M̅,H|_M̅ =  0

in the group 𝒩_n^G(Ẽ).
This shows that the map Φ_geom is injective.




In the classical literature, the reduced equivariant bordism groups 
of Ẽ usually appear in an isomorphic form, namely as
the groups 𝒩_*^G[,] of bordism classes of
smooth compact G-manifolds with boundary, but where there
are no fixed points on the boundary. A homomorphism

    𝒩_*^G[,] → 𝒩_n^G(Ẽ)

is defined as follows. We let M be a smooth compact G-manifold without
G-fixed points on the boundary.
The double of M is the smooth closed G-manifold

    D M  =  M∪_∂ M M

obtained by gluing two copies of M along their boundary.
A smooth structure in the neighborhood of the gluing locus
is provided by a choice of G-equivariant collar
(see <cit.>).
By Illman's theorem <cit.>,
D M admits the structure of a finite G-CW-complex.
Since the original manifold M had no G-fixed points on the boundary,
(D M)^G is the disjoint union of two copies of M^G, one from each of 
the two copies of M in the double.
There is thus a continuous G-map
h:D M→Ẽ, unique up to equivariant homotopy,
that takes the `left' copy of M^G to the fixed point 0
and the `right' copy of M^G to the fixed point ∞.
The pair (D M,h) is then a singular G-manifold over Ẽ,
and it represents a bordism class 

    [D M,h]   ∈ 𝒩_n^G(Ẽ) .

The fact that the map (<ref>) is an isomorphism
is Satz 3 in <cit.>.

For finite groups, Stong shows in <cit.>
that the geometric fixed point map

    𝒩_*^G[,] → ⊕_j≥ 0 𝒩_n-j (G r_j^G,⊥ )

is an isomorphism.
Combined with tom Dieck's isomorphism (<ref>)
this gives a different proof of Proposition <ref>
in the special case of finite groups.
I was unable to find a convenient reference for 
Proposition <ref> in the 
generality of compact Lie groups, which is the main reason for including a proof.



[Bordism of manifolds with involution]
We look at the geometric isotropy separation sequence 
in the simplest non-trivial case of the two-element group G=C_2=C,
the case originally considered by  Conner and Floyd <cit.>.
In this case  consists only of the trivial subgroup,
and so 𝒩_*^C(E)=𝒩_*^C(E C)
is the bordism ring of manifolds with free C-action.
If C acts freely and smoothly on M, then we can form the smooth manifold

    [-1,1] ×_C M  =  ([-1,1] × M ) / (x,m)∼ (-x,τ m)

with C-action by τ· [x, m] = [-x,τ m];
the boundary of this manifold is equivariantly diffeomorphic to the original
manifold M. So every C-manifold with free action is null-bordant,
and thus the forgetful map 𝒩_*^C(E)→𝒩_*^C
is zero. 
The long exact mapping cone sequence of the C-map E C→∗
(see Proposition <ref>) thus 
decomposes into short exact sequences.
For every compact Lie group G, the map

    𝒩_n^G(E G)  → 𝒩_n( B G)  ,  
    [M,h] ⟼ [G M, G h ]

is an isomorphism from the bordism group of G-manifolds with free
action to the non-equivariant bordism group of the classifying space B G=G E G.
So in the case at hand, 𝒩_*^C(E)=𝒩_*^C(E C)
is isomorphic to 𝒩_*( P^∞).

We use Proposition <ref>
to replace the group 𝒩_*^C(Ẽ C)
by the direct sum of non-equivariant bordism groups.
Since there is only one non-trivial irreducible
C-representation, the 1-dimensional sign representation,
every linear subspace of _C^⊥ is C-invariant.
Hence G r_j^C,⊥ is just a Grassmannian of j-planes
in an infinite dimensional -vector space,
hence a classifying space of the orthogonal group O(j).
Altogether, this yields a short exact sequence

    0 → 𝒩_*^C   Φ_geom  ⊕_j≥ 0 𝒩_*-j ( G r_j(^∞) )   J  𝒩_*-1( P^∞)  → 0  .

The map J is defined as follows. Given an (n-j)-dimensional 
singular manifold (F,η:F→ G r_j(^∞)),
we factor the continuous map η through a finite-dimensional Grassmannian
F→ G r_j(^m), choose a homotopic smooth approximation
η':F→ G r_j(^m) and let P(η') denote the 
projectivized bundle of the pullback of the tautological j-plane bundle along η'. 
The total space of P(η') is then a smooth closed (n-1)-manifold
equipped with a map to P^∞ that classifies 
the tautological line bundle over P(η'); this data represents the class J[F,η].

One can deduce from the above short exact sequence that 
𝒩_*^C is free as a module over the non-equivariant 
bordism ring 𝒩_*.
This is in fact true much more generally: Stong shows 
in <cit.>
that 𝒩_*^G(X) is free as an 𝒩_*-module
for every finite group G and every G-space X.
The paper <cit.> by Alexander exhibits an explicit 
𝒩_*-basis of 𝒩_*^C.
Some basic C-bordism classes are represented by the projective spaces
P^n equipped with the involution

    τ· [x_0:x_1:…:x_n]  =  [-x_0:x_1:…:x_n]  .

We denote by y_n = [ P^n,τ]
the bordism class of this C-manifold in 𝒩_n^C.
An 𝒩_*-linear map

    Γ :  𝒩_*^C  → 𝒩_*+1^C

of degree 1 is given by sending the class of
a manifold M with involution τ:M→ M to the manifold

    S()×_C M  =  (S()× M ) / (z,m)∼ (-z,τ m)  .

So S()×_C M is diffeomorphic to the
mapping torus of the involution τ. The involution on this manifold is by

    τ :  S()×_C M  → S()×_C M  ,  τ· [z,m]  =  [z̅, τ m] .

In our present notation, the operator can also be expressed as

    Γ =  ^O(2)_C ∘_O(1)× O(1)^O(2)∘ p^* ,

where we embed C into O(2) as a reflection, and
where p:O(1)× O(1)→ C is the epimorphism with kernel e× O(1).

Alexander shows in <cit.> that the multiplicative unit 1
together with the classes

    Γ^n(y_i_1·…· y_i_r)

for all n≥ 0, r≥ 1 and i_j≥ 2 form a basis of 𝒩_*^C
as a module over 𝒩_*.
Besides the trivial group and C, 
the equivariant bordism groups have been calculated for various
finite abelian groups, see <cit.>. 


subjectglobal Thom spectrum|(

Now we work our way towards the equivariant Thom-Pontryagin construction
that assigns to every equivariant bordism class over a G-space X
an equivariant homology class in _*^G(X_+).
We break the construction up into two steps, 
and we first discuss the normal class, 
a basic invariant associated to a closed  smooth G-manifold.

subjectnormal class!of a G-manifold|(
[Normal class of a G-manifold]
In Example <ref> we defined the ultra-commutative 
ring spectrum  that consists of the Thom spaces of the tautological
vector bundles over Grassmannians.
To every smooth closed G-manifold M we associate
a normal classsubjectnormal class!of a G-manifold

    M ∈ _0^G(M_+) .

This class records the equivariant homotopical information in the
stable normal bundle of M, and
it is the geometric input for the Thom-Pontryagin map
to equivariant -homology.
If M has dimension m, then the class lives in the homogeneous summand
^[-m] of .

The construction starts from the 
Mostow-Palais embedding theorem <cit.>
that provides a smooth G-equivariant embedding i: M→ V, 
for some G-representation V. 
We can assume without loss of generality that V is a subrepresentation
of the chosen complete G-universe _G.
We use the inner product on V to define the normal bundle ν 
of the embedding at x∈ M by

    ν_x  =  V - (d i)(T_x M) ,

the orthogonal complement of the image of the tangent space T_x M in V.
By multiplying with a suitably large scalar, if necessary, 
we can assume that the embedding is widesubjectwide embedding
in the sense that the exponential map

    D(ν)  → V  ,    (x,v) ⟼ i(x)   +   v

is injective on the unit disc bundle of the normal bundle, 
and hence a closed G-equivariant embedding.
The image of this map is a tubular neighborhood of radius 1 around i(M),
and it determines a G-equivariant Thom-Pontryagin collapse map

    c_M :  S^V  →  T h(G r(V) ) M_+  =  (V) M_+

as follows: every point outside of the tubular neighborhood is sent
to the basepoint, and a point i(x)+ v, for (x,v)∈ D(ν),
is sent to 

    c_M(i(x) +  v)  =  ( v/1-|v|, ν_x  )   x  .

The normal class M is the homotopy class of the collapse map c_M.



 The normal class of a smooth closed G-manifold
is independent of the choice of wide embedding into a G-representation.


If we enlarge the embedding i:M→ V by postcomposition
with a direct summand embedding (0,-):V→ U⊕ V,
then the collapse map associated to the composite embedding (0,-)∘ i
is equivariantly homotopic to the composite

    S^U⊕ V S^U c_M  S^U(V)  M_+
     σ_U,V (U⊕ V) M_+  .

So the resulting class in _0^G(M_+) does not change.
Two classes based on two different wide embeddings i:M→ V and j:M→ W
can be compared by passing to V⊕ W; in this larger representation,
the map

    M× [0,1] → V⊕ W  ,    
    (m,t)  ⟼ ( t· i(m), (1-t)· j(m))

is a smooth isotopy through wide embeddings. This isotopy induces a
homotopy between the two collapse maps and shows that altogether
the normal class M is independent of the wide embedding.




Part (ii) of the following proposition refers to an external
multiplication morphism

    μ_A,B : ( A_+) ( B_+)     () (A× B)_+
       μ (A× B)_+ (A× B)_+  ,

where A and B are two G-spaces
and μ:→ is the multiplication morphism.
In part (iv) we consider the k-th power M^k of a G-manifold M
as a (Σ_k≀ G)-manifold via the action

    (σ; g_1,…,g_k) · (x_1,…, x_k)  =  
    (g_σ^-1(1)x_σ^-1(1),…,g_σ^-1(k) x_σ^-1(k))  .

The morphism j=j^_:→ that shows up in part (v) was defined 
in (<ref>).



Let G be a compact Lie group, and let M and N be smooth closed G-manifolds. 
  
  
  * Let i^1:M→ M∪ N and i^2:N→ M∪ N denote the inclusions
    into a disjoint union. Then the relation
    
    M ∪ N =  i^1_*M + i^2_*N

    holds in the group ^G_0((M∪ N)_+).
  
  * The relation
    
    M× N =   (μ_M,N)_*(M×N)

    holds in the group _0^G((M× N)_+).
  
  * For every continuous homomorphism α:K→ G of compact Lie groups
    and every smooth closed G-manifold M the relation
    
    α^* M =  α^*M

    holds in ^K_0(α^*(M)_+).
  
  * For every k≥ 0, the relationsubjectpower operation!for G-manifolds
    
    M^k =  P^kM

    holds in the group _0^Σ_k≀ G(M^k).
  
  * Let B be a compact smooth G-manifold with boundary ∂ B.
    Then the class ∂ B is in the kernel of the homomorphism
    
    (jι_+)_*  :  _0^G(∂ B_+)  → ()_0^G(B_+)   ,

    where ι:∂ B→ B is the inclusion.



(i) 
We let p^1:(M∪ N)_+→ M_+ and p^2:(M∪ N)_+→ N_+ 
denote the two projections.
We choose any wide smooth equivariant embedding i:M∪ N→ V
and observe that the composite

    S^V  c_M∪ N (V) (M∪ N)_+  (V) p^1 (V) M

is on the nose the collapse map for M based on the restriction of
the embedding i to M.
We then obtain

    p^1_* M ∪ N =  M =  p^1_*( i^1_*M  +i^2_*N)  ;

the second relation uses that p^1∘ i^1 is the identity,
p^2∘ i^1 is the trivial map and p^1_* is additive.
The analogous argument shows that 
p^2_* M ∪ N =  p^2_*( i^1_*M  +i^2_*N).
Since equivariant homotopy groups are additive on wedges, the map 

    (p^1_*,p^2_*) :  ^G_0((M∪ N)_+) → ^G_0(M_+)×^G_0( N_+)

is bijective, and this proves the claim.

(ii) We choose wide smooth equivariant embeddings

    i  :  M → V      and     j  :  N  → W

into G-representations.
The product map

    i× j  :  M× N  → V⊕ W

is then another wide smooth equivariant embedding that we use for the
Thom-Pontryagin construction of M× N.
The normal bundle of i× j is the
exterior direct sum of the normal bundles of i and j.
The unit disc D(V⊕ W) of the direct sum is contained in the
product D(V)× D(W) of the unit discs, so the exponential tubular neighborhood
for i× j is contained in the product of the 
exponential tubular neighborhoods for i and j.
The collapse map

    S^V⊕ W  c_M× N  (V⊕ W) (M× N)_+

is equivariantly homotopic to the composite

    S^V S^W c_M c_N    ( (V)  M_+)  ((W)  N_+) 
        μ_M,N^V,W  (V⊕ W) (M× N)_+ .

This shows the desired relation.
Part (iii) is straightforward from the definitions.

(iv) We choose a wide smooth equivariant embedding i:M→ V
into a G-representation. Then

    i^k  :  M^k  → V^k

is a (Σ_k≀ G)-equivariant wide smooth embedding
that we use to calculate the class M^k.
The collapse map 

    c_M^k :  S^V^k →  T h(G r(V^k) ) M^k_+

based on i^k is (Σ_k≀ G)-equivariantly homotopic to the composite

    S^V^k    (c_M)^ k  ((V) M_+)^ k
       shuffle (V)^ k M^k_+  μ_V,…,V^ M^k_+ (V^k) M^k_+  .

This latter composite represents the power operation P^kM,
so altogether this shows the desired relation.

(v) We let C( j B_+) denote the mapping cone of the morphism 
j B_+: B_+→ B_+.
We define a relative normal class

    B^ ∈ π_1^G( C (j B_+))

such that the relation

    ∂( B^ )  =  (ι_+)_* ∂ B

holds in the group π_0^G( B_+), where
∂ is the connecting homomorphism (<ref>) 
in the long exact homotopy group sequence of the mapping cone.
Because two consecutive maps in the long exact homotopy
group sequence compose to zero, we can then conclude that

    (jι_+)_* ∂ B 
    =    (j B_+)_*( ∂ (B^))  =  0 .

It remains to construct the class B^ and establish 
the relation (<ref>).
We choose an equivariant collar, i.e., a smooth G-equivariant embedding

    c :  ∂ B× [0,1)  → B

such that c(-,0):∂ B→ B is the inclusion and the image of c is
an open neighborhood of the boundary inside B.
Then we choose a smooth function 

    κ :  [0,1] → [0,1]

that is the identity on [0,1/3], identically 1 on [2/3,1]
and whose restriction to [0,2/3) is injective.
We define the smooth function

    ψ :  [0,1) → [0,1)      by    ψ(t) =  κ(t)-1/t-1 ;

then ψ(t)=1 for t∈ [0,1/3] and κ(t)=0 for t∈[2/3,1).

The Mostow-Palais embedding theorem
provides a wide smooth G-equivariant embedding j: B→ V, 
for some G-representation V. Then the smooth G-map

    i  :  B  →  V⊕

defined by

    i(b) = 
    
    (ψ(t)·  j(x)+ (1-ψ(t))· j(b), κ(t))    for b=c(x,t) with (x,t)∈∂ B× [0,1),
         (j(b),1)    for b∉c(B× [0,1)),

is a new wide smooth equivariant embedding which satisfies

    i(∂ B) ⊂  V×{0}

and which is `orthogonal to V near the boundary', i.e.,
the set U=c(∂ B× [0,1/3)) is
an open neighborhood of ∂ B in B, and

    i(U)  =   i(∂ B) × [0,1) .

Since the embedding i is wide, the exponential map

    D(ν)  → V⊕ ,    (x,v,t) ⟼ i(x)   +   (v,t)

is injective on the unit disc bundle of the normal bundle ν of j, 
and hence a closed G-equivariant embedding.
We define a continuous map

    κ :   
    D(ν)  → C( j(V)(V)  →(V⊕))  B_+

to the reduced mapping cone of the embedding j(V) of
(V) B_+ into  (V⊕) B_+ as follows.
We consider (b,v,t)∈ D(ν) where b∈ B and (v,t)∈ V⊕ is
normal to i(B) at i(b). 
If b∈ U, then the normal vector must lie in V⊕ 0, i.e., t=0.
The map κ then takes (b,v,0) to

    [ (v/1-|v|, ν_b ) ,  i_2(b)  ]  b

in the cone of (V) B_+,
where i_2(b)∈[0,1) is the second component of i(b).
For b∉U, the map κ sends (b,v,t) to
((v,t)/1-|v,t|, ν_b) b 
in (V⊕) B_+.

The total space of the disc bundle D(ν) is a topological manifold with boundary,
and its boundary is the union of the sphere bundle S(ν) and 
the subspace D(ν)|_∂ B, the part sitting over the boundary of B.
The map κ sends the subspace D(ν)|_∂ B
to the cone point in the mapping cone, and it sends
the sphere bundle S(ν) to the basepoint.
So κ sends the entire boundary of D(ν) to the basepoint
of the reduced mapping cone.
So we can extend κ continuously to S^V⊕
by sending the complement of D(ν) to the basepoint.
The result is a continuous based G-map

    c̃_B  :  S^V⊕ → 
    C( j(V) B_+ : (V) B_+ →  ()(V)  B_+)  .

The map c̃_B represents the relative normal class B^  
in π_1^G( C (j B_+)).

It remains to establish the relation (<ref>).
The composite

    S^V⊕ c̃_B C( j(V)   B_+)  p   (V) B_+   S^1

is equivariantly homotopic to the map (((V)ι_+)∘ c_∂ B) S^1, 
where

    c_∂ B :  S^V → (V)  (∂ B)_+

is the collapse map for ∂ B
based on the restriction of i to an embedding ∂ B→ V.
Thus

    p_*(B^)  =  ι_* ∂ B S^1

in the group π_1^G( B_+ S^1).
The relation (<ref>) thus follows from
the definition of the connecting homomorphism (<ref>)
as the composite of p_* and the inverse suspension isomorphism.



The inverse Thom classsubjectinverse Thom class 

    τ_H,W ∈ _0^H(S^W)

of an H-representation W was defined in (<ref>).
The next theorem shows how the normal class of an
induced equivariant manifold G×_H M
is determined by the normal class of the H-manifold M 
and the inverse Thom class of the tangent H-representation L=T_e H(G/H).
The Wirthmüller isomorphism

    _H^G :  _0^G( (G×_H M)_+)     _0^H(M_+ S^L)

was established in Theorem <ref>.



Let H be a closed subgroup of a compact Lie group G and M a closed smooth
H-manifold. 
Then the relation

    _H^GG×_H M =  Mτ_H,L

holds in the group _0^H( M_+ S^L), where L=T_e H (G/H)
is the tangent H-representation.


The Wirthmüller map is the composite

    _0^G( (G×_H M)_+)     ^G_H_0^H((G×_H M)_+)   ( l_M)_* _0^H(M_+ S^L)  .

Here l_H^G:G→ S^L H_+ is the H^2-equivariant collapse maps
for the embedding of H into G, and l_M:(G×_H M)_+→ M_+ S^L
is the composite

    (G×_H M)_+ l_H^G_H M S^L M_+    M_+ S^L  ,

compare Construction <ref>.
So the claim is equivalent to the relation

    (l_H^G_H M)_*(^G_H G×_H M ) 
     =  τ_H,LM

in the group _0^H(S^L  M_+).

We choose a G-equivariant wide smooth embedding i:G/H→ V
into a G-representation; 
we also choose an H-equivariant wide smooth embedding j:M→ W 
into an H-representation underlying some other G-representation.
Then the map

    ψ :  G×_H M → V⊕ W  ,    
    [g,m] ⟼ (i(g H), g· j(m))

is a G-equivariant wide smooth embedding. We base the collapse map for
the G-manifold G×_H M on the embedding ψ.

The differential at the coset H of the embedding i is a linear embedding

    L  =  T_e H (G/H)   d i  V  ;

we define a scalar product on L so that this embedding becomes isometric.
Now we choose a slice as in the construction of the map l_H^G,
i.e., a smooth embedding

    s :  D( L )  → G

of the unit disc of L with s(0)=1, such that
s(h · l)= h· s(l)· h^-1
for all (h,l)∈ H× D(L), and such that 
the differential at 0 of the composite

    D(L)  G   G/H

is the identity of L.
After scaling the slice, if necessary, the map 

    s̅ :  D(L)× H  → G  ,    (l,h) ⟼ s(l)· h

is a smooth embedding whose image is an H^2-equivariant tubular neighborhood
of H inside G. Here the group H^2 acts on the source  by

    (h_1,h_2)· (l, h) =  (h_1 l,  h_1 h h_2^-1)   .

The map l_H^G: G → S^L H_+ is then the collapse map 
for this tubular neighborhood.

The following H-equivariant composite is thus a representative for the class
(l_H^G_H M)_*(^G_H G×_H M)

    S^V⊕ W  c_G×_H M  (V⊕ W) (G×_H M_+)
     (l_H^G_H M) (V⊕ W) S^L  M_+ .

The map l_H^G takes the complement of the tubular neighborhood s̅(D(L)× H)
to the basepoint. So the map

    l_H^G_H M_+ :  (G×_H M)_+  →  S^L  M_+

takes the complement of the subset s̅(D(L)× H)×_H M
to the basepoint. The composite

    D(L)× M   s̅  G   ψ  V⊕ W

is given by the formula

    (l,m) ⟼ ( i(s(l)· H),  s(l)· j(m)) .

We define a homotopy of smooth wide H-equivariant embeddings

    [0,1]× D(L)× M  → V⊕ W
       by    (t,l,m) ⟼ ( i(s(l)· H),  s(t l)· j(m))  .

For every time t∈[0,1], the corresponding embedding has an associated
collapse map 

    S^V⊕ W → (V⊕ W) S^L M_+  .

So the homotopy induces a homotopy of based H-equivariant maps between
the above representative for (l_H^G_H M)_*(^G_H G×_H M)
and the collapse map for the product embedding

    D(L)× M  → V⊕ W
       by    (l,m) ⟼ ( i(s(l)· H),  j(m))  .





We identify the normal classes of some equivariant manifolds
in terms of classes that were previously defined.
In the case where M=∗ is a single point, Theorem <ref>
specializes to the relation

    _H^GG/H =  τ_H,L

in the group _0^H(S^L), where L=T_e H (G/H) is the tangent H-representation.
In other words, the normal class of the homogeneous G-manifold G/H
is the inverse, under the Wirthmüller isomorphism,
of the inverse Thom class.

We let V be a representation of a compact Lie group G.
Then the unit sphere S(⊕ V) in ⊕ V is a smooth closed
G-manifold. We recall that

    Π_V : S(⊕ V) →  S^V
     ,    (x,v)  ⟼ v/1-x

is the G-equivariant stereographic projection.subjectstereographic projection
We claim that 

    S(⊕ V) =   (Π_V^-1)_*(τ_G,V)  .

Since both sides of equation (<ref>) 
commute with restriction along continuous
homomorphisms, it suffices to show the relation
for the tautological m-dimensional O(m)-representation ν_m.
The composite

    O(m+1)/O(m)  ψ  S(⊕ν_m)  Π_ν_m S^ν_m

is O(m)-equivariantly homotopic to the map l_O(m)^O(m+1):O(m+1)/O(m)→ S^ν_m
that appears in the Wirthmüller isomorphism,
where ψ(A· O(m))=A· (0,…,0,1).
So we can argue:

    (Π_ν_m)_*    S(⊕ν_m)
       =  (Π_ν_m)_*((ψ_*)^O(m+1)_O(m) O(m+1)/O(m)) 
       =  ( l_O(m)^O(m+1))_*(^O(m+1)_O(m)O(m+1)/O(m)) 
       =  _O(m)^O(m+1)O(m+1)/O(m) =  τ_O(m),ν_m  .

Inverting (Π_ν_m)_* gives the desired relation (<ref>) 
in the universal example.

subjectnormal class!of a G-manifold|)





[Equivariant Thom-Pontryagin construction]
subjectThom-Pontryagin construction
The equivariant Thom-Pontryagin construction defines a natural transformation
of G-homology theories

    Θ^G  =  Θ^G(X) :  𝒩_*^G(X)   → _*^G(X)  ,

as we now recall.
We let (M,h) be an m-dimensional singular G-manifold over a based G-space X.
The way we have set things up, all the geometry is already encoded
in the normal class M∈_0^G(M_+);
the rest is a formal procedure: we push the normal class forward 
along the morphism b:→ defined in (<ref>),
and use the periodicity of  to move into the homogeneous
summand  of degree 0.
While the normal class is not yet a bordism invariant, 
pushing it forward to  makes it one, 
see Proposition <ref> below.
The periodicity class t∈π_-1^e(^[-1]) was defined in (<ref>),
and we let σ∈π_1^e(^[1]) be its inverse.
We define

    Θ^G[M,h] =  (b h)_* M· p_G^*(σ^m)  ∈ _m^G(X) ,

i.e., we take the image of the normal class of M under the homomorphism

    (b h)_* :  _0^G(M_+) → _0^G(X)

and multiply by the unit p_G^*(σ^m) in π_m^G(^[m]).
Since M has dimension m, the normal class lies in the 
homogeneous summand ^[-m], whereas σ^m lies in the
summand ^[m]; so the product indeed lies in the
homogeneous degree 0 summand =^[0].



 
The class Θ^G[M,h] in _m^G(X)
  only depends on the bordism class of the singular G-manifold (M,h).


The morphisms j_=j^_:→
and j_=j_^:→ 
were defined in (<ref>) respectively (<ref>).
The square

    @C=15mm[r]^-j_[d]_b    [d]^ b
    [r]_-j_^-≃

commutes and the lower horizontal morphism is a homotopy equivalence of
orthogonal spectra by Proposition <ref> (i).
Now we consider a smooth compact G-manifold B with boundary ∂ B. 
 Proposition <ref> (v) shows that

    (j_ B_+)_*( (bι_+)_*∂ B)     =     ( b B_+)_* ( (j_ι_+)_*∂ B)  = 0 ,

where ι:∂ B→ B is the inclusion.
Since j_ is a homotopy equivalence, this implies 
(bι_+)_*∂ B=0.
We let (M,h:M→ X) be a singular G-manifold that is null-bordant.
We choose a null-bordism (B,H:B→ X,ψ:M∂ B), 
so that H∘ι∘ψ=h. Then

    (b h)_*M    =    (b (H∘ι∘ψ))_*M 
    =  ( H)_*( (bι_+)_*∂ B ) =  0 .

Multiplying by p_G^*(σ^m) gives that Θ^G[M,h]=0.
Since the normal class is additive on disjoint unions 
(Proposition <ref> (i)), naturality
then implies that Θ^G[M,h] only depends on the bordism class of (M,h).




We let G be a compact Lie group and V a G-representation of dimension m.
We claim that then 

    Θ^G(d_G,V)  =  τ̅_G,V

in the group _m^G(S^V).
In other words, the Thom-Pontryagin construction matches the distinguished 
geometric bordism class d_G,V in 𝒩_m^G(S^V)
with the shifted inverse Thom class in the 
Thom spectrum .subjectshifted inverse Thom class!in  

Indeed, the class d_G,V is represented by the singular
G-manifold (S(⊕ V),Π_V), where Π_V:S(⊕ V)→ S^V
is the stereographic projection.subjectstereographic projection
So

    Θ^G(d_G,V)     =  (bΠ_V)_*S(⊕ V)· p_G^*(σ^m)
     
    _(<ref>)   =  (bΠ_V)_*(  (Π_V^-1)_*(τ_G,V) )· p_G^*(σ^m)
       =  (b S^V)_*( τ_G,V) · p_G^*(σ^m)  =  τ̅_G,V .




The next theorem says, roughly speaking, that the Thom-Pontryagin construction
is compatible `with all global structure'.
There is one caveat, though, namely in how the geometric induction 
in equivariant bordism compares with the homotopy theoretic transfer.
Indeed, the geometric induction
increases the dimension by the dimension of G/H,
whereas the Wirthmüller isomorphism
increases the dimension in a twisted way,
namely by the sphere of the tangent representation L=T_e H(G/H) of H in G.
Multiplication by the inverse Thom class is needed to compensate
this `twist' on the homotopy theory side.
In the special case where H has finite index in G, then the
tangent representation is zero, so in this special case τ̅_H,L=1 and part (v)
of the following theorem specializes to the simpler relation
    
    Θ^G( G×_H  y)  =  G⋉_H Θ^H(y) .



 
  
  
  * The Thom-Pontryagin map Θ^G is additive.  
  
  * The Thom-Pontryagin map is multiplicative, i.e., for
    all classes x∈𝒩_m^G(X) and
    y∈𝒩_n^G(Y), the relation
    
    Θ^G(x y) =  Θ^G(x) Θ^G(y)

    holds in _m+n^G(X Y).
  
  * The Thom-Pontryagin map is compatible 
    with the boundary maps in the mapping cone sequences in equivariant bordism
    and equivariant -homology, i.e., Θ^G is a transformation of
    equivariant homology theories.
  
  * For every continuous homomorphism α:K→ G of compact Lie groups,
    every based G-space X and all x∈𝒩_m^G(X)
    the relation
    
    Θ^K(α^*(x))   =  α^*(Θ^G(x))

    holds in _m^K(α^*(X)).
  
  * If H is a closed subgroup of G, then
    for every H-space Y and all y∈𝒩_m^G(Y), 
    the relation
    
    Θ^G(G×_H y)  =  G⋉_H (Θ^H(y)τ̅_H,L )

    holds in _m+d^G( (G×_H Y)_+), where L=T_e H(G/H) is the 
    tangent H-representation and d=(G/H).
  


(i) The two functors

    X ⟼ 𝒩_n^G(X)      and     
    X⟼ _n^G(X)

from the category of based G-spaces to the category of abelian groups
are reduced and additive.
Proposition <ref> thus shows that the
Thom-Pontryagin map is additive.

Part (ii) is a formal consequence of the multiplicativity of the normal classes.
We consider singular G-manifolds
(M,h:M→ X) and (N,g:N→ Y). The class M,hN,g
is then represented by the singular G-manifold
(M× N,q∘ (h× g)),
where q:X× Y→ X Y is the quotient map. 
Then

    (b (q∘(h× g)))_* M× N    =  
    (b (q∘(h× g)))_* ( (μ_M,N)_*(M×N) )
       =  (μ_M,N)_* ((b h)_* M× (b g)_* N)  
       =  (b h)_* M  (b g)_* N

in the group _0^G(X Y).
The first equation is Proposition <ref> (ii);
the second equation exploits that b:→ 
is a homomorphism of E_∞-orthogonal ring spectra.
Now we multiply with the class p_G^*(σ^m+n) and obtain
the desired relation 

    Θ^G M× N,q∘ (h× g)    =  (b (q∘(h× g)))_* M× N· p_G^*(σ^m+n) 
       =   ( (b h)_* M  (b g)_* N ) · p_G^*(σ^m+n) 
       =  ((b h)_* M· p_G^*(σ^m))  ( (b g)_* N· p_G^*(σ^n) ) 
       =  Θ^GM,hΘ^GN,g .

    
(iii) 
We let f:X→ Y be a continuous G-map.
Compatibility of the Thom-Pontryagin construction with the boundary homomorphism 
amounts to the commutativity of the following square:

    @C=18mm𝒩_m+1^G(C f)[r]^-p_*[d]_Θ^G   𝒩_m+1^G(X_+ S^1)[r]^-(- d_G,)^-1_-[d]_Θ^G   𝒩_m^G(X)[d]^Θ^G
    _m+1^G(C f)[r]_-p_*   _m+1^G(X_+ S^1)[r]_-(-τ̅_G,)^-1^-   _m^G(X_+)

Here p:C f→ X_+ S^1 is the projection defined in (<ref>).
Indeed, the upper composite agrees with the boundary map in equivariant bordism by 
Proposition <ref>;
the lower composite is the homotopy theoretic boundary map by the definition in
Construction <ref> and the fact that the 
suspension isomorphism in G-equivariant -homology is
exterior multiplication with the class τ̅_G,∈_1^G(S^1).

The Thom-Pontryagin construction is natural for continuous G-maps,
so it remains to show the commutativity of the right square above.
However, multiplicativity and (<ref>) give

    Θ^G(x d_G,)   =  Θ^G(x)Θ^G(d_G,)
     =  Θ^G(x)τ̅_G,

for all x∈𝒩_m^G(X).

Part (iv) is straightforward from the definitions.  

(v)
This is a formal consequence of the formula for the normal class
of an induced manifold in Theorem <ref>.
We consider a singular H-manifold
(M,h:M→ Y). The class G×_H[M,h]
is then represented by the singular G-manifold
(G×_H M,G×_H h). 
Then
    
    _H^G ( (b (G⋉_H h))_*G×_H M)    =  (b h S^L)_*( _H^G G×_H M) 
       =  (b h S^L)_*( Mτ_H,L) 
       =   (b  h)_*M  (b S^L)_*(τ_H,L) 
       =  (b h)_*Mτ_H,L .

Now we multiply with the class p_G^*(σ^m+n) and obtain
    
    _H^G ( Θ^G(G×_H y))     =  _H^G ( (b (G⋉_H h))_*G×_H M· p_G^*(σ^m+d)) 
       =  _H^G ( (b (G⋉_H h))_*G×_H M) · p_H^*(σ^m+d)
       =  ( (b h)_*Mτ_H,L ) · p_H^*(σ^m+d)
       =  ( (b h)_*M· p_H^*(σ^m)) 
    ( τ_H,L· p_H^*(σ^d))
       =  Θ^H(y)τ̅_H,L .

By Theorem <ref> the Wirthmüller isomorphism is inverse 
to the composite of ε_L:_m+d^H(Y_+ S^L) →_m+d^H(Y_+ S^L)
(the effect of -_L on S^L) and the exterior transfer. 
Proposition <ref> (i) implies that
the inverse Thom class τ_H,L is fixed by the involution ε_L.
So this last relation is equivalent to
the desired relation
Θ^G(G×_H y) = G⋉_H( Θ^H(y)τ̅_H,L).


As before we let E be a universal G-space 
for the family of proper closed subgroups of G,
and Ẽ denotes its unreduced suspension.
Then (Ẽ)^G=S^0, consisting of the two cone points,
and (Ẽ)^H is contractible for every proper subgroup of G.


The Thom-Pontryagin map

    Θ^G  :  𝒩_*^G(Ẽ) → _*^G(Ẽ)

is an isomorphism for every compact Lie group G. 


We claim that the following diagram commutes:

    @C=12mm𝒩_n^G(Ẽ)  [d]_Θ^G[rr]^-Φ_geom_-      ⊕_j≥ 0 𝒩_n-j( G r_j^G,⊥ ) [d]^⊕Θ^e
    _n^G(Ẽ)[r]_-Φ^-   Φ_n^G() [r]^-_-(<ref>)   ⊕_j≥ 0 _n-j( (G r_j^G,⊥ )_+)

Here G r_j^G,⊥=( G r_j(_G^⊥))^G,
and the geometric fixed point homomorphism Φ_geom
was defined in (<ref>).
To see this we let (M,h) be an n-dimensional singular
G-manifold over Ẽ.
We choose a smooth G-equivariant wide embedding i:M→ V into
a G-representation and let

    c_M  :   S^V  → 
    T h(G r_|V|-n(V))  Ẽ_+

be the associated collapse map.
As before we let M^(j) denote the (n-j)-dimensional component of the
fixed point manifold M^G_0 over the point 0, and

    ν_j  :  M^(j) → ( G r_j(V^⊥))^G

the classifying map for the normal bundle of M^(j) inside M,
sending a fixed point x∈ M^(j) to the orthogonal complement
of (d i)(T_x (M^(j))) inside (d i)(T_x M).

The trick is now to base the Thom-Pontryagin construction for M^(j) 
on the non-equivariant embedding

    M^(j) incl M^G_0   i^G   V^G

into the G-fixed points of V.
As explained in Example <ref>,
the G-fixed points of the Thom space that occurs in the target of c_M 
decompose as a wedge. The composite 

    S^V^G (c_M)^G   ( T h(G r_|V|-n(V)) )^G

with the projection to the j-th summand is then on the nose the map 

    S^V^G c_M^(j) (ν_j)_+  
     T h(G r_(V^G)+j(V^G)) ( G r_j(V^⊥))^G_+  ,

the smash product of the collapse map for the non-equivariant manifold M^(j),
based on the embedding i^G, and ν_j.
This shows that Φ(Θ^GM,h) is the 
non-equivariant Thom-Pontryagin construction applied to Φ_geomM,h.

The upper map in the original diagram is an isomorphism
by Proposition <ref>.
The lower left homotopical geometric fixed point map Φ identifies
the equivariant homotopy groups of Ẽ
with the geometric fixed point groups Φ_*^G(),
as shown in Proposition <ref>. 
The  lower right isomorphism is the calculation of these geometric fixed point groups 
in (<ref>).
The right vertical map is a direct sum of non-equivariant Thom-Pontryagin maps,
hence an isomorphism by Thom's theorem <cit.>.
Since the diagram commutes, the left vertical map is also an isomorphism.


Now we come to the main result of this section, showing that the
equivariant homology theory represented by the global Thom spectrum 
is equivariant bordism, at least for products of finite groups and tori.
The result is usually credited to Wasserman,
because it can be derived from his equivariant transversality
theorem <cit.>.

 
Let G be a compact Lie group that is isomorphic to a product of
a finite group and a torus. Then for every cofibrant G-space X,
the Thom-Pontryagin map

    Θ^G(X) :  𝒩_*^G(X)     → _*^G(X_+)
 
is an isomorphism.


We adapt tom Dieck's proof given in <cit.> to our setting.
Tom Dieck only considers finite groups, where homotopy theoretic transfer  
and geometric induction match up under the Thom-Pontryagin construction.
The general case has a new ingredient, namely the observation that for compact Lie groups
of positive dimensions the difference between homotopy theoretic transfer  
and geometric induction is controlled by the inverse Thom class
of the tangent representation,
compare Theorem <ref> and 
Theorem <ref> (v).

We prove the statement by double induction over the dimension and the
number of path components of G. The induction starts with the trivial
group, i.e., the non-equivariant statement, which is Thom's 
celebrated theorem <cit.>.
Now we let G be a non-trivial compact Lie group that is a product of a finite 
group and a torus, and we assume that the theorem has been established 
for all such groups of smaller dimension, 
and for all groups of the same dimension but with fewer path components.

Every cofibrant G-space is equivariantly homotopy equivalent to
a G-CW-complex.
We can thus assume without loss of generality that X is a G-CW-complex.
To show that Θ^G is an isomorphism,
we exploit that 𝒩_*^G and _*^G 
are both equivariant homology theories and Θ^G 
is a morphism of homology theories. This reduces the claim to
the special case X=G/H of an orbit for a closed subgroup H of G.
The argument for an orbit falls into two cases, depending on whether H 
is a proper subgroup or H=G.

When H is a proper closed subgroup of G, the following diagram
commutes by Proposition <ref> (v)
for the one-point H-space:

    @C=12mm𝒩_m^H(∗) [rr]^-G×_H -[d]_Θ^H(∗)      𝒩_m+d^G( G/H) [d]^Θ^G(G/H)
    _m^H(S^0)  [r]_--·τ̅_H,L   _m+d^H(S^L) [r]_- G⋉_H -   _m+d^G(G/H_+)

In that diagram, the left vertical map is an isomorphism by the inductive hypothesis.
The induction map G×_H -:𝒩_m^H→𝒩_m+d^G(G/H)
is an isomorphism, with inverse given by sending 
a singular G-manifold f:M→ G/H to the fiber over
the coset H of an equivariant smooth approximation of f.
The lower right horizontal map is an isomorphism
by Theorem <ref>.subjectWirthmüller isomorphism
Now we use the hypothesis that the group G is a product
of a finite group and a torus.
In this situation the group H acts trivially on the tangent representation L, 
so multiplication by the class τ̅_H,L in _d^H(S^L) 
is the suspension isomorphism, hence bijective, 
by Proposition <ref>.
Hence the right vertical Thom-Pontryagin map Θ^G(G/H) is an isomorphism.

Now we treat the case H=G. 
We let E be a universal G-space for the family of proper subgroups of G,
and Ẽ its unreduced suspension.
We then get compatible long exact isotropy separation sequences:

    @C=5mm⋯[r]    𝒩_*^G(E) [r]^-p_*[d]_Θ^G   𝒩_*^G(∗) [r]^-i_*[d]_Θ^G   𝒩_*^G(Ẽ) [r]^-∂[d]^Θ^G   𝒩_*-1^G(E) [r] [d]^Θ^G   ⋯
    ⋯[r]    _*^G(E_+) [r]_p_*   _*^G(S^0) [r]_-i_*   _*^G(Ẽ) [r]_-∂   _*-1^G(E_+) [r]    ⋯

The map Θ^G is an isomorphism 
for the cofibrant and fixed point free G-space E by the previous paragraph,
and Θ^G is an isomorphism for Ẽ by Proposition <ref>.
So the five lemma shows that the Thom-Pontryagin map for the one-point G-space 
is an isomorphism.
This completes the inductive step, and hence the proof of the theorem.





In dimension 0, Theorem <ref> gives an explicit description
of the global functor _0()
as the quotient of the Burnside ring global functor  
by the global subfunctor generated by _e^O(1)∈(O(1)).
Equivariant manifolds of dimension 0 are easy to understand,
and this allows us to present the groups 𝒩_0^G in a global fashion 
very similar to (but different from) this presentation of _0().
We use this description to give a direct verification that the map

    Θ^G :  𝒩_0^G     → π_0^G()
 
is an isomorphism when G is a product of a finite group and a torus.
We will also use the calculation of _0()
to show that the map Θ^G is not an isomorphism in general.

As we mentioned above, the groups 𝒩_0^G
enjoy the structure of a restricted global functor,
i.e., a group-like global power monoid.
Moreover, the interval [-1,1] with C_2-action by reflection at the origin
is a C_2-equivariant null-bordism of the free transitive C_2-set.
This shows that

    _e^C_2(1)  =  0    in   𝒩_0^C_2 ,

where 1∈𝒩_0^e is the bordism class of a point.
The action on the class 1 thus factors over a morphism of restricted global functors

    ^ /_e^C_2^ → 𝒩_0  ,

from the quotient of the represented restricted global functor ^=^(e,-)
by the restricted global subfunctor generated by _e^C_2∈^(C_2).

We claim that the morphism (<ref>) is an isomorphism.
Indeed, a smooth G-manifold of dimension 0 is just a finite G-set,
so  (<ref>) is surjective. 
By the same algebraic argument as for unrestricted global functors in 
Proposition <ref>,
the value of the restricted global functor _e^C_2^ at G
is the subgroup of ^(G) generated by 2·^(G) and
the classes _H^G∘ p_H^* for those finite index subgroups H
with Weyl group of even order.
So the source of the map (<ref>) at a group G
is an _2-vector space with basis the classes _H^G∘ p_H^* 
for those conjugacy classes of finite index subgroups H with Weyl group of odd order.
The same classes form a basis for the bordism group 𝒩_0^G;
this is shown for finite groups in <cit.>,
and the general case follows because restriction along the projection p:G→π_0(G) 
induces an isomorphism p^*:𝒩_0^π_0(G)→𝒩_0^G
for bordism of 0-manifolds.

Summing up, we have calculated both sides of the Thom-Pontryagin map Θ
in dimension 0; under these isomorphisms Θ becomes the map

    ^ /_e^C_2^ →  /_e^C_2 ,

i.e., the same kind of quotient in the category of restricted versus unrestricted 
global functors. So the only difference between 𝒩_0
and _0() is that the left side only has finite index transfers,
whereas the right hand side also has transfers for infinite index inclusions
with finite Weyl group. In finite and abelian compact Lie groups,
every subgroup inclusion with finite Weyl group is necessarily of finite index,
so for finite and abelian compact Lie groups, there is no difference in
the two kinds of quotients. 
This is an independent verification of Theorem <ref> 
in dimension 0. Moreover, we conclude that the Thom-Pontryagin map
Θ^G:𝒩_0^G→π_0^G() in dimension 0
is always injective.

On the other hand, the map Θ^G is not generally surjective
in dimension 0.
A specific example is the group G=S U(2):subjectspecial unitary group!S U(2)
the normalizer N=N_S U(2) T of a maximal torus T of S U(2)
is self-normalizing, so N has trivial Weyl group in S U(2).
So the classes 1 and _N^S U(2)(1) are linearly independent in π_0^S U(2)().
On the other hand 𝒩_0^S U(2)=/2 because S U(2)
is connected.








[Stable equivariant bordism and localized ]

We showed in Corollary <ref> that the localized 
equivariant -theory ^G_*[1/τ]
is isomorphic to the theory ^G_*.
Our next task is to show that [1/τ],
and hence also , 
has a geometric interpretation as stable equivariant bordism.
subjectequivariant bordism!stable
In contrast to Theorem <ref>, 
this stable interpretation works for all compact Lie groups,
not only for products of finite groups and tori.

In <cit.>,
Bröcker and Hook define the stable equivariant bordism groups
𝔑̃^G:S_*(X) of a based G-space X
as the localization of
the geometric bordism group 𝒩̃_*^G(X)
by formally inverting all the classes d_G,V.
More precisely, their definition comes down to

    𝔑̃^G:S_m(X) =  _V∈ s(_G) 𝒩_m+|V|^G(X S^V)  ;

for V⊂ W, the structure map in the colimit system is the multiplication

    𝒩_m+|V|^G(X S^V)  - d_G,W-V 𝒩_m+|W|^G(X S^V S^W-V)  𝒩_m+|W|^G(X S^W)   .

As we explained in Example <ref>,
the Thom-Pontryagin construction takes the distinguished 
geometric bordism class d_G,V to the shifted inverse Thom class, i.e., 
Θ^G(d_G,V) = τ̅_G,V in _|V|^G(S^V).
Since the Thom-Pontryagin maps take the geometric product in equivariant
bordism to the homotopy theoretic product in , we conclude that
for every compact Lie group G, 
every G-representation V and every based G-space X, the following
square commutes:

    @C=12mm𝒩_m^G(X) [d]_- d_G,V[r]^Θ^G   _m^G(X) [d]^-·τ̅_G,V
    𝒩_m+|V|^G(X S^V) 
    [r]_-Θ^G   _m+|V|^G(X S^V)
  
The Thom-Pontryagin maps thus assemble into a natural transformation

    Θ^G  :  𝔑̃^G:S_m(X) → ^G_m(X)[1/τ]

between the localized theories, for which we use the same letter.


If G is a product of a finite group and a torus, then the next theorem
is a direct consequence of Theorem <ref>.
The point, however, is that the following localized version holds
without any restriction on the compact Lie group G.
Morally, the reason for this is that formally inverting the classes d_G,V 
forces the Wirthmüller isomorphism to hold,
so in stable equivariant bordism this potential obstruction
to representability by a global homotopy type vanishes. 

 
For every compact Lie group G and every cofibrant based G-space X,
the map

    Θ^G  :  𝔑̃^G:S_*(X) → ^G_*(X)[1/τ]

is an isomorphism of graded abelian groups.  


Since filtered colimits of abelian groups are exact, the localized theories 
𝔑̃^G:S_*(-) and ^G_*(-)[1/τ]
are both equivariant homology theories for every fixed group G.
We can run the same inductive argument
as in the proof of Proposition <ref>, i.e., show the
claim by double induction over the dimension and the
number of path components of G. 
But this time the case of an orbit X=G/H for a proper closed subgroup H of G
works without any restriction on G and H. Indeed, in the commutative diagram

    @C=12mm𝔑̃_m^H:S(S^0) [rr]^-G×_H -[d]_Θ^H      𝔑̃_m+d^G:S( G/H_+) [d]^Θ^G
    _m^H(S^0)[1/τ]  [r]_--·τ̅_H,L   _m+d^H(S^L)[1/τ] [r]_- G⋉_H -   _m+d^G(G/H_+)[1/τ]

multiplication by the class τ̅_H,L is now invertible
in the localized theory _*^G[1/τ]. So all horizontal maps in the diagram
are isomorphisms. Since the left vertical map is an isomorphism by induction, 
the right vertical map is an isomorphism as well.

The case X=G/G is essentially taken care of
by Proposition <ref>.
We let V be any G-representation, 
and observe that the fixed point inclusion i:V^G→ V
induces a G-homotopy equivalence

    Ẽ i  :  Ẽ S^V^G → Ẽ S^V  .

In the commutative diagram

    @C=13mm@R=7mm𝒩_m+|V^⊥|^G(Ẽ)[r]^-Θ^G[d]_- d_G,V^G   _m+|V^⊥|^G(Ẽ) [d]^-·τ_G,V^G
    𝒩_m+|V|^G( Ẽ S^V^G)[r]^-Θ^G[d]_(Ẽ i)_*   _m+|V|^G(Ẽ S^V^G) [d]^(Ẽ i)_*
    𝒩_m+|V|^G(Ẽ S^V)[r]_-Θ^G   _m+|V|^G(Ẽ S^V)

the lower vertical maps are thus isomorphisms.
The upper vertical maps are suspension isomorphisms,
hence both vertical composites are isomorphisms.
The upper horizontal map is an isomorphism by Proposition <ref>,
hence so is the lower horizontal map.
As a colimit of isomorphisms, the localized Thom-Pontryagin map

    Θ^G : 𝔑_*^G:S(Ẽ) →_*^G(Ẽ)[1/τ]

is also an isomorphism.
Now we finish the argument as in Proposition <ref>:
we compare the isotropy separation sequence for
𝔑̃^G:S_*(-) to that for ^G_*(-)[1/τ],
and the five lemma concludes the inductive step.





[Stable equivariant bordism and ]
Corollary <ref> and Theorem <ref> 
provide natural isomorphisms

    𝔑̃^G:S_*(X)  Θ^G ^G_*(X)[1/τ]    ^G_*(X)

for cofibrant based G-spaces X.
Hence the composite is an isomorphism, which provides an alternative proof 
that stable equivariant bordism
agrees with equivariant -homology, which is the main result 
of the paper <cit.> by Bröcker and Hook. 
Strictly speaking there is a bit more work involved
in the translation, because our group _*^G(X) is not literally the same
as the homotopy theoretic equivariant bordism group Ñ_n^G(X)
in <cit.>; we invite the reader to spell out
an isomorphism, which boils down to a certain rewriting of colimits,
such that our composite isomorphism 
becomes the map Φ^S considered by Bröcker and Hook.


subjectglobal Thom spectrum|)
subjectequivariant bordism|)





§ CONNECTIVE GLOBAL K-THEORY
 



In this section we define and discuss the ultra-commutative ring spectrum ,
the connective global K-theory spectrum, 
see Construction <ref>.
Our construction is an elaboration of a model of non-equivariant connective K-theory 
by Segal <cit.>,
constructed from certain -spaces of 
`orthogonal subspaces in the symmetric algebra'.
The degree zero equivariant cohomology theory represented by 
tries hard to be equivariant K-theory.
Indeed, Theorem <ref> exhibits a natural transformation
from the G-equivariant K-group that is an isomorphism
for finite G-CW-complexes with finite isotropy groups.
A special case is a ring homomorphism (G)→π_0^G()
from the complex representation ring that is an isomorphism whenever G is finite.
For varying G, these homomorphisms almost form a morphism of global power functors,
in the sense that they are compatible
with finite index transfers, but not with general degree zero transfers
across an infinite index inclusion, see Remark <ref>.   

The Bott class is a preferred element β∈π_2^e() 
that we define in Construction <ref>.
This class becomes invertible in the periodic global K-theory spectrum ,
by Theorem <ref> below.
More generally, every G-^c-re­presen­tation has an associated
equivariant Bott class

    β_G,W ∈ ^0_G(S^W)  ,

see Construction <ref>,
that becomes an R O(G)-graded unit in the periodic theory .





 
We let  be a complex vector space of countable dimension (finite or infinite)
equipped with a hermitian inner product.
We recall a certain -space () of `orthogonal subspaces in ',
due to Segal <cit.>.subjectGamma-space@-space
For a finite based set A we let (,A) be the space of tuples
(E_a), indexed by the non-basepoint elements of A,
of finite-dimensional, pairwise orthogonal -subspaces of . 
The topology on (,A) 
is that of a disjoint union of subspaces of a product
of Grassmannians.
The basepoint of  (,A) is the tuple where each E_a
is the zero subspace.
For a based map α:A→  B the induced map
(,α):(,A)→(,B) sends (E_a) to (F_b) where

    F_b  =  ⊕_α(a)=b  E_a  .

Then () is a -space whose underlying space is 

    (,1_+)  =  ∐_n≥ 0  G r_n^() ,

the disjoint union of the different Grassmannians of .

Every -space can be evaluated on a based space 
by the coend construction (<ref>).
We write (,K)=()(K) for the value of the -space
() on a based space K.
When A is a finite based set with discrete topology, 
then the prolongation ()(A) is canonically
homeomorphic to the original space (,A), 
compare Remark <ref>.
Elements of (,K) can be interpreted as `labeled configurations': 
a point is represented by an unordered tuple 

    [ E_1,…,E_n; k_1,…,k_n ]

where (E_1,…, E_n) is an n-tuple of finite-dimensional,
pairwise orthogonal subspaces of , 
and k_1,…,k_n are points of K, for some n. 
The topology is such that,
informally speaking, the labels sum up whenever two points collide,
and a label disappears whenever a point approaches the basepoint of K.



When K is compact, the space (,K) can be described differently,
compare again <cit.>, namely as the space

    _V∈ s()   C^∗(C_0(K),_(V) )  ,

where the colimit runs over all finite-dimensional -subspaces of .
Here  C_0(K) is the C^∗-algebra of continuous -valued functions
on K that vanish at the basepoint,
_(V) is the endomorphism C^∗-algebra of V
(which is isomorphic to a matrix algebra over ),
and C^∗(-,-) is the space of C^∗-algebra homomorphisms,
with the zero homomorphism as the basepoint.
For U⊂ V the map in the colimit system is induced by
the ∗-homomorphism _(U)→_(V)
that extends an endomorphism by 0 on the orthogonal complement.
A homeomorphism

    (,K) → _V∈ s()   C^∗(C_0(K),_(V))

is given by sending a configuration [E_1,…,E_n; k_1,…,k_n]
in (,K) to the homomorphism that takes a function φ∈ C_0(K) to 

    ∑_i=1^n φ(k_i)· p_E_i ;

here V is chosen large enough to contain all the spaces E_i, and
p_E_i:V→ V is the orthogonal projection onto the subspace E_i.



[Eigenspace decomposition]
If  is infinite dimensional, then the -space () is special,
compare Theorem <ref> (i) below.
The orthogonal spectrum ()()
is then a positive Ω-spectrum by the general theory. In particular,
the space ()()()=(,S^1) is an infinite loop space. 
In fact, (,S^1) is a familiar space, 
namely the infinite unitary group U(), i.e., the group of 
linear self-isometries of  that are the identity on the orthogonal complement
of some finite-dimensional subspace.
This eigenspace decompositionsubjecteigenspace decomposition
works for every hermitian inner product space , of finite or countable
dimension, as follows.
As before we identify S^1 with the unit circle U(1) in the complex numbers via
the Cayley transformsubjectCayley transform

    c  :  S^1    U(1) ,    x  ⟼ (x+i)(x-i)^-1 .

This homeomorphism sends the basepoint at infinity to 1. 
Given a tuple of pairwise orthogonal subspaces (E_1,…,E_n) 
of  and a point (x_1,…,x_n)∈ (S^1)^n,
we let ψ(E_1,…,E_n; x_1,…,x_n)
be the isometry of  that is multiplication by c(x_i) on E_i 
and the identity on the orthogonal complement of ⊕_i=1^n E_i. 
In other words: E_i is the eigenspace 
of ψ(E_1,…,E_n; x_1,…,x_n) for the eigenvalue c(x_i).
As n varies, these maps are compatible
with the equivalence relation and so they assemble into a continuous map

    subjecteigenspace decomposition(,S^1)  =  ∫^n_+∈(,n_+) × (S^1)^n    →   U()
    
    [E_1,…,E_n; x_1,…,x_n]    ⟼ψ(E_1,…,E_n; x_1,…,x_n) .

This map is a homeomorphism because every unitary transformation
is diagonalizable with eigenvalues in U(1) 
and pairwise orthogonal eigenspaces. 



Now we let  and  be two hermitian vector spaces
(of countable dimension, but possibly infinite dimensional).
Here and below we write =_ for the tensor product over .
We endow   with a hermitian scalar product by declaring

    (u v,  u' v') =  (u,u')· (v,v')

on elementary tensors and extending biadditively.
If E,F are orthogonal subspaces of 
and  E' is a subspace of , then
E E' and F E' are orthogonal subspaces of ,
and similarly in the second variable.
For based spaces K and L we can thus define 
a continuous multiplication map

    (,K)(,L)    → (,K L)
    
    [E_i;k_i] [F_j;l_j]     ⟼   [E_i F_j;k_i l_j] .

These multiplication maps are associative, and commutative
in the sense that the following square commutes:

    @C=12mm(,K)(,L)[r][d]_τ_(,K),(,L)   (,K L)[d]^(τ_,,τ_K,L)
    (,L)(,K)[r]    (,L K)

   

Our construction of connective global K-theory
needs induced inner products on symmetric powers.
We explain the complex version; the real version 
works in much the same way.
For a -vector space V we denote by

    ^n(V)  =  V^ n/Σ_n

the n-th symmetric power of V and by (V)=⊕_n≥ 0^n(V)
the symmetric algebra of V.
If W is another -vector space, then
the two direct summand embeddings of V and W into V⊕ W
induce homomorphisms of symmetric algebras 
that combine (by multiplying in the target) into a natural
-algebra isomorphism

    (V)(W)  (V⊕ W)  .

If V is equipped with a hermitian inner product, 
then the symmetric powers inherit a preferred inner product:



  For every hermitian inner product space V there is a unique 
  inner product on ^n(V) that satisfies 
  
    ( v_1·…· v_n, v̅_1·…·v̅_n)  =  ∑_σ∈Σ_n 
      (v_1,v̅_σ(1))·…· (v_n,v̅_σ(n))

  for all v_i,v̅_i∈ V.
  This inner product on ^n(V) is natural for -linear isometric embeddings
  and it makes the algebra isomorphism (<ref>) into an isometry.


  Uniqueness of the scalar product follows from the fact that the
  symmetric products v_1·…· v_n generate ^n(V) 
  as a -vector space.
   The tensor product V W of two hermitian inner product spaces V and W 
   has a preferred hermitian inner product as in (<ref>).
   By iteration, the n-fold tensor product V^ n inherits an inner product. 
   There is thus a unique inner product
   on ^n(V) that makes
   the normalized -linear `symmetrization' embedding
  
    ^n(V)     →        V^ n
    
        v_1·…· v_n     ⟼ 1/√(n!)·∑_σ∈Σ_n v_σ(1)… v_σ(n)

   a linear isometric embedding. 
   We omit the straightforward verification that this inner product is indeed
   given by the formula in the statement of the proposition,
   and that it is natural for linear isometric embeddings.

  The algebra isomorphism (<ref>)  is the sum of the embeddings
  
    ^m(V)^n(W)     →  ^m+n(V⊕ W) 
    
        v_1·…· v_m  w_1·…· w_n    ⟼    (v_1,0)·…· (v_m,0)·(0,w_1)·…·(0, w_n)  .

  The relation
  
    ( v_1·…· v_m     w_1·…· w_n, v̅_1·…·v̅_mw̅_1·…·w̅_n) 
       =     (v_1·…· v_m, v̅_1·…·v̅_m)
        · (w_1·…· w_n, w̅_1·…·w̅_n)
       =  ∑_(σ,τ)∈Σ_m×Σ_n 
      ( v_1,v̅_σ(1))·…· (v_m,v̅_σ(m))
      ·  (w_1,w̅_τ(1))·…· (w_n,w̅_τ(m))
       =  ( (v_1,0)·…· (v_m,0)·(0,w_1)·…·(0, w_n),
        ( v̅_1,0)·…· (v̅_m,0)·(0,w̅_1)·…·(0,w̅_n) )

then proves that (<ref>) preserves the inner product.
The last equation uses that (v,0) and (0,w) are orthogonal
in V⊕ W for all v∈ V and w∈ W.




  The case n=2 gives an idea of the induced inner product on ^n(V):
  if {e_1,…,e_k} is an orthonormal basis of V, then the vectors
   
    1/√(2)· e_i^2    (1≤ i≤ k)     and     
       e_i· e_j    (1≤ i< j≤ k)

   form an orthonormal basis of ^2(V).



subjectK-theory@K-theory!connective global|(

 [Connective global K-theory]
We can now define an ultra-commutative ring spectrum , 
the connective global K-theory spectrum.subjectconnective global K-theorysubjectK-theory@K-theory!connective globalsymbol - connective global K-theory  
The value of  on a euclidean inner product space V is

    (V) =  ((V_),S^V) ,

the value of the -space ((V_))
on the one-point compactification of V.
Here V_ is the complexification of V with the
induced hermitian inner product, and the inner product on the symmetric algebra described 
in Proposition <ref>.
The action of O(V) on V then extends to a unitary action on (V_). 
We let the orthogonal group O(V) act diagonally, 
via the action on the sphere S^V and the action 
on the -space ((V_)). 
Using the tensor product pairing (<ref>)
we define an (O(V)× O(W))-equivariant multiplication map

    μ_V,W :  (V)(W) =      ((V_),S^V)((W_),S^W)
    
    _(<ref>)→    ((V_)(W_),S^V S^W)
     
    _(<ref>)      (((V⊕ W)_),S^V⊕ W) = (V⊕ W) .

The maps μ_V,W  are associative and commutative.
An O(V)-equivariant unit map is given by

    ι_V  :  S^V → ((V_),S^V) = (V)  ,    
    v  ⟼ [· 1;  v] ,

where · 1 is the homogeneous summand of degree 0
in the symmetric algebra, i.e., the line spanned by the multiplicative unit.
This structure makes  into an ultra-commutative ring spectrum.

The space _0=(· 1,S^0) consists of all subspaces of
(0)=· 1, so it has two points, the basepoint 0
and the point · 1. The unit map ι_0:S^0→_0 is 
thus a homeomorphism.





[Complex conjugation on ]subjectcomplex conjugation!on 
The ultra-commutative ring spectrum  comes with an involution 
by `complex conjugation'
that preserves all the structure available.
Indeed, for every euclidean inner product space V the 
complex symmetric algebra (V_) of the complexification
is canonically isomorphic to __(V), 
the complexification of the real symmetric algebra of V.
So (V_) comes with an involution ψ_(V)
that is -semilinear and preserves the orthogonality relation. 
Applying this involution elementwise to tuples of orthogonal subspaces
gives an involution (ψ_(V)):((V_))→((V_))
of the -space and hence a homeomorphism

    ψ(V) =  (ψ_(V),S^V) :  (V)  → (V)

of order 2.
As V varies, the maps ψ(V) form an automorphism 
ψ:→ of the ultra-commutative ring spectrum .





[Connective real global K-theory]
There is a straightforward real analog  of the 
complex connective global K-theory spectrum . 
subjectconnective global K-theory!realsubjectK-theory@K-theory!real connective globalsymbol - real connective global K-theory  
The value of  on an inner product space V is

    (V) =  _((V),S^V) ,

where now (V) is the symmetric algebra of V over the real numbers, 
and _((V)) is the -space of tuples of
pairwise orthogonal, finite-dimensional -subspaces of (V).
As V varies, the spaces (V) again come with the structure 
of an ultra-commutative ring spectrum. 

Complexification defines a morphism of ultra-commutative ring spectra
subjectcomplexification morphism!from  to 

    c  :   →  .

In more detail: if V is a real inner product space, then the map

    c(V)  :  (V) = _((V),S^V)  → ((V_),S^V) = (V)

sends a configuration [E_1,…,E_n; v_1,…,v_n] 
to the complexified configuration  

    [(E_1)_,…,(E_n)_; v_1,…,v_n]   .

As V varies, these maps form the morphism (<ref>) 
of ultra-commutative ring spectra.
The complexification of a real subspace of (V) is invariant under 
the complex conjugation involution ψ_(V) of (V_),
so the image of the complexification morphism 
is invariant under the complex conjugation involution ψ of 
defined in (<ref>).
Even more is true: a complex subspace of (V_)
is ψ-invariant if and only if it is the complexification of
its real part (the +1 eigenspace of ψ_(V)). 
This means that   `is'
the ψ-fixed orthogonal ring subspectrum of ; more formally,
the complexification morphism (<ref>) is an isomorphism
from  onto the ψ-fixed orthogonal ring subspectrum of .




The next proposition justifies the adjective `connective'
that we attached to the names of the orthogonal spectra  and .
The proof uses a certain cofibrancy property of the -space ((V_))
that we now introduce.


[Latching map]
We let (n) denote the power set of the set {1,…,n}, 
i.e., the set of subsets.
We also write (n) for the associated poset category, i.e., with object set (n)
and exactly one morphism U→ T whenever U⊆ T.
Given a -space F, we obtain a functor from (n) 
to based spaces by sending a subset U
to F(U_+), with the maps F(U_+)→ F(T_+) induced by the inclusions.
We obtain a latching mapsubjectlatching map!of a -space

    l_n :  _U⊊{1,…,n}  F(U_+) →  F(n_+)  .

The map l_n is equivariant for the action of Σ_n
given on the target by functoriality of F. The action of a permutation
σ∈Σ_n on the source is induced by
sending F(U_+) to F(σ(U)_+) via the map

    F( (σ· -)_+)  :  F(U_+) → F(σ(U)_+) .

The latching map l_n is always a closed embedding, 
by Proposition <ref> (ii).


We will now consider a -G-space for a compact Lie group G,
i.e., a reduced functor F:→ G_* to the category of based G-spaces.
Then G acts on source and target of the latching map l_n, which
is thus (Σ_n× G)-equivariant.
The following cofibrancy condition on a -G-space ensures that
the prolongation is homotopically meaningful.


Let G be a compact Lie group. A -G-space F is G-cofibrant
subjectGamma-space@-space!cofibrant
if for every n≥ 1 the latching map l_n 
is a (Σ_n× G)-cofibration.


  
We let G be a compact Lie group and  a unitary G-representation,
of finite or countably infinite dimension.
We argue that the -G-space (,-) is G-cofibrant.
The actions of G respectively Σ_n on an n-tuple of
orthogonal subspaces are componentwise respectively by permuting the entries, i.e., 

    (σ,g)· (E_1,…,E_n)  =  
    (g· E_σ^-1(1),…,g· E_σ^-1(n))

for (σ,g)∈Σ_n× G.
The topology on (,n_+) is that of a disjoint union of subspaces of a product
of Grassmannians:

    (,n_+)  =  ∐_(i_1,…,i_n)∈^n (;i_1,…,i_n)  ,

where
(;i_1,…,i_n) is the subspace of those tuples
such that (E_j)=i_j for all 1≤ j≤ n.

The summands are invariant under the G-action, 
and (;i_1,…,i_n) is G-equivariantly homeomorphic to

    ^(^i_1+…+i_n,) /  U(i_1)×…× U(i_n)  ,

where ^(-,-) is the space of -linear isometric embeddings.
This G-space is G-cofibrant by the unitary analog of
Proposition <ref> (ii).
We let Γ⊂Σ_n 
be the stabilizer group of the dimension vector (i_1,…,i_n),
i.e., the group of those σ∈Σ_n such that i_j=i_σ(j) 
for all 1≤ j≤ n. Then Γ acts trivially on (;i_1,…,i_n),
so this space is (Γ× G)-cofibrant.
Hence the induced (Σ_n× G)-space

    Σ_n ×_Γ(; i_1,…,i_n)

is (Σ_n× G)-cofibrant.
The full configuration space (,n_+)
decomposes as the disjoint union of (Σ_n× G)-invariant subspaces
indexed by the Σ_n-orbits on ^n,

    (,n_+)  =  ∐_i Σ_n ∈^n/Σ_n Σ_n×_Γ_i(; i)  ,

where Γ_i⊂Σ_n is the stabilizer of i=(i_1,…,i_n).
Altogether this shows that (,n_+) is a disjoint union of 
(Σ_n× G)-cofibrant spaces. 
The latching map

    l_n  :  _U⊊{1,…,n} (,U_+) → (,n_+)

is a closed embedding 
by Proposition <ref> (ii),
and its image is the disjoint union of the
summands (; i_1,…,i_n) such that i_j=0 for some j∈{1,…,n}.
Since all additional summands not in the image are (Σ_n× G)-cofibrant, 
the map l_n is a (Σ_n× G)-cofibration.




The orthogonal spectra  and  are globally connective.subjectglobally connective  


  The arguments for  and  are completely parallel, and we concentrate
  on the complex case.  We let G be any compact Lie group, and we show 
  that the group π_-k^G() is trivial for all k≥ 1.
  We let V be a G-representation and set =((^k⊕ V)_),
  a unitary G-representation of countably infinite dimension.
  We let H be a closed subgroup of G.
  The -H-space (,-) is H-cofibrant by
  Example <ref>.  
  So the space (,S^^k⊕ V)^H
  is (k+(V^H)-1)-connected by Proposition <ref> (i). 

  The cellular dimension of S^V at H,
  in the sense of <cit.>,
  is the topological dimension of the space (S^V^H)/N_G H;
  this cellular dimension is at most (V^H). 
  Because k is positive, the cellular dimension of S^V at H does not exceed
  the connectivity of (^k⊕ V)^H.
  So every based continuous G-map S^V→(^k⊕ V) is equivariantly
  null-homotopic by <cit.>,
  and the set [S^V,(^k⊕ V)]^G has only one element.
  Passage to the colimit over V∈ s(_G) proves the claim.  


Every -linear isometric embedding u:→ 
of complex inner product spaces induces a morphism of -spaces
(u):()→() by applying u elementwise
to a tuple of orthogonal subspaces.
So if a compact Lie group G acts on  by linear isometries
(for example if  is a G-universe), 
then the -space () inherits a G-action, 
so it becomes a -G-space.


Let G be a compact Lie group and  and  two isomorphic
complex G-universes. Then for every G-equivariant linear isometric embedding
u:→ and every based G-space K the map

    (u,K) : (,K) → (,K)

is a G-homotopy equivalence.


We start with the special case where =.
The space of G-equi­variant linear isometric embeddings from 
to itself is contractible. A homotopy from  u
to the identity then induces a G-homotopy from (u,K)
to the identity of the G-space (,K).
In the general case we choose a 
G-equivariant linear isometry v:.
Then the two G-maps

    (v u,K) (,K) → (,K)  and  (u v,K) (,K) → (,K)

are G-homotopic to the respective identity maps.




If F is any -space and S a finite set, then we define the map

    P_S  :   F(S_+) → (S, F(1_+))

by P_S(x)(s)= F(p_s)(x), where p_s:S_+→ 1_+ sends s to 1 and all other elements
of S_+ to the basepoint. 

Now we suppose that G is a compact Lie group and F is a -G-space.
When we evaluate F (or rather its prolongation)
at a G-space K, it comes with a (G× G)-action;
one action comes from the `external' action on F, the other one from the
G-action on K via the continuous functoriality of F. In such a situation,
we always consider F(K) as a G-space 
via the diagonal action of this (G× G)-action.
The map P_S is natural for bijections in S;
so whenever the group G acts on S, then P_S:F(S_+)→(S,F(1_+)) 
is G-equivariant for the diagonal action on the source 
and the conjugation action on the target.

We recall from Definition <ref>
that a -G-space F is specialsubjectGamma-space@-space!specialsubjectGamma-space@-space!equivariant  
if for every closed subgroup H of G and every finite H-set S the map

    (P_S)^H  :  F(S_+)^H → ^H(S,F(1_+))

is a weak equivalence. It goes without saying that H must act continuously
on S with the discrete topology; so S is a disjoint union of H-sets
of the form H/K for finite index subgroups K of H.
Equivalently, F is special if and only if the map

    P_n  :  F(n_+) → ({1,…,n},F(1_+))  =  F(1_+)^n

is an (G;Σ_n)-equivalence for every n≥ 1, where
(G;Σ_n) is the family of graph subgroups, 
see Proposition <ref>.subjectgraph subgroup




 
Let G be a compact Lie group and  a complete complex G-universe.


  * The -G-space (,-) is special.

  * If G is finite and V and W are two G-representations with W^G{0}, 
then the adjoint assembly map

    α̃ :  (,S^W) → _*(S^V, (,S^V⊕ W))

is a G-weak equivalence.



(i)
We show that for every n≥ 1 the (G×Σ_n)-map

    P_n  :  (,n_+) → (,1_+)^n

is an (G;Σ_n)-weak equivalence, where (G;Σ_n)
is the family of graph subgroups of G×Σ_n.
We define a morphism of (G×Σ_n)-spaces

    λ_n :  (,1_+)^n  → (^n,n_+)  ;

here the Σ_n-action on the target is diagonally, from
the permutation action on n_+ and on the tensor factor ^n.
The map λ_n sends an n-tuple ( E_1,…, E_n)
of finite-dimensional -subspaces to the configuration 

    ( e_1 E_1,…, e_n E_n )

of pairwise orthogonal subspaces of ^n, 
where e_j=(0,…,0,1,0,…,0) is the j-th vector of the
standard basis of ^n. 

Now we consider the two composites λ_n∘ P_n 
and P_n'∘λ_n, where P_n' is the map P_n
for the universe ^n (as opposed to ):

    (,n_+)  P_n (,1_+)^n  λ_n (^n,n_+)   P_n'  (^n,1_+)^n

We claim that both composites are (G;Σ_n)-weak equivalences.
We start by investigating the composite λ_n∘ P_n.
For 1≤ j≤ n, we define a 1-parameter family of unit vectors

    u_j  :  [0,1]  → ^n
         by    
    u_j(t) =  
    t· e_j + √(1-t^2/n-1)·∑_k j e_k .

This provides a homotopy

    H  :  (,n_+)× [0,1] → (^n,n_+)

by

    H( (E_1,…,E_n),t)  =  ( u_1(t) E_1,…,u_n(t) E_n )  .

We have

    u_1(1/√(n)) =  … =  u_n(1/√(n)) =  
     1/√(n)· (1,…,1)

and 

    H( (E_1,…,E_n),1)  =  ( e_1 E_1,…,e_n E_n)  =  
    (λ_n∘ P_n)(E_1,…,E_n) .

Moreover, each of the maps H(-,t) is (G×Σ_n)-equivariant,
so the composite λ_n∘ P_n is (G×Σ_n)-equivariantly homotopic 
to the map (u,n_+), where
u:→^n is the (G×Σ_n)-equivariant
linear isometric embedding 

    u(v)  =  1/√(n)·(1,…,1)  v .

Now we let α:H→Σ_n be a continuous homomorphism
defined on a closed subgroup H of G.
Since  is a complete complex G-universe,
both  and α^*(^n)
are complete complex H-universes, 
and u is an H-equivariant linear isometric embedding

    u  :  ^G_H() → α^*(^n)^G_H() .

Interpreted in this way, the map (u,n_+)^H is a homotopy equivalence by 
Proposition <ref>.
This shows that the morphism (u,n_+) is an (G;Σ_n)-weak equivalence,
hence so is the morphism λ_n∘ P_n.

Now we show that the composite P_n'∘λ_n is an
(G;Σ_n)-weak equivalence.
We let H be a closed subgroup of G,
α:H→Σ_n a continuous homomorphism,
and Γ={ (h,α(h))   |  h∈ H}≤ G×Σ_n
the graph of α.
We let a_1,…, a_k∈{1,…,n} be a set of representatives
of the orbits of the H-action on {1,…,n} through α.
We let H_i≤ H be the stabilizer group of a_i.
Then projection to the factors indexed by 
a_1,…, a_k provides homeomorphisms

    ((,1_+)^n)^Γ      ∏_i=1^k   (,1_+)^H_i   and
    
      ((^n,1_+)^n)^Γ     ∏_i=1^k (α^*(^n),1_+)^H_i .

Under these identifications, the map (P_n'∘λ_n)^Γ 
becomes the product of the maps

    ((,1_+))^H_i → (α^*(^n),1_+)^H_i

induced by the H_i-equivariant linear isometric embeddings

    → α^*(^n) ,   v  ⟼ e_a_i v .

Since  is a complete complex G-universe,
both  and α^*(^n)
are complete complex H_i-universes.
So the map (<ref>) is a homotopy equivalence by 
Proposition <ref>.
Since λ_n∘ P_n and P_n'∘λ_n are
(G;Σ_n)-weak equivalences, 
so is the map P_n:(,n_+)→(,1_+)^n.

Part (ii) is essentially a special case of Segal and Shimakawa's equivariant
delooping machine based on  -G-spaces <cit.>.
There is a bit of extra work necessary, though, because our claim is about
the prolongation (,S^V⊕ W) (which is a categorical coend),
whereas Segal and Shimakawa formulate their results in terms of a bar construction
(which amounts to a homotopy coend).  
We provide the additional arguments 
in Theorem <ref>;
to apply this, we need that the -G-space (,-)
is special by part (i) and G-cofibrant by Example <ref>.




The orthogonal spectrum  is trying to be a -global Ω-spectrum.
However, the global Ω-spectrum condition on the adjoint structure maps
only holds for `sufficiently large' representations.

 
Let G be a compact Lie group. An orthogonal G-re­presen­tation W
is amplesubjectample G-representationsubjectG-representation@G-representation!ample|seeample G-representation
if the complex symmetric algebra (W_)
is a complete complex G-universe.


 
Every ample G-representation is non-zero and the action of G is faithful.
Examples of ample representations are 
non-zero faithful permutation representations
(which in particular forces the group to be finite). 
Indeed, we let A denote the permutation representation
of a non-empty faithful finite G-set A.
Then the complexified symmetric algebra ( A)
is a complex permutation representation, namely of the infinite
G-set ^A=(A,) of functions from A to .
Since G acts faithfully on A, every injective map A→ 
generates a free G-orbit in the G-set ^A. 
Since A is non-empty, there are infinitely many injections from A to  with
pairwise disjoint images, and these generate infinitely many
distinct free G-orbits in ^A. So ( A) contains
infinitely many copies of the complex regular G-representation,
and is thus a complete complex G-universe.  

A more general class of ample G-representations are non-zero representations
containing a vector with trivial isotropy group.
Indeed, given a non-zero G-free vector w_0∈ W, 
we let ν be the normal space at w_0 to the orbit G w_0.
A choice of G-equivariant tubular neighborhood of the orbit G w_0
allows us to G-equivariantly embed the Hilbert space L^2(G;)(ν)
into L^2(W;).
Since L^2(G;)(ν) contains every finite-dimensional G-representation
infinitely often, so does L^2(W;). 
But L^2(W;) is equivariantly isomorphic to 
the Hilbert space completion of (W^∗_):
elements of (W^∗_) are polynomial functions on W,
which we can map to L^2-functions by scaling with the function
w↦exp(-|w|^2). This gives a G-equivariant isometric embedding
(W^∗_)→ L^2(W;) with dense image.
So if there was any complex G-representation that did not embed
equivariantly into (W^∗_), then it would also
not embed into L^2(W;), a contradiction.






Let G be a finite group and W an ample orthogonal G-representation.
Then for every G-representation V the adjoint structure map

    σ̃_V,W :  (W) → _*(S^V,(V⊕ W))

is a G-weak equivalence. 


The adjoint structure map σ̃_V,W factors as the composite

    (W)    =  ((W_),S^W) ((i_),S^W) (((V⊕ W)_),S^W)
        α̃ _*(S^V, (((V⊕ W)_),S^V⊕ W)) =  _*(S^V,(V⊕ W));

here i:W→ V⊕ W is the inclusion of the second summand and
α̃ is the adjoint assembly map for the -G-space
(((V⊕ W)_)).
Since W is ample, the map (i_):(W_)→((V⊕ W)_)
is an equivariant linear isometric embedding between complete complex
G-universes. So the first map is a G-homotopy equivalence by 
Proposition <ref>.
The second map is a G-weak equivalence by Theorem <ref> (ii).





Now we will justify that for every finite group G 
the underlying orthogonal G-spectrum of  represents 
connective G-equivariant topological K-theory.



We define a morphism of orthogonal spaces

    c  :  ^ → Ω^∙ ,

where ^ is the complex additive Grassmannian
introduced in Example <ref>.subjectadditive Grassmannian
The value at an inner product space V is the continuous map

    c(V)  :  G r^(V_)  → _*(S^V,(V)) =  (Ω^∙)(V)

that sends a complex subspace L⊂ V_ to the continuous based map

    [L;-] :  S^V  → ((V_),S^V) =   (V)  ,  
    v  ⟼ [L; v] .

Here we consider L⊂ V_ as sitting in the linear summand of
the symmetric algebra of V_.


It will be useful to also define a delooping of the morphism c, namely
a morphism of orthogonal spaces

    :   → Ω^∙() ,

where  is the ultra-commutative monoid of unitary groups
(compare Example <ref>),
and =^ is the shift of  as 
defined in (<ref>).
The name `eig' refers to the fact that the morphism records the eigenspace
decomposition of a unitary isomorphism, compare Remark <ref>.
The definition uses the homeomorphism

    h  :  U(1)  S^1  , 
       h(λ) =  i·λ+1/λ-1 ,

the inverse of the Cayley transform.subjectCayley transform
Every unitary automorphism of a finite-dimensional hermitian inner
product space is diagonalizable with eigenvalues in U(1)
and pairwise orthogonal eigenspaces.
So given an inner product space V we define

    (V)  :  (V) =  U(V_)  → _*(S^V, (V⊕) )
    = ( Ω^∙() )(V)

by

    (V)(A)(v)  =  
    [E(λ_1),…, E(λ_n); (v,h(λ_1)),…,(v,h(λ_n)) ]  .

Here λ_1,…λ_n∈ U(1) are the eigenvalues of 
A and E(λ_i) is the eigenspace of λ_i.
Strictly speaking, E(λ_i) is a subspace of V_, which we embed into
the linear summand of ((V⊕)_).



The morphism

    :   → Ω^∙()

is a -global equivalence of orthogonal spaces.


  We let  denote the orthogonal space with

    (V) =  U(((V⊕)_)) .

The structure map induced by φ:V→ W 
is given by extending by the identity on the orthogonal complement
of ((φ⊕)_):((V⊕)_)→((W⊕)_).
The eigenspace decomposition map then factors as the composite

    →     Ω^∙()  ,

where  is defined in the same way as , recording the 
set of eigenvalues and eigenspaces.
Since (V) is the colimit over n≥ 0, along closed embeddings,
of the spaces U(^≤ n((V⊕)_)),
the morphism → is a global equivalence of orthogonal spaces
by Theorem <ref>
and Proposition <ref> (ix). 
We may thus show that the morphism

    :   → Ω^∙()

is a -global equivalence.
We show the stronger statement that for every finite group G and
every ample G-representation V the map
(V) is a G-weak equivalence. 
Indeed, the map (V) factors as the composite

    U(((V⊕)_))        (( (V_⊕)_), S^1) 
        ∼  _*(S^V, (( (V⊕)_), S^V⊕))  .

The first map is the eigenspace decomposition, hence a homeomorphism
by Remark <ref>.
The second map is adjoint to the assembly map
 
    S^V(((V⊕)_), S^1)  → (( (V⊕)_), S^V⊕)

of the -G-space (((V⊕)_)).
Since V is ample, ((V⊕)_) is a complete complex G-universe.
So the adjoint assembly map is a G-weak equivalence 
by Theorem <ref> (ii).



The set [A,^]^G has an abelian monoid structure arising from
the ultra-commutative multiplication of ^ 
as explained in (<ref>).
The set [A, Ω^∙]^G has an abelian group structure 
as an equivariant stable homotopy group, i.e., through the
adjunction bijection

    [A, Ω^∙]^G  π_0^G((A,))  ;

so this group structure arises from concatenation of loops.
The morphism of orthogonal spaces c: ^→Ω^∙
defined in (<ref>) is not a homomorphism of ultra-commutative monoids,
nor is it a loop map; so it is not a priori clear whether
the induced map on equivariant homotopy sets is a monoid homomorphism.


Let G be a compact Lie group and A a finite G-CW-complex.
Then the map

    [A,c]^G :  [A,^]^G  →   [A, Ω^∙]^G

is a monoid homomorphism. If all isotropy groups of A are finite, then
[A,c]^G is a group completion of abelian monoids.


To show that [A,c]^G is a monoid homomorphism we exhibit a `delooping' of c,
namely the eigenspace morphism (<ref>).
In (<ref>)
we defined a morphism of ultra-commutative monoids
β:^→Ω that is a global group completion 
by Theorem <ref>.
Now we link the monoid homomorphism

    [A,β]^G :  [A,^]^G → [A,Ω]^G

to the set map [A,c]^G:[A,^]^G →[A, Ω^∙]^G.
The global equivalence

    λ̃_ :   → Ω( )

was defined in (<ref>).
The adjunction isomorphisms

    (Ω^∙ (Ω()))(V)    =  _*(S^V,Ω(V⊕))
        _*(S^V⊕, (V⊕)) =  (_⊕(Ω^∙))(V)

provide an isomorphism of orthogonal spaces between
Ω^∙ (Ω()) and
_⊕(Ω^∙), where _⊕=_⊕^
is the additive shift defined in Example <ref>.
Under this isomorphism, the morphism

    Ω^∙λ̃_ :  Ω^∙ →Ω^∙(Ω())

becomes the morphism

    (Ω^∙)∘ i  :  Ω^∙ → _⊕(Ω^∙)

given by precomposition with the embeddings i_V:V→ V⊕.
This morphism is a global equivalence by Theorem <ref>.
So the morphism Ω^∙λ̃_ is a global equivalence
of orthogonal spaces as well.

The rigorous statement of the delooping property of the eigenspace morphism 
is the following commutative square of orthogonal spaces:

    @C=12mm^[d]_β[r]^-c    Ω^∙[d]^Ω^∙λ̃__≃
    Ω[r]_-Ω   Ω (Ω^∙ ())

This square induces a commutative square of set maps

    @C=15mm
    [A, ^]^G [d]_[A,β]^G[r]^-[A,c]^G    
    [A,Ω^∙]^G[d]^[A,Ω^∙λ̃_]^G_
    
    [A,Ω]^G [r]_-[A,Ω]^G    
    [A,Ω( Ω^∙ ())]^G

The set [A,Ω]^G can be endowed with a monoid structure 
in two ways, via the ultra-commutative multiplication
as in (<ref>), and by concatenation of loops. 
Since the ultra-commutative monoid structure of Ω 
is `pointwise' (i.e., induced by the ultra-commutative monoid structure of ),
these two monoid structures satisfy the interchange law,
so they coincide, and both are abelian group structures. 
The morphism β is a homomorphism of ultra-commutative monoids,
so it induces an additive map on [A,-]^G.
The morphisms Ω and Ω^∙λ̃_
are loop maps, so they induce homomorphisms with respect to the
group structure by concatenation of loops. This shows that three of the four
maps in (<ref>) are homomorphisms of abelian monoids.
Since the right vertical map is an isomorphism, the map 
[A,c]^G is a homomorphism as well.

For the rest of the proof we suppose that the isotropy groups of A are finite.
The morphism Ω is a -global equivalence
by Theorem <ref>, so the map [A,Ω]^G
is bijective, and hence an isomorphism of abelian groups,
by Proposition <ref> (ii).
The morphism Ω^∙λ̃_:Ω^∙→Ω (Ω^∙())
is a global equivalence of orthogonal spaces,
so the map [A,Ω^∙λ̃_]^G is bijective, 
and hence an isomorphism of abelian groups,
again by Proposition <ref> (ii).
Since [A,β]^G is a group completion of abelian monoids
(by Corollary <ref>),
the commutative square (<ref>) 
shows that [A,c]^G is a group completion.



We draw an important consequence of Theorem <ref>,
namely that the equivariant cohomology theory represented by the 
connective global K-theory spectrum  is essentially equivariant K-theory.
There is a caveat, however, as this is not true on arbitrary
finite G-CW-complexes, but only under the hypothesis of finite stabilizer groups.

We let G be a compact Lie group and A a compact G-space. 
We define the 0-th G-equivariant -cohomology group of A as

    ^0_G(A_+) =  [A,Ω^∙]^G  ,

the equivariant homotopy set into the orthogonal space Ω^∙.
This set is an abelian group by concatenation of loops,
i.e., via the adjunction bijection

    [A,Ω^∙]^G   π_0^G((A,))

to an equivariant stable homotopy group. 
The set also has a multiplication, i.e., another
commutative binary operation arising from the ring spectrum structure
of , which turns Ω^∙ into a `multiplicative'
ultra-commutative monoid 
as in Example <ref>. 
A conjugation involution of the ultra-commutative ring spectrum 
was defined 
in Construction <ref>.subjectcomplex conjugation!on  

We denote by _G(A)symbol_G(A) - equivariant K-group of Asubjectequivariant K-theory
the G-equivariant K-group of A, 
i.e., the group completion (Grothendieck group) of the abelian monoid
of isomorphism classes of complex G-vector bundles over A.
The map 

    - :  [A,^]^G  → _G(A)  , 
       [f:A→ G r^(V)] ⟼ [f^⋆(γ_V^)]

that takes the pullback of the tautological vector bundles
over Grassmannians is a group completion of abelian monoids by
Theorem <ref> (or rather its complex analog).
Since [A,c]^G:[A,^]^G→ [A,Ω^∙]^G
is a monoid homomorphism  (by Theorem <ref>)
to an abelian group, the universal property of group completion
provides a unique additive extension 

    [-] :  _G(A) → _G^0(A_+)

such that the composite

    [A,^]^G   -  _G(A)  [-]  _G^0(A_+) 
     =  [A,Ω^∙]^G

is the map induced by the morphism of orthogonal spaces c:^→Ω^∙.


subjectequivariant K-theory|(
 
Let G be a compact Lie group and A a finite G-CW-complex.


  * For every G-representation V and every 
continuous G-map f:A→ G r^((V_)), the homomorphism [-]
sends the class of the G-vector bundle f^⋆(γ^_(V_))
to the homotopy class of the G-map

    A  [f(-);  -] _*(S^V,(V))  ,  
    a  ⟼ { v ⟼ [f(a);v] } .


  * The additive map [-] is a ring homomorphism, natural for G-maps in A,
natural for restriction homomorphisms in G,
and compatible with complex conjugation.

  * If A has finite isotropy groups, 
then the homomorphism [-] :_G(A)→_G^0(A_+) is an isomorphism.



(i)
We recall from Example <ref>
the multiplicative Grassmannian _^subjectmultiplicative Grassmannian
with values

    _^(V) =  ∐_n≥ 0  Gr_n^((V_))  ,

the disjoint union of all Grassmannians in the symmetric algebra of V_.
We let i:V_→(V_) be the embedding as the linear summand of the
symmetric algebra. Then as V varies, the maps

    i(V) :  ^(V) → _^(V) ,    L ⟼ i(L)

form a global equivalence i:^→_^ of orthogonal spaces,
see Example <ref>.
The morphism of orthogonal spaces
c :^→Ω^∙ defined in (<ref>)
has an extension

    c_ :  _^ → Ω^∙ ,

defined by the same formula as for c, namely

    c_(V)  :  _^(V)  → _*(S^V,(V)) =  (Ω^∙)(V) ,    
    L  ⟼ [L;-]

with

    [L;-] :  S^V  → ((V_),S^V) =   (V)    given by  
    v  ⟼ [L; v] .

The `pullback bundle' map -:[A,^]^G→_G(A)
also has a straightforward extension

    -_ : [A,^_]^G → _G(A)

again given by the same recipe: we send the class represented by
a G-map f:A→ G r^((V_)) to the class of the pullback
f^⋆(γ_(V_)^) of the tautological bundle over 
G r^((V_)).
The outer square in the diagram

    @C=15mm 
    [A,^]^G[r]^--[d]_[A,i]^G^   _G(A)[d]^[-]
    
    [A,^_]^G[r]_-[A,c_]^G[ur]^(.4)-_   [A,Ω^∙]^G

commutes because c_∘ i=c and [A,c]^G=[-]∘-.
The upper left triangle commutes because 
the tautological bundle on ^_(V)
restricts to the tautological bundle on ^(V)
along the map i(V):^(V)→^_(V).
The left vertical map is bijective
by Proposition <ref> (ii), 
because i:^→_^ is a global equivalence.
So the lower right triangle commutes as well.
The map

    [A,c_]^G :  [A,^_]^G  → 
    [A, Ω^∙]^G  =  ^0_G(A_+)

sends the homotopy class of a continuous G-map f:A→ G r^((V_)) 
to the class of the map [f(-);-]:A→_*(S^V,(V)), 
by the very definition of c_.
So this proves the claim.

(ii) 
Naturality in G-maps is straightforward.
For a G-map h:A'→ A and every G-map f:A→^(V),
the two G-vector bundles h^⋆(f^⋆(γ_V^))
and (f h)^⋆(γ_V^) over A' are isomorphic, 
so the following square commutes:

    @C=10mm 
    [A,^]^G [r]^--[d]_[h,^]^G   _G(A)[d]^_G(h)
    
    [A',^]^G [r]_--   _G(A')

Together with the defining property of the homomorphism [-], 
this implies that the two group homomorphisms

    [-]∘_G(h) ,  _G^0(h_+)∘ [-] :  _G(A) → _G^0(A'_+)

coincide after precomposition with -:[A,^]^G→_G(A).
Since this map is a group completion of abelian monoids, already
the homomorphisms [-]∘_G(h) and _G^0(h_+)∘ [-] agree, 
by the universal property of group completions.

The compatibility with complex conjugation and restriction along
group homomorphisms follow the same pattern.
The conjugation morphism of orthogonal spectra ψ:→
deloops the conjugation morphism of ultra-commutative monoids ψ:^→^,
in the sense that the square of orthogonal spaces commutes:

    ^[d]_ψ[r]^-c     Ω^∙[d]^Ω^∙ψ
    ^[r]_-c     Ω^∙

In particular, the homomorphism [A,c]^G:[A,^]^G→_G^0(A_+)
commutes with complex conjugation.
On the other hand, for every G-map f:A→^(V),
the bundle (ψ(V)∘ f)^⋆(γ_V^) is isomorphic
to the complex conjugate of the bundle f^⋆(γ_V^).
So the following square commutes:

    @C=10mm 
    [A,^]^G [r]^--[d]_[A,ψ]^G   _G(A)[d]^ψ
    
    [A,^]^G [r]_--   _G(A)

Thus the two group homomorphisms

    [-] ∘ψ ,  ψ∘ [-] :  _G(A) → _G^0(A_+)

agree after precomposition with a group completion, hence they coincide.
The analogous argument works for restriction
along a continuous homomorphism α:K→ G,
using that the underlying K-vector bundle of f^⋆(γ_V^)
equals the bundle α^*(f)(γ_α^*(V)^), and 
the effect of c (being a morphism of orthogonal spaces) commutes with restriction.

Now we show that the additive map [-]:_G(A)→_G^0(A_+)
is a ring homomorphism. The multiplicative unit of _G(A) is the class
of the trivial line bundle A×. 
This bundle is isomorphic to f^⋆(γ_(0)^) for
the constant map f:A→ G r^((0)) with value ,
the constant (and only non-trivial) summand in the symmetric algebra associated 
to the 0-dimensional G-representation.
The associated G-map [f(-);-]:A→_*(S^0,(0))
is constant with image the unit map ι_0:S^0→(0);
so by part (i) the class A× of the trivial line bundle
is the multiplicative unit in the ring _G^0(A_+).
So the map [-] preserves multiplicative units.

It remains to show that the map [-] preserves products.
Because [-] is additive and the group _G(A) is generated by
classes of actual vector bundles (as opposed to virtual  bundles),
it suffices to show multiplicativity for two classes in _G(A) represented 
by G-vector bundles. We may assume that the two bundles are classified by
continuous G-maps

    f  :  A  → G r^(V_)    respectively  
     g  :  A  → G r^(W_)  ,

where V and W are G-representations.
The tensor product bundle f^⋆(γ^_V)  g^⋆(γ^_W) 
is then classified by the composite

    A   (f,g)  G r^(V_)× G r^(W_)    -   - G r^(V_ W_) 
       G r^(j) G r^(( (V⊕ W)_))  ,

where 

    j  :  V_ W_ → ((V⊕ W)_)    is defined by  
    j(v w)=(v,0)·(0,w) .

By part (i) the associated homotopy class 
[f^⋆(γ^_V)  g^⋆(γ^_W)]
is represented by the G-map 

    [ (j∘ (f g))(-);-]  :  
    A → _*( S^V⊕ W,(V⊕ W)) .

This map is the adjoint of the composite 

    A_+ S^V⊕ W a v w↦ [f(a);v] [g(a);w] (V)(W) μ_V,W (V⊕ W) .

This composite represents the product
of the classes [f^⋆(γ_V^)] and [g^⋆(γ_W^)], 
so this establishes the relation

    [f^⋆(γ_V^)] · [g^⋆(γ_W^)] =   
    [f^⋆(γ^_V)  g^⋆(γ^_W)]

in the group _G^0(A).

(iii) If A is a finite G-CW-complex with finite isotropy groups,
then the map [A,c]^G:[A,^]^G →^0_G(A_+) is a group completion
of abelian monoids by Theorem <ref>.
The map -: [A,^]^G→_G(A) is also a group completion
(by Theorem <ref>, or rather its complex analog),
so the unique extension [-]:_G(A)→^0_G(A_+) is an isomorphism.

subjectequivariant K-theory|)



We specialize Theorem <ref> 
to the case A=∗, i.e., when the base is a single point.
In this case the bundle projection is no information,
G-vector bundles specialize to G-representations,
and the ring _G(∗) becomes the unitary representation ring (G).subjectrepresentation ring!unitary
On the other hand, _G^0(S^0) specializes to π_0^G().
The ring homomorphism

    [-] :  (G) → π_0^G()

is then determined by its effect on the classes
of actual representations. If W is a unitary G-representation,
we get an explicit representative for the class [W] by choosing
a G-equivariant -linear isometric embedding j:W→ V_
into the complexification of an orthogonal G-representations.
For example, the map

    j_W  :  W  → ( u W)_ ,    
    j_W(w)  ⟼ 1/√(2)· ( 1 w - i(i w)  )

into the complexification of the underlying orthogonal G-representation of W
does the job.
Then [W] is the homotopy class of the G-map 

    S^V → ((V_),S^V)  =   (V)
     ,    v  ⟼   [j(W);  v ]  .

Both (G) and π_0^G() have restriction maps,
transfers and multiplicative power operations in G, 
i.e., they are global power functors
in the sense of Definition <ref>.
The maps [-]:(G)→π_0^G() preserve
most of this additional structure, but that is not apparent from
what we discussed so far. 



For every compact Lie group G, the map

    [-] :  (G)  → π_0^G()

is a ring homomorphism.
As G varies over all compact Lie groups, the
homomorphisms (<ref>)
are compatible with restriction maps, with complex conjugation, 
with finite index transfers, and with multiplicative power operations.
Moreover, the map [-] is an isomorphism whenever the group G is finite.


The fact that the map is a ring homomorphism, compatible with restrictions,
compatible with complex conjugation
and an isomorphism for finite groups is a special case of 
Theorem <ref> for a one-point G-space.

Now we show that the maps [-] are compatible with finite index transfers.
We let H be a finite index subgroup of G and introduce 
an orthogonal G-spectrum [G/H] by

    [G/H](V)  =  ((V_), S^V (G/H)_+)   .

The structure maps of [G/H] are defined in much the same way as for ,
with the extra smash factor (G/H)_+ acting as a dummy;
the G-action comes entirely from the translation action on G/H.
As V varies over all inner product spaces, the assembly maps

    (V) (G/H)_+ =     ((V_),S^V) (G/H)_+ 
       → ((V_),S^V (G/H)_+)  =  [G/H](V)

form a morphism of orthogonal G-spectra α: (G/H)_+→[G/H].

We let ∇:(G/H)_+→ 1_+ be the G-equivariant fold map
that sends all of G/H to 1. We let l:G/H_+→ 1_+ be the
H-equivariant `projection' onto the preferred coset, i.e., l(e H)=1 
and l(g H)=0 for g∉H.
Then the following diagram commutes:

    @C=11mmπ_0^H()    π_0^H( G/H_+)[d]^π_0^H(α)[l]^-π_0^H( l)   π_0^G( G/H_+)[d]^π_0^G(α)[l]^-_H^G[r]^-π_0^G(∇)@<-.4ex>@/_1pc/[ll]_(.7)_H^G   π_0^G() 
       π_0^H([G/H])@/^1pc/[ul]^π_0^H([l])   π_0^G([G/H]) [l]^^G_H@/_1pc/[ur]_π_0^G([∇])

The upper left horizontal composite
is the Wirthmüller map (<ref>).
Since the Wirthmüller map is an isomorphism 
(Theorem <ref>),
the map π_0^H([l])∘^G_H is surjective.
Now we let f:S^V→[G/H](V) be a G-map that represents an element
in the kernel of the map π_0^H([l])∘^G_H,
where V is a G-representation.
After increasing V, if necessary, 
we can assume that V is ample and that the composite

    S^V  f  [G/H](V) [l](V) (V)

is H-equivariantly null-homotopic. By adjointness, this means that the composite

    S^V  f  [G/H](V)    =  ((V_);S^V (G/H)_+)
        l^♭  ^H(G,((V_);S^V)) =  ^H(G,(V))

is G-equivariantly null-homotopic,
where l^♭ is adjoint to the H-equivariant map

    ((V_);S^V l) :  ((V_);S^V (G/H)_+) →((V_);S^V)  .

The map l^♭ coincides with the Wirthmüller map 

    ω_S^V : ((V_);G⋉_H S^V) → ^H(G,((V_);S^V))

defined in (<ref>),
up to the effect of the shearing isomorphism S^V (G/H)_+ G⋉_H S^V.
Since V is ample, the -G-space
((V_)) is special by Theorem <ref> (i), 
and it is G-cofibrant by Example <ref>.  
So the Wirthmüller map ω_S^V, and hence also the map l^♭,
is a G-weak equivalence by Theorem <ref> (ii).
Hence already f is G-equivariantly null-homotopic. 
Altogether this proves that the map
π_0^H([l])∘^G_H is also injective, hence bijective.
Since π_0^H([l])∘^G_H and the Wirthmüller map
are both bijective, the commutativity of (<ref>) 
shows that the map π_0^G(α):π_0^G((G/H)_+)→π_0^G([G/H]) 
is also bijective.

Now we let W be any unitary H-representation
and j:^H(G,W)→ V_ a G-equivariant -linear embedding into
the complexification of some G-representation.
We define a class

    [W]_H^G  ∈ π_0^G([G/H])

by specifying a representative G-map

    S^V  → ((V_), S^V (G/H)_+)
     =  [G/H](V)

by

    v ⟼ [j(^H(H g_1^-1,W)),…,j(^H(H g_m^-1,W));  v g_1 H,…,v g_m H]  .

Here g_1,…,g_m is a set of coset representatives for H in G.
The relation

    ((V_), S^V l_+)
    [j(^H(H g_k,W)); v g_k H   ]_k=1,…,m
       = [j(^H(H,W));  v]

shows that

    π_0^H([l])(^G_H([W]_H^G)) =  [W]  .

The commutativity of the left part of diagram (<ref>) then shows that

    [W] =  π_0^H([l])(^G_H([W]_H^G))  =  _H^G(π_0^G(α)^-1( [W]_H^G))  .

Since the Wirthmüller map is inverse to the external transfer
(Theorem <ref>),
this is equivalent to the relation

    [W]_H^G  =  π_0^G(α)(G⋉_H [W] )

in the group π_0^G([G/H]).

On the other hand, if we fold S^V (G/H)_+ onto S^V, then

    ((V_), S^V∇_+)
       [^H(H g_1,W),…,^H(H g_m,W);  v g_1 H,…,v g_m H] 
       = [^H(H g_1,W),…,^H(H g_m,W);  v,…,v] 
       = [^H(H g_1,W)⊕…⊕^H(H g_m,W);  v] 
       = [^H(G,W);  v]  .

This shows that

    π_0^G([∇])([W]_H^G) =  [^H(G,W)]  .

The commutativity of the right part of diagram (<ref>) 
then yields the desired relation for classes of actual representations:

    [^H(G,W)] =  π_0^G([∇])([W]_H^G)     =  π_0^G([∇])(π_0^G(α)(G⋉_H [W] )) 
       =  π_0^G(∇)(G⋉_H [W])  =  _H^G[W]  .

Since transfer maps are additive,
the relation persists to classes of virtual representations.

Now we treat the compatibility with power operations,
i.e., that the following square commutes for every compact Lie group G
and all m≥ 1:

    @C=20mm(G)[r]^-[-][d]_P^m   π_0^G()[d]^P^m
    (Σ_m≀ G) [r]_-[-]   π_0^Σ_m≀ G()

We consider a unitary G-representation W and a G-equivariant
-linear isometric embedding j:W→ V_ into the complexification
of an orthogonal G-representation.
The class [W] is then represented by the G-map

    [j(W);-] :  S^V → (V) .

The map

    J  :  W^ m     →    (V^m_) 
    
    w_1⋯ w_m    ⟼  
    (j(w_1),0,…,0)· (0,j(w_2),0,…,0)·… · (0,…,0,j(w_m))

is a (Σ_m≀ G)-equivariant -linear isometric embedding.
So the class [W^ m]=[P^mW]
is represented by the G-map

    [J(W^ m);-]  :  S^V^m → ( V^m)  .

This map coincides with the composite

    S^V^m [j(W);-]^ m(V)^ m μ_V,…,V (V^m)
 
which represents the power operation P^m[W], so we have shown the relation

    P^m[W] =  [W^ m] =  [P^mW]

in the group π_0^Σ_m≀ G(  ).
Since the map [-] is additive, compatible with finite index transfers
and the classes of actual representations generate (G) as an abelian group, 
the additivity formula for power operations 

    P^m(x+y)  =  ∑_k=0^m _k,m-k (P^k(x)× P^m-k(y))

implies that the map [-] is compatible with
power operations of virtual representations.




[Dimension homomorphism]
The dimension homomorphism issubjectdimension homomorphism 
a homomorphism of ultra-commutative ring spectra

    :   → S p^∞

from the connective global K-theory spectrum to the infinite symmetric product spectrum
defined in Example <ref>.subjectinfinite symmetric product spectrum
The value of this homomorphism at an inner product space V is the map

    (V) :  (V)=((V_),S^V)     →   S p^∞(S^V)  
    
    [E_1,…, E_n;  v_1,…,v_n]    ⟼ ∑_i=1^n (E_i)· v_i ,

i.e., a configuration of vector spaces is mapped to the configurations
of its dimensions (i.e., the sum in the abelian monoid structure of S p^∞(S^V)). 
The map is multiplicative because dimension
is multiplicative on tensor products.  

The infinite symmetric product spectrum includes 
by a global equivalence of ultra-commutative ring spectra S p^∞→
into the Eilenberg-Mac Lane spectrum
of the integers,subjectEilenberg-Mac Lane spectrum!of the integers
see Proposition <ref> (ii).
So we sometimes take  (instead of S p^∞) as the target of
the dimension homomorphism.


We record that the effect of the dimension homomorphism 
on equivariant homotopy groups `is' the augmentation

    :  (G) →

that sends a virtual representation to its virtual dimension.
We warn the reader that the two vertical maps in
the following commutative square are isomorphisms for finite groups,
but not for general compact Lie groups.



For every compact Lie group G, 
the following square of commutative rings commutes:

    @C=15mm(G)[r]^-[d]_[-]   [d]^n↦ n· 1
    π_0^G()[r]_-π_0^G()   π_0^G(S p^∞)



For n≥ 1 we consider the `geometric' degree n morphism

    δ^n  :   → S p^∞ ,    x  ⟼ n· x .

In level 0, the map

    δ^n(0) :  S^0  → S p^∞(S^0)  {0}

sends the non-basepoint 0∈ S^0 to n· 0, the n-fold
multiple of the corresponding generator of the free abelian monoid {0}.
Since the ring homomorphism →π_0^G(S p^∞)
sends n to the class of n· 0, this shows that
δ^n_*(1)= n· 1 in the group π_0^G(S p^∞).

Now we let W be a unitary G-representation of dimension n.
A representative [j_W(W);-] for the class [W] in (G)
was specified in (<ref>). The composite 

    S^u W [j_W(W);-] (u W) (u W)  S p^∞(S^u W)

is the map 

    δ^n(u W) : S^u W→ S p^∞(u W) .

Hence we conclude that

    π_0^G()[W]  =  [δ^n(u W)] =  δ^n_*(1) =  n· 1 
    =  (W)· 1 .

This proves the proposition for the classes of actual representations.
Since all four maps in the square are homomorphisms of abelian groups,
the relation also holds for the classes of virtual representations.



 

Segal <cit.>
discusses a `smooth transfer' i_!:(H)→(G)
where i:H→ G is the inclusion of a closed subgroup of a compact Lie group G.
This smooth transfer is not to be confused with the holomorphic transfer, 
which is defined when G/H is a complex algebraic variety. 
One could hope that the maps [-]:(G)→π_0^G (  ) take the smooth transfer
to the homotopy theoretic transfer 

    _H^G  :  π_0^H (  )  → π_0^G (  )  .

This (false!) expectation is suggested by the fact that the smooth
transfer has all the right formal properties, and the complex representations
rings do form a global functor.
As we now illustrate by a specific example,
the collection of ring homomorphisms [-]:(G)→π_0^G (  ) 
comes close to being a morphism of global functors, but it fails to commute with
infinite index transfers.
For periodic global K-theory , 
discussed below in Construction <ref>, 
the two transfers do correspond also for infinite index inclusions;
in fact, _0 (  ) is isomorphic, as a global power functor,
to the representation ring global functor,
see Theorem <ref> below.

To illustrate the issue we recall a specific smooth transfer
to G=S U(2) from the normalizer N=N_S U(2)T of the diagonal maximal torus

    T  =  {( [    λ    0;    0 λ^-1 ])
     :  λ∈ U(1) } .
subjectspecial unitary group!S U(2)
The maximal torus normalizer N is generated by T and the matrix  
([ -0  1; -1  0 ]).
We use Segal's character formula <cit.>
to calculate the character of the smooth transfer i_!(1) of the trivial
1-dimensional N-representation, where i:N→ S U(2) is the inclusion. 
An element g∈ S U(2) is regular 
in the sense of <cit.>
if and only if it has infinite order, and then the closed subgroup
generated by g is a conjugate ^γ T of T.
The fixed set of such a regular element g on G/N then
consists of a single coset, namely γ N. So

    χ_i_!(1)(g) =  χ_1(γ^-1gγ) =  1  .

Since the character is continuous and regular elements are dense,
the character of i_!(1) is constant with value 1.
Since the character determines the representation up to isomorphism, we have
i_!(1) = 1 in (S U(2)).
We claim that in contrast to this relation in (S U(2)),
the elements _N^S U(2)(1) and 1 are linearly independent 
(so in particular different) in π_0^S U(2) (  ). 
We can detect this through the 
dimension homomorphismsubjectdimension homomorphism
: → introduced in Example <ref>.
As we discussed in Example <ref>,
the elements _N^S U(2)(1) and 1 are linearly independent in π_0^S U(2)(H),
so they must also be linearly independent in π_0^S U(2) (  ).

The dimension homomorphism can also be used to detect some
odd-dimen­sional classes in the coefficient ring π_*^U(1)():
we showed in Theorem <ref> that the group π_1^U(1)()
is isomorphic to , and that the dimension shifting transfer
from the trivial group to U(1), applied to the suspension of the multiplicative unit
1∈π_0^e(), is non-zero in π_1^U(1)().
So the class

    _e^U(1)(1 S^1)  ∈ π_1^U(1)()

has infinite order, and is in particular non-zero.










There is a  morphism of orthogonal spaces
c : →Ω^∙
defined in much the same way as its complex analog (<ref>).
Derived from this are ring homomorphisms

    [-] :  _G(A) → _G^0(A_+)

for every compact G-space A, analogous to the homomorphism (<ref>) 
in the complex case.subjectequivariant K-theory
For A=∗ this specializes to a ring homomorphism

    [-] :  (G) → π_0^G (  )  ,

where (G) is the orthogonal representation ring of G.subjectrepresentation ring!orthogonal
The analog of Theorem <ref> then says that these homomorphisms are
compatible with restriction, finite index transfer and multiplicative power operations, 
and an isomorphism for finite groups G.
However, in the real situation there is no 
eigenspace decomposition, hence no direct analog of
the delooping :→Ω^∙() of the morphism c.
So to establish the real analog of Theorem <ref>
(which is used in the proof of Theorem <ref>),
one has to use a different proof.



[Rank filtrations]subjectrank filtration!of subjectrank filtration!of 
The connective global K-theory spectra come with exhaustive filtrations 

    ^[1]→^[2]→…→^[m]→… →

respectively

    ^[1]→^[2]→…→^[m]→… →

by orthogonal subspectra.
We define the rank filtration for , the case of  being similar.
For m≥ 1  we let

    ^[m](V) =  ^[m]((V_),S^V)  ⊂ ((V_),S^V) =  (V)

be the subspace 
of those configurations [E_1,…, E_n; v_1,…,v_n]
such that

    ∑_i=1^n  (E_i) ≤ m  .

As V varies, the spaces ^[m](V) form an 
orthogonal subspectrum ^[m] of .

The dimension function is multiplicative on tensor products, so
the multiplication of  restricts to associative and unital pairings

    ^[m]^[n] → ^[m n] .

For m=1 this gives ^[1] the structure 
of an ultra-commutative ring spectrum 
and it gives ^[n] a module structure over ^[1].
The inclusion ^[1]→ is multiplicative, i.e., a morphism
of ultra-commutative ring spectra.

The first pieces ^[1] and
^[1] are multiplicative models for the suspension spectra
of global classifying space 
of the cyclic group C_2=O(1) respectively the circle group U(1).
Indeed, the complex global projective space ^=_^,[1],
introduced in (<ref>), is a multiplicative model of the 
global classifying space of U(1).
A configuration of points labeled by vector spaces 
of total dimension 1 has to be concentrated on at most one point.
So the map

    S^V^(V)_+ =  S^V P((V_))_+    → ^[1]((V_),S^V)
     =  ^[1](V)
    
     v L        ⟼   [L; v]

is a homeomorphism. 
As V varies through real inner product spaces,
these maps form an isomorphism of ultra-commutative ring spectra

    Σ^∞_+ ^  ^[1] .

subjectglobal projective spacesymbol - global projective space
Similarly, the ultra-commutative ring spectrum ^[1] 
is globally a suspension spectrum of a global classifying space 
of the group C_2.

The underlying non-equivariant spectrum of Σ^∞_+ B_ U(1),
hence of ^[1], 
has the stable homotopy type of the suspension spectrum of P^∞; 
moreover, on underlying non-equivariant spectra,  the morphism 

    ^[1] →

is a rational stable equivalence.
However, the morphism (<ref>) 
is not a rational global equivalence, which can be seen already
on the level of _0 for the group C_2.
Indeed, by Proposition <ref>  
the homotopy group global functor _0( Σ^∞_+ B_ U(1) ),
and hence also the global functor _0( ^[1]),
is the represented global functor ( U(1),-). The inclusion 
^[1]→ induces a morphism of global power 
functors (U(1),-) _0( ^[1] )→_0(  ).
An element of infinite order in the kernel of the map 
(U(1),C_2)→π_0^C_2() is 

    _e^C_2∘^U(1)_e - z^* - ^U(1)_C_2 ∈ (U(1),C_2)  ,

where z:U(1)→ C_2 is the trivial homomorphism.
This element maps trivially to _0() because
the regular representation of C_2 splits as the sum of the 1-dimensional
trivial and sign representations.




For finite groups G, 
the ring homomorphism [-]:(G)→π_0^G() is an isomorphism.
It will follow from Theorem <ref> 
below that the map [-]:(G)→π_0^G() is always a split monomorphism,
also when G has positive dimension.
On the other hand, Remark <ref>
shows that the map is not always surjective, as the class
_N^S U(2)(1) in π_0^S U(2)() is not in the image.
In <cit.>, Hausmann and Ostermayr give a complete
calculation of _0() as a global functor.subjectspecial unitary group!S U(2)
The strategy of <cit.> is to identify the 
global homotopy types of the subquotients of the rank filtration 
and deduce from it a presentation _0() by generators and relations.
The final answer is that _0() is generated as a global
functor by the classes

    x_m  =  [τ_m] ∈ π_0^U(m)()  ,

where τ_m denotes the tautological U(m)-representation on ^m.
There are two kinds of relations; on the one hand, the relations

    ^U(m+n)_U(m)× U(n)(x_m+n)  =  p^*(x_m) + q^*(x_n)
      ∈π_0^U(m)× U(n)()

for all m,n≥ 1, where p:U(m)× U(n)→ U(m)
and q:U(m)× U(n)→ U(n) are the two projections.
These relations follow from the fact that the maps [-]
are additive and compatible with restrictions.
The other set of relations equates finite index transfers
of representations with the corresponding finite index transfer in
homotopy theory.




[Bott class]subjectBott class!in connective K-theory
The Bott class

    β ∈ π_2^e()

is an important non-equivariant homotopy class of the spectrum 
that we recall now. As we shall show in Theorem <ref>
below, the Bott class becomes invertible in the homotopy ring
of the periodic K-theory spectrum .symbolβ - Bott class in π_2^e()

We define the Bott class by specifying an explicit representative.
We define a continuous map m:S^⊕→ S U(2) by the formula

    m(v) =  m(x,z) =  1/|v|^2+1[ |v|^2-1 - i 2 x          2 i z̅;           2 i z |v|^2-1 + i 2 x ] .

The eigenspace decomposition 

    :   U(2)   (^2;S^1)

is the inverse of the homeomorphism (<ref>);subjecteigenspace decomposition
it sends a unitary matrix to the configuration of its eigenvalues,
with the inverse Cayley transform applied, 
labeled by the corresponding eigenspaces.
The Bott class is then represented by the composite

    S^3  =  S^⊕  m   U(2)    []   (^2;S^1)
       incl (();S^1) = ()  .

In the last step we identify ^2 with the subspace of () spanned
by the constant and linear summand in the symmetric algebra,
in terms of the preferred basis 1∈^0() and 1∈^1().





  The  group π_2^e() is infinite cyclic, 
  and the Bott class β is a generator.


The map m:S^⊕→ S U(2) is bijective;
indeed, an explicit formula for the inverse is 

    S U(2) → S^⊕ ,   [   a  b̅; - b  a̅ ] ⟼ Im(a)/Re(a)-1i b/1-Re(a) ,

where a,b∈^2 are complex numbers satisfying |a|^2+|b|^2=1.
As a continuous bijection between compact spaces, m is thus a
homeomorphism from S^3 to S U(2).
Since the inclusion S U(2)→ U(2) induces an isomorphism on π_3,
the map m represents a generator of the 
infinite cyclic group π_3(U(2),1).
The standard embedding U(2)→ U into the infinite unitary group is 4-connected, 
so the composite (<ref>)
represents a generator of the infinite cyclic group π_3((),∗).
Theorem <ref> says that  
is a positive Ω-spectrum (in the non-equivariant sense),
so in particular the stabilization map

    π_3((),∗)  =  [S^3,()] → _n≥ 0  [S^n+2,(^n)] =  π_2^e()

is an isomorphism. So the group π_2^e() is infinite cyclic, 
and the Bott class is a generator.



[Equivariant Bott classes]
There are more general equivariant Bott classessubjectBott class!of a ^c-representationsymbolβ_G,W - equivariant Bott class of a ^c-representation

    β_G,W ∈ ^0_G(S^W)

defined for G-^c-representations, i.e., 
hermitian inner product spaces W
equipped with a lift G→^c(u W) of the representation homomorphism G→ O(u W)
through the adjoint representation
(u W) :^c(u W)→ SO(u W), see (<ref>).
These equivariant Bott classes become
invertible in the R O(G)-graded homotopy ring of ,
compare Remark <ref> below;
this equivariant generalization of Bott periodicity goes back to
Atiyah <cit.>.subjectBott periodicity!equivariant 

We recall the definition as given, 
for example, in <cit.>.
The construction depends heavily on a canonical isomorphism,
for every hermitian inner product space W, 
between the complexified Clifford algebra 
_(u W) of the underlying euclidean vector space of W, 
and _(Λ^*(W)), the endomorphism algebra of the exterior algebra of W. 
The underlying -vector space u W of the given hermitian inner product space
has a euclidean inner product defined by

    w,w' =  Re (w,w') ,

the real part of the complex inner product.
In what follows, we use the hermitian inner product 
on the exterior algebra Λ^*(W) characterized by the formula

    ( v_1… v_n, w_1… w_n ) =  ( ( v_i,w_j )_i,j)

for all v_i,w_j∈ W. Another way to say this is that 
if (e_i)_i=1,…, k is an orthonormal basis of W, then the vectors

    e_i_1… e_i_n

form an orthonormal basis of Λ^*( W) 
as the indices run through all tuples with
1≤ i_1<…< i_n≤ k.

For w∈ W we let

    d_w : Λ^*(W) → Λ^*(W)  ,    d_w(x) =  w x

denote left exterior multiplication by w. 
We let d_w^∗:Λ^*(W)→Λ^*(W) denote the adjoint of d_w,
i.e., the -linear map characterized by

    (x, d^∗_w(y)) =  ( d_w(x) ,y )  =  ( w x ,y )

for all x,y∈Λ^*(W).

The exterior algebra Λ^*(W) is /2-graded by
even respectively odd exterior powers. The endomorphism
algebra _(Λ^*(W)) then inherits a /2-grading by
even respectively odd operators.
Exterior multiplication by w takes Λ^n(W) to Λ^n+1(W),
so the operator d_w is odd for every w∈ W. 
Hence the adjoint d^*_w is odd, and so is the endomorphism

    δ_w  =  d_w - d_w^∗ .





Let W be a hermitian inner product space.
For every w∈ W, the endomorphisms d_w and δ_w 
of Λ^*(W) satisfy the relations

    d_w∘ d_w^* + d_w^*∘ d_w  =  |w|^2·     and    δ_w∘δ_w =  -|w|^2· .



If we multiply w by a real scalar λ∈, then both sides
of both equations scale by λ^2. So it suffices to show the claims
for unit vectors, i.e., we may assume that |w|=1.

If w is a unit vector, then Λ^*(W) decomposes as 
an orthogonal direct sum Λ^*(W^⊥)⊕ (wΛ^*(W^⊥)),
where W^⊥ is the orthogonal complement of w in W.
Moreover, d_w is an isometry from the summand Λ^*(W^⊥) 
onto the summand wΛ^*(W^⊥), and it vanishes on wΛ^*(W^⊥).
So the adjoint d_w^* is inverse to d_w on the summand
 wΛ^*(W^⊥), and it vanishes on Λ^*(W^⊥).
Hence d_w^*∘ d_w is the orthogonal projection onto the summand Λ^*(W^⊥), 
and d_w∘ d_w^* is the orthogonal projection onto the other summand 
wΛ^*(W^⊥). This shows that 
d_w∘ d_w^* + d_w^*∘ d_w is the identity.

We have d_w∘ d_w=0, hence also d_w^*∘ d_w^*=0.
This gives 

    δ_w∘δ_w  =  (d_w -d_w^*)∘(d_w -d_w^*) =  
    - ( d_w∘  d_w^* + d_w^* ∘ d_w ) =  -  .




Because of the previous lemma, the -linear map

    u W  → _(Λ^*(W))  ,    w  ⟼ δ_w

satisfies δ_w^2 =  -|w|^2·;
so the universal property of the Clifford algebra provides a unique
homomorphism of /2-graded -algebras 

    δ_W :  _(u W)  → _(Λ^*(W))

sending w∈ W to δ_w. 
The homomorphism is also compatible with passage to adjoints;
indeed, for w∈ W we have

    δ_w^*  =  (d_w-d_w^∗)^∗ =  d^∗_w-d_w =  
    d_-w -d_-w^∗ =  δ_-w =  δ_w^∗ .

Since _(u W) is generated by the elements of u W, the compatibility
with adjoints holds in general.
The homomorphism δ_W thus makes the exterior algebra Λ^*(W)
into a /2-graded _(u W)-module.


[The complexified Clifford algebra of u]
We make the homomorphism δ_W explicit in the simplest non-trivial case,
namely for W=. In this case the elements 1, i∈ form
an orthonormal -basis of u; we let

    e  =  1 [1]     respectively     f  =  1 [i]

be their images in the complexified Clifford algebra _(u).
This Clifford algebra then has a -basis (1, e, f, e f),
with multiplicative relations

    e^2  =  f^2  =  -1      and     f e  =  - e f .

On the other hand, the exterior algebra Λ^∗() 
is two dimensional, and an orthonormal -basis is given by
the classes 1∈Λ^0() (the multiplicative unit)
and x∈Λ^1(), the class of 1∈.
A direct calculation shows that in terms of the basis (1,x)
of Λ^*(), the map δ_ is given by

    δ_e  = [  0 -1;  1  0 ] ,   δ_f  =  [ 0 i; i 0 ]   and  δ_e∘δ_f  =  [ - i   0;   0   i ] .

In particular, the homomorphism δ_ sends the basis (1,e, f, e f)
of _(u) to a basis of _(Λ^*()),
so δ_ is an isomorphism.


We recall that the homomorphism δ_W is in fact an isomorphism
in general. 
Indeed, given two hermitian inner product spaces V and W, the following
square commutes: 

    @C=12mm@R=7mm_(u V⊕ u W) [r]^-δ_V⊕ W[dd]_   _(Λ^*(V⊕ W)) [d]_
       _(Λ^*(V)Λ^*(W)) 
    
     (_(u V))_ (_(u W)) 
    [r]_-δ_Vδ_W   _(Λ^*(V))_(Λ^*(W)) [u]_^

The left vertical isomorphism sends 1 [v,w]
to (1 v) (1 1)+(1 1) (1 w).
The upper right vertical map is induced by the isomorphism 
of /2-graded -algebras

    Λ^*(V⊕ W)    Λ^*(V)Λ^*(W)

that extends the linear map

    V⊕ W → Λ^*(V)Λ^*(W)   ,   
    (v,w) ⟼ v 1 +  1 w .

The vertical maps in the above diagram are isomorphisms, and every 
hermitian inner product space is isometrically isomorphic
to ^n with standard inner product; so this reduces the
claim to the special case W=, in which case the morphism δ_
is an isomorphism by Example <ref>.



To construct the equivariant Bott class
we now start with a hermitian inner product space W and a continuous homomorphism
G→^c(u W).
The group G then acts on the underlying euclidean inner product space of W
via the adjoint representation (u W):^c(u W)→ S O(u W).
The equivariant Bott class most naturally lives in the
relative K-group _G(D(W), S(W));
elements in this group are represented by triples
(ξ,η,α), where ξ and η are G-vector bundles
over D(W), and α:ξ|_S(W)η|_S(W) is an 
equivariant isomorphism between the restrictions of the two bundles
to the unit sphere S(W).
In this description, the Bott class β_G,W is represented by the triple

    ( D(W)×Λ^(W), D(W)×Λ^(W),α)

consisting of the trivial vector bundles over D(W)
with fibers Λ^(W) respectively Λ^(W),
and the  equivariant bundle isomorphism α is given 
by the Clifford action (<ref>):

    α :  S(W)×Λ^(W)  →  S(W)×Λ^(W)  , 
       (w, x) ⟼ (w, δ_w(x)) .

Here the group G acts on Λ^(W) 
and Λ^(W) via the given ^c-structure, i.e., 
through the composite

    G  → ^c(u W) ⊂ _(u W)
     [] δ_W  _(Λ^*(W)) .

Equivariant Bott periodicity <cit.>subjectBott periodicity!equivariant 
says that for every compact G-space A, exterior product
with the Bott class is an isomorphism

    -×β_G,W  :  _G(A) → _G(A× D(W),A× S(W)) .

To represent the Bott class in _G^0(S^W)
we interpret the Clifford action as a clutching function for a vector bundle.
We define a G-vector bundle ξ(W) over S^W
by gluing the trivial bundle with fiber Λ^(W) over D(W)
with the trivial bundle with fiber Λ^(W) over S^W-  D(W)
using the map

    S(W)  → ^(Λ^(W),Λ^(W)) ,    
    w ⟼ δ_w .

The homomorphism

    [-]  :  _G(S^W) → _G^0(S^W_+)

defined in (<ref>) turns this G-vector bundle
into an unreduced equivariant -cohomology class.
By construction, the fiber of ξ(W) over the basepoint at infinity
is the G-representation Λ^(W),
with G acting via the composite (<ref>).  
So by subtracting the class of the trivial vector bundle with fiber 
Λ^(W) we obtain the equivariant Bott class as a reduced 
equivariant -cohomology class, 

    β_G,W =  [ ξ(W) ] - [ S^W×Λ^(W)]   ∈ _G^0(S^W)  .

We invite the reader to perform a reality check and establish the relation

    β_e, =  - β

in π_2^e(). In other words,
the equivariant Bott class of the trivial group acting on 
specializes to the non-equivariant Bott class 
as defined in Construction <ref>, up to a sign.


subjectK-theory@K-theory!connective global|)




§ PERIODIC GLOBAL K-THEORY
 


Our main object of study in this section is periodic global K-theory , 
an ultra-commutative ring spectrum whose G-homotopy type realizes 
G-equi­variant periodic K-theory, see Construction <ref>.
The model we use is due to M. Joachim <cit.>,
and made of spaces of homomorphisms of /2-graded C^∗-algebras.
Corollary <ref> shows that the 
equivariant cohomology theory represented by  on finite G-CW-complexes `is'
equivariant K-theory;
Theorem <ref> shows that _0() is isomorphic,
as a global power functor, to the complex representation ring functor .
Periodic global K-theory receives a morphism of ultra-commutative ring spectra
j:→ from connective global K-theory defined in the previous section.
Theorem <ref> shows that  is Bott periodic,
i.e., the Bott class in π_2^e() becomes invertible in π_2^e().

Global connective K-theory 
is a certain homotopy pullback of the periodic theory , 
its associated global Borel theory,
and the global Borel theory of connective K-theory, 
see Construction <ref>.
The global homotopy type ^c is a refinement 
of Greenlees `equivariant connective K-theory' <cit.>.
One should  note the different order of the adjectives `global' and `connective',
indicating that  and ^c are quite different global homotopy types
(with the same underlying non-equivariant homotopy type).




The spectrum  consists of spaces of homomorphisms 
of /2-graded C^∗-algebras.
Consequently, in this section we will use basic results 
from the theory of C^∗-algebras; the textbooks <cit.> 
can serve as general references.
Since we are not assuming the reader to be fluent with C^∗-algebras, 
we recall a certain amount of material in some detail.


[The C^∗-algebra s]
  The construction of the orthogonal spectrum  is based
  on a certain C^∗-algebra and on spaces of ∗-homo­morphisms out of it. 
  We consider graded C^∗-algebras, i.e., C^∗-algebras A
  equip­ped with a ∗-automorphism α:A→ A such that α^2=.
  We can then decompose A into the ± 1 eigenspaces of α
  and obtain a /2-grading of the underlying -algebra by setting
  
    A_ =  { a∈ A  |  α(a)=a}   and   
      A_ =  { a∈ A  |  α(a)=-a} .

  The conjugation of A preserves the grading into even and odd parts.

  We let s denote the C^∗-algebra of complex valued continuous functions on 
   vanishing at infinity; this is a /2-graded C^∗-algebra with involution 
  α:s→ s defined by
  
    α(f)(t) =  f(-t) .

  With respect to this grading, `even' and `odd' have their usual meaning:
  a function f∈ s is even (respectively odd)
  if and only if f(-t)=f(t) (respectively f(-t)=-f(t)) for all t∈.
  
  Graded ∗-morphisms out of s correspond to special elements
  in the target C^∗-algebra. The continuous function
  
    r(x)  =  2i/x-i :   →

  is an element of s that satisfies
  
    r r^∗ +  r  +  r^∗ =  0      and    α(r) =  r^∗ .

  The element r is not homogeneous; its even respectively odd components
  are given by
  
    r_+(x)  =  -2/x^2+1   respectively  
      r_-(x)  =  2 x i/x^2+1 .

  Moreover, s is the universal /2-graded C^∗-algebra generated by
  such an element, i.e., evaluation at r is a bijection
  
    C^∗_(s, A)   { x∈  A  :  x x^∗ = x^∗ x = - x - x^∗ , α(x)=x^∗}

  for every graded C^∗-algebra A.
  Indeed, after adjoining a unit to the algebra s and identifying S^1
  with U(1) via the Cayley transformsubjectCayley transform
  c:S^1 U(1) given by c(x)=(x+i)(x-i)^-1,
  the algebra ⊕ s becomes isomorphic to the unital C^∗-algebra C(U(1))
  of continuous -valued functions on U(1);
  this isomorphism takes 1+r to the inclusion z:U(1)→,
  which is a unitary element in C(U(1)).
  So the universal property of s
  follows from the fact that C(U(1)) is freely generated,
  as an ungraded, unital C^∗-algebra, by the unitary element z;
  indeed, unitary elements are in particular normal and have their spectrum
  contained in U(1), so the bijectivity of (<ref>) 
  becomes a special case of functional calculus for normal elements in unital
  C^∗-algebras, see for example <cit.>.

  In what follows,  denotes the spatial tensor productsubjectspatial tensor product!of C^∗-algebras
  of graded C^∗-algebras, see <cit.>, 
  <cit.> or <cit.>.
  The spatial tensor product is the completion of the algebraic tensor product
  with respect to the `spatial norm'; it is also called the minimal tensor product
  because the spatial norm is minimal among all C^∗-norms on
  the algebraic tensor product, compare <cit.>. 
  A comprehensive discussion of the spatial and other tensor products of
  C^∗-algebras can be found in <cit.>.
   We need the graded tensor product to /2-graded algebras, which involves
  a sign in the formula for multiplication.
  If  a,a'∈ A and b,b'∈ B  are homogeneous elements of two
  /2-graded algebras, then the multiplication in A B is defined by
  
    (a b)·(a' b') =  (-1)^|b||a'|· (a a') (b b') .

  In other words, if a∈ A and b∈ B are odd, then
  a 1 and 1 b anti-commute in the graded tensor product.

  

  The underlying algebra of the graded tensor of two 
  /2-graded algebras is not the tensor product of the underlying
  ungraded algebras, as soon as both factors have non-zero odd elements.
  For example, while the ungraded C^∗-algebra underlying s is commutative,
  the underlying C^∗-algebra of the  graded tensor product ss 
  is not commutative anymore.  In particular, ss 
  does not embed into the algebra of continuous functions on any space.

  

  The algebra s has another important piece of extra structure,
  namely a comultiplication Δ:s→ ss
  in the category of /2-graded C^*-algebras.
  The algebra s is also generated by the self-adjoint functions 
  
    u_+(t) = e^-t^2     and    
        u_-(t) = t· e^-t^2 ,

  and the comultiplication is completely determined by the values on these;
  indeed, there is a unique graded ∗-homomorphism 
  Δ:s→ ss that satisfies
  
    Δ(u_+) =  u_+ u_+    respectively  Δ(u_-) = u_- u_+ + u_+ u_-  ,

  see for example Lemma 1.3.1 and Remark 1.3.3 of <cit.>.
  This characterization readily implies
  the cocommutativity and coassociativity of Δ.
  The cocommutativity of Δ is with respect to the
  graded symmetry automorphism of ss,
  which involves a sign whenever two odd elements are interchanged.
  An explicit definition of the diagonal morphism Δ
  can be found in <cit.>.

  

  
[Complex Clifford algebras]
  We let V be a euclidean inner product space. We define the complex
  Clifford algebra (V) bysubjectClifford algebra|(symbol(V)@(V) - complexified Clifford algebra
  
    (V)  =  (T V)_ / ( v v - |v|^2· 1)  ,

  the quotient of the complexified tensor algebra of V
  by the ideal generated by the elements v v -  |v|^2· 1
  for all v∈ V.  We write [-]:V→(V) for the
  -linear and injective composite
  
    V  linear summand  T V 1-  (T V)_ → (V)   .

  With this notation the relation [v]^2=|v|^2· 1 holds in (V)
  for all v∈ V.
  The Clifford algebra construction is functorial for -linear isometric embeddings,
  so in particular (V) inherits an action of the orthogonal group O(V). 
  The Clifford algebra is /2-graded, coming from the grading of
  the tensor algebra by even and odd tensor powers.

  The complex Clifford algebra is in fact a /2-graded O(V)-C^∗-algebra.
  The ∗-involution on (V) is defined by declaring [v]^∗=[v]
  for all v∈ V and extending this to a -semilinear anti-automorphism. 
  This makes the elements [v] for v∈ S(V) into unitary elements of (V).
  The norm on (V) arises from an embedding
  into the endomorphism algebra of the exterior algebra Λ^*( V_).
  Indeed, we consider the -linear map
  
    V  → _(Λ^*(V_)) ,   
      v  ⟼ i·δ_1 v ,

  where δ_1 v was  defined in (<ref>).
  Lemma <ref> provides the relation
  
    (i·δ_1 v)∘ (i·δ_1 v)  =  
        - δ_1 v^2 =  |v|^2· .

  The universal property of (V) provides a morphism of /2-graded -algebras
  
    (V) → _(Λ^*(V_))

  that sends [v] to i·δ_1 v.
  This homomorphism is an embedding and compatible with passage to adjoints.
  The target is a C^∗-algebra via
  the operator norm, so we can endow the Clifford algebra with a C^∗-norm
  via this embedding. For this norm, the embedding 
  [-]:V→(V) is isometric. Indeed, 
  
    δ_1 v^*∘δ_1 v =  
      (d^*_1 v -d_1 v)∘(d_1 v - d_1 v^*) =  
       d_1 v∘  d_1 v^* + d_1 v^* ∘ d_1 v =  |v|^2· ,

  by Lemma <ref>, and so
  
    i·δ_1 v^2 
       =  δ_1 v^*∘δ_1 v =  |v|^2  .



  The Clifford algebra functor sends orthogonal direct sum to 
  graded tensor product, in the following sense.
  The -linear map
  
    V⊕ W  → (V)(W)  ,    
      (v,w)  ⟼ [v]  1 + 1 [w]

  satisfies
  
    ([v]  1 + 1 [w])^2  =  [v]^2 1 + 1 [w]^2  =  
      (|v|^2+|w|^2)· 1 1

  because the elements [v] 1 and 1 [w] anti-commute
  in (V)(W). The universal property then provides
  a unique extension to a morphism of graded -algebras
  
    μ_V,W :  (V⊕ W)   (V)(W)

  characterized by μ_V,W[v,w] =  [v]  1 + 1 [w]
  for all (v,w)∈ V⊕ W. Moreover, this morphism is an isomorphism,
  and the square 
  
    @C=15mm@R=6mm(V⊕ W)[d]_(τ_V,W)[r]^-μ_V,W   (V)(W) [d]^
    (W⊕ V)[r]_-μ_W,V   (W)(V)

  commutes, where the right vertical map is the symmetry isomorphism
  for /2-graded -algebras (which involves a sign whenever two
  odd degree elements interchange places).



  The complex Clifford algebra (V)
  is isomorphic to the complexification of the real Clifford algebra (V) considered 
  in the definition of the ultra-commutative monoids  and  in 
  Example <ref>. However, the isomorphism involves
  a slight twist that we want to make explicit.
  Indeed, the real Clifford algebra (V) was defined from the tensor algebra
  by imposing the relations v v=- |v|^2· 1, so that unit vectors of V
  square to -1 in (V). 
  Over the field  it makes a difference whether  we make
  v v equal to |v|^2· 1 or - |v|^2· 1, 
  but the presence of the imaginary unit makes the difference disappear over .
  Indeed, the -linear map
  
    ψ :  V   → (V)  ,    v ⟼ i· [v]

  satisfies ψ(v)^2=-|v|^2· 1, so it 
  extends to a morphism of -algebras _(V)→(V)
  by the universal property of the former, and this morphism is an isomorphism.subjectClifford algebra|)

  

  
  Another key piece of structure is a continuous based map
  
    :  S^V  →  C^∗_(s,(V))  ,    v ⟼ (-)[v]  ,

  often referred to as `functional calculus'.
  For v∈ V the ∗-homomorphism (v) is given on homogeneous elements
  of s by
  
    f[v] =  (v)(f) =     f(|v|)· 1     when f is even, and
     f(|v|)/|v|· [v]     when f is odd.

  For v=0 the formula for odd functions is to be interpreted as f[0]=0; 
  this is continuous because for v 0, 
  the norm of f(|v|)/|v|· [v] is f(|v|), which tends to f(0)=0 
  if v tends to 0. If the norm of v tends to infinity, then f(|v|) tends to 0, so 
  (v) tends to the constant ∗-homomorphism 
  with value 0, the basepoint of C^∗_(s,(V)).
  Hence  extends to a continuous
  map on the one-point compactification S^V. 
  The functional calculus map is O(V)-equivariant for 
  the O(V)-action on the mapping space C^∗_(s,(V))
  through the action on the target.
  The functional calculus maps are multiplicative in the sense that
  the following diagram commutes:
  
    @C=5mm@R=8mm
            S^V  S^W [dd]_[rr]^-      
            C^∗_(s,(V)) C^∗_(s,(W))[d]^
           C^∗_(ss,(V)(W)) 
            [d]^Δ^*
    
            S^V⊕ W[r]_-   
            C^∗_(s,(V⊕ W))[r]^-(<ref>)_-(μ_V,W)_*    
            C^∗_(s,(V)(W))

   To see this we consider v∈ V and w∈ W, chase the element v w
   both ways around the square and compare the results 
   on the two functions (<ref>) that generate the algebra s.
   Indeed, for the even generating function we have
   
    μ_V,W((v,w)(u_+))    =  μ_V,W( e^-|v|^2-|w|^2· 1) =  
         ( e^-|v|^2· 1) (e^- |w|^2· 1)
       = (v)(u_+)(w)(u_+) 
         =     ((v)(w))(u_+ u_+)
       =  ((v)(w))(Δ(u_+))  ;

   and similarly for the odd generating function:
   
    μ_V,W(   (v,w)(u_-)) =  μ_V,W(  e^-|v|^2-|w|^2· [v,w])
       =     e^- |v|^2-|w|^2· ([v] 1+1 [w])
       =  
          ( e^-|v|^2· [v]) (e^- |w|^2· 1) +    ( e^-|v|^2· 1) ( e^- |w|^2· [w])
       = (v)(u_-)(w)(u_+)  +  (v)(u_+)(w)(u_-)
       =     ((v)(w))(u_- u_+ + u_+ u_-) 
         =  ((v)(w))(Δ(u_-))  .


subjectK-theory@K-theory!periodic global|(
Now we have all ingredients for the periodic global K-theory spectrum .

[Periodic global K-theory]
We let V be a euclidean inner product space.
As we explained in Proposition <ref>,
the symmetric algebra (V_) of the complexification
inherits a hermitian inner product and an O(V)-action by -linear isometries.
The inner product space (V_) 
is usually infinite dimensional (unless V=0),
but it is not complete. We denote by _V 
the Hilbert space completion of (V_).
Since the action of O(V) on (V_)
is by linear isometries, it extends to an analogous
action on the completion _V.
So _V becomes a complex Hilbert space representation
of the orthogonal group O(V). 
We denote by _V the C^∗-algebra
of compact operators on the Hilbert space _V,
see for example <cit.>.subjectcompact operators!on a Hilbert space

The orthogonal spectrum  assigns to a euclidean inner product space V
the spacesymbol - periodic global K-theorysubjectK-theory@K-theory!periodic global

    (V)  =  C^∗_(s,(V)_V)

of /2-graded ∗-homomorphisms from s to
the tensor product, over , of (V) and _V.
Here we consider _V as evenly graded, so the grading comes
entirely from the grading of the Clifford algebra.
The topology is the topology of pointwise convergence in the 
operator norm of _V; the basepoint is the zero ∗-homomorphism.

The continuous action of the orthogonal group O(V) by linear isometries of _V 
induces an action on the algebra _V by conjugation,
and together with the action on (V) it gives 
an O(V)-action on (V)_V by graded ∗-auto­morphisms.
This induces an O(V)-action on the mapping space (V).


The multiplication of the spectrum  starts from the 
(O(V)× O(W))-equivariant isometric isomorphism

    (V_)(W_)     (V_⊕ W_)    ((V⊕ W)_)   ,

compare (<ref>).
This extends to an isometry of the Hilbert space completions

    _V_W  =  (V_)(W_)      ((V_)(W_))^∧
        ((V⊕ W)_)   =  _V⊕ W .

Tensor product of compact operators and conjugation with this isometry 
is an injective homomorphism of -algebras

    _V_W → ( _V_W)   (_V⊕ W) = _V⊕ W .

The source of this map is the algebraic tensor product, which is not complete; 
the homomorphism (<ref>) extends to an isomorphism of C^∗-algebras

    _V_W  _V⊕ W

from the spatial (minimal) tensor product; this is in fact tautologically
true, as _V acts on the Hilbert space _V and
so the spatial norm on the algebraic tensor product can be defined via
the embedding (<ref>).
Combining (<ref>) with
the isomorphism between (V)(W)
and (V⊕ W) specified in (<ref>),
we obtain an isomorphism of graded C^∗-algebras

    ((V)_V)((W)_W)     
    ((V)(W))(_V_W) 
        (V⊕ W)_V⊕ W .

The first step is the symmetry isomorphism,
which in this special case does not involve any signs because _V 
is concentrated in even grading.
The multiplication map

    μ_V,W :  (V)(W) → (V⊕ W)

is now defined as the composite

    C^∗_(s,(V)_V)    C^∗_(s,(W)_W)
          
    C^∗_(ss, ((V)_V)((W)_W)) 
       C^∗_(Δ, (<ref>))   
    C^∗_(s,(V⊕ W)_V⊕ W) .

These multiplication maps are associative and commutative.
The unit map 

    η_V  :  S^V  →  C^∗_(s,(V)_V)  =  (V)

is defined as the composite

    S^V     C^∗_(s,(V)) (- p_0)_* C^∗_(s,(V)_V) =  (V)  ,

where p_0∈_V is the orthogonal projection onto the
constant summand in the symmetric algebra. In other words, 

    η_V(v)(f) =  f[v] p_0 .

The multiplicativity of the unit maps follows from 
the multiplicativity (<ref>)
of the functional calculus maps and the fact that 
the isomorphism (<ref>) sends p_0 p_0 to p_0.



The connective and periodic global K-theory spectra are related by a morphism 
of ultra-commutative ring spectra j:→.subjectK-theory@K-theory!connective global
To define the value 

    j(V) :  (V) =  ((V_),S^V)  → 
    C^∗_(s,(V)_V)   =  (V)

at an inner product space V we consider a configuration 

    [E_1,…,E_n; v_1,…,v_n]  ∈ ((V_),S^V)

of pairwise orthogonal, finite-dimensional subspaces on S^V.
The associated ∗-homomorphism 

    j(V)[E_1,…,E_n; v_1,…,v_n] : s → (V)_V

is then defined on a function f∈ s by

    j(V)[E_1,…,E_n; v_1,…,v_n](f)  =  ∑_i=1^n f[v_i] p_E_i ,

where f[v_i]=(v_i)(f) is the functional calculus (<ref>)
and p_E denotes the orthogonal projection onto a subspace E.
The fact that the morphism j(V)[E_k; v_k] is /2-graded
is a direct consequence of the same property for the functional calculus map.
The verification that the map j(V) is O(V)-equivariant
is straightforward, and we omit it.
The verification that the maps j(V) are multiplicative
amounts to the multiplicativity (<ref>)
of the functional calculus maps and the fact that 
the isomorphism (<ref>)
sends p_E p_F to p_E F.
Moreover, the composite of j(V) with the unit map S^V→(V)
is the unit map of , so the maps j indeed form a morphism
of ultra-commutative ring spectra. 





We have defined (V) in a slightly
different way, compared to the presentation of Joachim <cit.>. 
We use the Hilbert space completion of (V_),
instead of the Hilbert space L^2(V) of -valued square integrable
functions on V. These two Hilbert spaces are naturally isomorphic,
as follows. We use the inner product on V to identify it
with its dual space V^∗, and hence the symmetric algebra
(V_) with (V^∗_). Elements of (V^∗_)
are complex valued polynomial functions on V.
We make them square integrable by multiplying with the rapidly
decaying function ϵ:V→, ϵ(v)=e^-|v|^2. 
This provides a linear isometric embedding

    (V^∗_)  → L^2(V) ,   
    f  ⟼ f·ϵ

with dense image. 
Altogether this exhibits L^2(V) as a Hilbert space
completion of (V_).

The reader should beware that 
for us the C^∗-algebra _V 
is evenly graded, whereas Joachim
uses a non-trivial grading on L^2(V) and (L^2(V)) 
by even and odd functions.
Our grading convention is necessary to ensure that the 
map j(V): (V)→ C^∗_(s,(V)_V)=(V)
defined in Construction <ref> 
takes values in graded ∗-homomorphisms.




[Homotopy type of (V)]
Up to isomorphism, there are only
three different graded C^∗-algebras of the form (V)_V,
and only three different homeomorphism types of spaces (V).
The case V=0 is degenerate in that (0) is just the copy of  generated by 1.
This is already complete, so _0= and (0)=_0=. 
The space (0)=C^∗_(s,(0)_0) is thus
discrete with two points: the zero homomorphism as basepoint
and the augmentation 

    s → (0)_0 ,    f  ⟼ f(0) 1  .

This non-basepoint is a unit for the multiplication maps μ_V,W.
As we explain in Remark <ref>
below, the space (^n) is non-canonically homeomorphic to (^n+2)
for all n≥ 1. So all odd-dimensional terms of  are homeomorphic,
and all even-dimensional terms in positive degrees are homeomorphic.

We can also identify the space () as follows. We claim that the map

    j() :  ()  → ()

is a homotopy equivalence.
Indeed, if A is an (ungraded) C^∗-algebra, then 
∗-homomorphisms from s to A (in the ungraded sense) 
biject with graded ∗-homomorphisms 
from s to () A, via 

    C^∗_(s,() A)    C^∗(s, A)
     ,  φ ⟼ π∘φ ,

where π : () A → A is the ∗-homomorphism 
that sends both 1 a and [1] a to a.
The composite

    S^1     C^∗_(s,())  []π∘ - 
     C^∗(s,)

sends x∈ S^1 to evaluation at x.
This implies that the following square commutes:

    @C=10mm((),S^1) =  ()[r]^-j()[d]_   () = C^∗_(s, ()_) [d]_^π∘-
    _n C^∗(s, _(^[n]())) [r]     C^∗(s, _)

The lower horizontal map is induced by the ∗-homomorphisms
_(^[n])→_ that extend an endomorphism by zero
on the orthogonal complement, and 
the left vertical map
sends a configuration [E_1,…,E_n;x_1,…,x_n] to 
the ∗-homomorphism

    f ⟼ ∑_i=1^n   f(x_i)· p_E_i .

The two vertical maps are homeomorphisms
(compare Remark <ref>) and 
the lower horizontal map is a homotopy equivalence 
by <cit.> 
or <cit.>
(or rather its complex analog).
So the map j() is a homotopy equivalence, and ()
is homotopy equivalent to the infinite unitary group U.

The spectrum  is positive Ω-spectrum (in the non-equivariant sense)
by Theorem <ref> below.
So the adjoint structure map (^2)→Ω(^3)
is a weak equivalence, and the target is homeomorphic to
Ω()≃Ω U. So
(^2) has the weak homotopy type of × B U,
by Bott periodicity.






The analysis of the global homotopy type of  depends 
on the operator theoretic formulation of equivariant Bott periodicity
that we now recall.
We let G be a compact Lie group and V an orthogonal G-representation.
We denote by C_0(V, (V))
the G-C^∗-algebra of continuous (V)-valued functions on V
that vanish at infinity.
Since the Clifford algebra is finite-dimensional, the  map

    C_0(V)(V) → C_0(V, (V))  ,    
        f x  ⟼  f(-)· x

is an isomorphism of C^∗-algebras.
Functional calculus (<ref>) 
provides a distinguished graded ∗-homomorphism

    β_V  :  s  →  C_0(V,(V))  ,   β_V(f)(v) =  f[v]  .

Since the functional calculus map is G-equivariant, 
β_V takes values in the G-fixed points of C_0(V,(V)) 
for the conjugation action of G.

We let _G be any complete G-Hilbert space universe, i.e.,
a Hilbert G-representation that is isometrically isomorphic
to the completion of a complete unitary G-universe.
We let _G be the G-C^∗-algebra of 
(not necessarily equivariant) compact operators on _G, with
G acting by conjugation.




Let G be a compact Lie group and V an orthogonal G-representation.
Then for every graded G-C^∗-algebra A, the map

    β_V· -  :  
    C^∗_(s,  A_G ) → 
    C^∗_(s, C_0(V,(V))   A_G )

is a G-weak equivalence.


When G=e is a trivial group, _G reduces to the C^∗-algebra
of compact operators on a separable Hilbert space,
and then this formulation of Bott periodicity can 
be found in <cit.>.
Unfortunately, we are lacking a reference in the generality of
compact Lie groups. If one specializes <cit.> 
to finite-dimensional representations of finite groups, 
one obtains a formulation very close to 
(but not exactly the same as) Theorem <ref>.


Since the unit map η_V:S^V→(V) is essentially the functional
calculus map, a direct consequence of the formulation of equivariant Bott periodicity
in Theorem <ref> is that  is `eventually'
a global Ω-spectrum, see Theorem <ref> below.





We showed in Proposition <ref>
that an equivariant linear isometric embedding between
two complete complex G-universes induces a G-homotopy equivalence
between configuration spaces with labels in the two universes.
We will now establish an analogous property for the C^∗-algebras
of compact operators on the Hilbert space completions.


We recall that a -linear isometric embedding φ:→'
between complex separable Hilbert spaces gives rise to a preferred ∗-homomorphism
(φ) between the C^∗-algebras of compact operators.
We refer to (φ) as the conjugation homomorphismsubjectconjugation homomorphism!between compact operators
induced by φ.
We emphasize that there is no further hypothesis on φ besides
-linearity and the requirement

    (x,y)_ =  (φ(x),φ(y))_'

for all x,y∈;
in particular, φ is not assumed to be bounded, complemented, or adjointable.
To characterize (φ) we need the following notation:
for a finite-dimensional subspace L of  we write
p_L∈() for the orthogonal projection onto L.
There is then a unique ∗-homomorphism

    (φ)  :  () → (') 
         such that    (φ)(p_L)   =  p_φ(L)

for all finite-dimensional L in .
The uniqueness is a consequence of the fact that 
the linear span of the finite rank projections is the space of finite rank
operators <cit.>
and the finite rank operators are dense in () <cit.>.
The construction of (φ) can
be found in <cit.>.
 
Informally, one can think of (φ) as `conjugation by φ',
combined with extension by 0 on the orthogonal complement, which justifies the name.
Indeed, if φ happens to be complemented, i.e.,
' is the direct sum of φ() and the orthogonal complement
φ()^⊥, then this is literally true.
In particular, if φ is a linear isometric isomorphism, 
then (φ) is conjugation by φ.

The construction is covariantly functorial in the linear isometric embedding.
Indeed, if ψ:'→” is another linear isometric embedding,
then 

    (ψ)∘(φ) =  (ψ∘φ)

by the uniqueness clause.

The construction generalizes to the 
G-equivariant context, where G is any compact Lie group.
More precisely, we assume that  and ' are 
separable complex G-Hilbert space representations. 
The C^∗-algebra () then inherits a continuous
G-action by conjugation with the G-action on . 
If φ:→' is G-equivariant,
then the functoriality of the homomorphism (φ)
as a function of φ directly implies that
the ∗-homomorphism (φ):()→(')
is also G-equivariant. 




Let G be a compact Lie group and
u:→ a G-equivariant -linear isometric embedding
between two complete complex G-universes. Then the morphism of G-C^∗-algebras

    (û) : () → ()

is a G-equivariant homotopy equivalence.


We start with the special case where =.
The space of G-equivariant linear isometric embeddings from 
to itself is contractible.
By passage to completions, a G-equivariant homotopy from u
to the identity induces a G-homotopy

    Φ :  × [0,1] →  ,

i.e., the map Φ(x,-):[0,1]→ is continuous for every x∈,
and the map Φ(-,t) is a linear isometric embedding for every t∈[0,1].
The map 

    [0,1] → C^*((),()) ,    t  ⟼ (Φ(-,t))

is then continuous, see for example <cit.>.
So (û):()→()
is homotopic, through G-equivariant ∗-homomorphisms,
to the identity.

In the general case we choose an equivariant linear isometry v:.
By the previous paragraph the G-C^∗-homomorphisms
(v̂)∘(û)=(v u)
and (û)∘(v̂)=(u v)
are G-homotopic to the respective identity maps.



We recall from Definition <ref> that an orthogonal G-representation
is amplesubjectample G-representation
if its complexified symmetric algebra is a complete complex G-universe.


 
For every orthogonal G-representation V 
and every ample G-representation W, the adjoint structure map

    σ̃_V,W :  (W)  → _*(S^V, (V⊕ W) )

is a G-weak equivalence.


We let u:(W_)→( (V⊕ W)_) be 
the linear isometric embedding
induced by the direct summand embedding W→ V⊕ W.
Since W is ample, u is an equivariant 
-linear isometric embedding between complete 
complex G-universes.
So the induced map (û):_W→_V⊕ W
of compact operators is a G-equivariant homotopy equivalence of C^∗-algebras
by Proposition <ref>.

The adjoint structure map σ̃_V,W factors as the composite:

    (W)  =     C^∗_(s, (W)_W)
    β_V· -    C^∗_(s, C_0(V,(V))(W)_W)
    (û)_*    C^∗_(s, C_0(V,(V)) (W)_V⊕ W)
    [](<ref>)   C^∗_(s, C_0(V)(V⊕ W)_V⊕ W)
         _*(S^V, C^∗_(s, (V⊕ W)_V⊕ W)) =_*(S^V, (V⊕ W) )

Since W is ample, _W is a complete Hilbert G-universe 
and so _W is a _G. 
Bott periodicity (Theorem <ref>)
for the G-C^∗-algebra (W)
shows that the first map is a G-weak equivalence.
The second map is a G-homotopy equivalence by the first paragraph,
so altogether σ̃_V,W is a G-weak equivalence.


The shift X=^ X of an orthogonal spectrum X was
defined in (<ref>).
Shifting an orthogonal spectrum provides a delooping by 
Proposition <ref> (i).
The eigenspace morphism was defined in (<ref>).


The composite 

    Ω^∙()  Ω^∙( j)Ω^∙()

is a global equivalence of orthogonal spaces.


As in the proof of Theorem <ref>
we let  denote the orthogonal space with
(V)= U(((V⊕)_)),
and we factor the eigenspace decomposition morphism as the composite

    →     Ω^∙()  ,

where the first morphism is the global equivalence of orthogonal spaces
induced by the natural linear isometric embedding of V_
into the linear summand in ((V⊕)_).

We claim that composite

    Ω^∙() 
     Ω^∙( j) Ω^∙()

is an `eventually strong level equivalence'. More precisely,
we show that for every ample G-representation V the map

    U(((V⊕)_)) =  (V)     → (Ω^∙())(V)  
       =  _*(S^V, C^∗_(s, (V⊕)_V⊕))

is a G-weak equivalence.
This map factors through the G-map

    U(((V⊕)_))    (((V⊕)_),S^1)  → (_V⊕,S^1)
         C^∗(s, _V⊕)  
    C^∗_(s, ()_V⊕)  .

Here (_V⊕,S^1) is the space of configurations,
not necessarily finite, of pairwise orthogonal finite-dimensional 
subspaces of _V⊕,
that are allowed to accumulate around ∞.
This map is a G-equivariant homotopy equivalence;
the non-equivariant argument (or rather its real analog)
can be found in <cit.> 
or <cit.>,
and the explicit homotopies in <cit.>
work just the same way in the presence of an isometric action of a compact Lie group.

By the operator theoretic equivariant 
Bott periodicity theorem (Theorem <ref>), 
multiplication by the graded ∗-homomorphism β_V:s→ C_0(V,(V)) 
is a G-weak equivalence 

    β_V · - :  
    C^∗_(s, ()_V⊕ ) → 
    C^∗_(s, C_0(V,(V))()_V⊕ )  .

We use here that ()_V⊕ is a _G because
the G-representation V is ample.
This proves the claim because the target is G-equivariantly homeomorphic to 

    C^∗_(s, C_0(V)(V)()   _V⊕) 
     
    _(<ref>)    
     C^∗_(s, C_0(V)(V⊕)_V⊕) 
        _*(S^V, C^∗_(s, (V⊕)_V⊕))  
       =  _*(S^V, (V)) .





We emphasize that in contrast to Theorem <ref>,
the composite in Theorem <ref>
is a global equivalence for all compact Lie groups (as opposed to 
only a -global equivalence). Another key difference is that the
spectrum  is globally connective; in contrast, we
will see in Theorem <ref> below that  is Bott periodic,
so applying Ω^∙ is losing information.


[ globally deloops ]
A corollary of the previous theorem is that  globally deloops
the orthogonal space  defined in Example <ref>.
Indeed,  the global formulation of Bott periodicity in 
Theorem <ref> provides a global equivalence
of ultra-commutative monoids β̅:→Ω(_).
Combined with Theorem <ref> this yields a chain of global equivalences
of orthogonal spaces

    [≃] β̅  Ω(_)  [ ≃ ] 
     Ω [≃]Ω( (Ω^∙( j))∘)Ω (Ω^∙()) 
     [≃]Ω^∙(λ̃_) Ω^∙ .



We spell out another corollary of Theorem <ref>.
We recall that _G(A) denotes the equivariant K-group of a G-space A,
i.e., the Grothendieck group of isomorphism classes of G-vector bundles over A.
A ring homomorphism [-] from this Grothendieck group
to the equivariant cohomology group ^0_G(A_+) 
was defined in (<ref>).



For every compact Lie group G and every finite G-CW-complex A the composite

    _G(A)   [-] ^0_G(A_+)   j_*  ^0_G(A_+)

is an isomorphism.subjectequivariant K-theory


We contemplate the commutative diagram of orthogonal spaces:

    @C=13mm^[d]_β[r]^-c    Ω^∙[d]^Ω^∙λ̃__≃[r]^-Ω^∙ j   Ω^∙[d]^Ω^∙λ̃__≃
    Ω[r]_-Ω   Ω( Ω^∙ ()) 
    [r]_-Ω (Ω^∙( j))   Ω( Ω^∙())

The morphism (Ω^∙( j)) ∘
is a global equivalence by Theorem <ref>,
so the lower horizontal composite is also a global equivalence. 
The same argument as for Ω^∙λ̃_ in
the proof of Theorem <ref>
shows that the right vertical morphism Ω^∙λ̃_
is a global equivalence of orthogonal spaces.
In the commutative diagram of abelian monoids

    @C=12mm
    [A,^]^G  [d]_[A,β]^G[r]^- [A,c]^G    
    [A,Ω^∙]^G   [r]^- [A,Ω^∙ j]^G    
    [A, Ω^∙]^G = _G^0(A_+) 
    [d]^ [A, Ω^∙λ̃_]^G _
    
    [A, Ω]^G 
    [rr]_-  [A, Ω( (Ω^∙( j))∘)]^G^-       
    [A, Ω (Ω^∙()) ]^G

the lower horizontal and right vertical maps are thus isomorphisms
by Proposition <ref> (ii). 
The left vertical map [A,β]^G is a group completion of abelian monoids
by Corollary <ref>.
So the upper horizontal composite 
[A,(Ω^∙ j)∘ c]^G:[A,^]^G→_G^0(A_+)
is also a group completion of abelian monoids. 
On the other hand,
the homomorphism -:[A,^]^G→_G(A)
is yet another group completion of abelian monoids, by 
the complex analog of Theorem <ref>.
The universal property of group completions yields
an isomorphism of abelian groups

    Ψ :  _G(A) → _G^0(A_+)

such that 
Ψ∘-=[A,(Ω^∙ j)∘ c]^G:[A,^]^G→_G^0(A_+).
The defining relation [A, c]^G=  [-]∘- 
of the homomorphism (<ref>) then yields

    Ψ∘- =  [A, (Ω^∙ j)∘ c]^G 
    =  [A,Ω^∙ j]^G∘ [-]∘- .

Since -:[A,^]^G→_G(A) is a group completion, this
forces the relation Ψ=[A,Ω^∙ j]^G∘ [-].



The special case A=∗ of the previous corollary is worth spelling out explicitly.
In this case the group _G(∗) becomes
the unitary representation ring (G), and _G^0(∗) becomes
the 0-th equivariant homotopy group π_0^G().


subjectrepresentation ring!unitary 
As G ranges of all compact Lie groups, the composite maps

    (G)  [-]  π_0^G()  π_0^G(j)  π_0^G()

form an isomorphism of global power functors between  and _0().


By Corollary <ref> for A=∗ 
the composite is a ring isomorphism for every compact Lie group G.
The second map is induced by a homomorphism j:→ of ultra-commutative
ring spectra, so the maps π_0^G(j) form a morphism of global power functors.

The maps [-]:(G)→π_0^G()
are ring homomorphisms, compatible with restriction homomorphisms,
with finite index transfers  and multiplicative power
operations by Theorem <ref>. 
There is still something to show, though, because the maps [-]
are definitely not compatible with general transfers (i.e., of infinite index). 
So an additional argument is needed to see that the composite does
commute with all transfers.

We consider a closed subgroup H of a compact Lie group G, not necessarily of
finite index, and we want to show that

    _H^G(j_* [x] )   =  j_*[_H^G(x)]

in π_0^G() for all classes x∈(H).
Since representations are detected by characters, two classes 
in (G) are equal already if their restrictions to all
finite abelian subgroups of G coincide. 
Since the composite maps  (G)→π_0^G() 
are all isomorphisms and compatible with restriction,
the analogous property holds for the global functor _0().
So it suffices to show that the relation (<ref>)
holds after restriction to every finite abelian subgroup A of G.
The double coset formula

    ^G_A∘_H^G  =  ∑_[M]χ^♯(M)·_A∩^g H^A  ∘ g_⋆∘^H_A^g∩ H

holds for  by <cit.>,
and it holds in the homotopy global functor of every orthogonal spectrum.
Since A is finite, 
the right hand side of the double coset formula only involves finite index transfers. 
Since the maps from  to _0() under consideration
do commute with finite index transfers, they commute with the right hand
side of the double coset formula, hence also with the left hand side.
This shows that (<ref>) holds after restriction
to every finite abelian subgroup, so it holds altogether.




In Construction <ref> we discussed 
the rank filtrationsubjectrank filtration!of  
of the connective global K-theory spectrum ,
and we identified the first stage ^[1] with the suspension spectrum
of the ultra-commutative monoid ^.
On _0, the morphisms of ultra-commutative ring spectra

    Σ^∞_+^  ^[1] incl   j

induce morphisms of global power functors. 
Since ^ is a global classifying space of the circle group U(1),
the global functor _0(Σ^∞_+ ^)
is representable by U(1), by Proposition <ref>.  
On the other hand, _0() is isomorphic to the representation
ring global functor , by Theorem <ref>.
Under these identifications, the composite morphism 
Σ^∞_+^→
of ultra-commutative ring spectra becomes the morphism

    _x :  (U(1),-) →

that sends the generator 1_U(1)∈(U(1), U(1)) to the class of the tautological
U(1)-representation on . 
As we recalled in Remark <ref>, this morphism is surjective,
and `explicit Brauer induction' provides a specific section.




Our language allows a reformulation of the generalization, 
due to Adams, Haeberly, Jackowski 
and May <cit.>,
of the Atiyah-Segal completion theorem:
the global K-theory spectrum  
is right induced from the global family 
of finite cyclic groups.symbol - global family of finite cyclic groupssubjectglobal family!of finite cyclic groups 

If G is a compact Lie group, then a virtual G-representation
that restricts to zero on every finite cyclic subgroup is already zero.
In other words, the intersection of the kernels 
of all restriction maps ^G_C:R(G)→ R(C) for all
finite cyclic subgroups C of G, is trivial. 
Then by <cit.>, 
the projection A× E(∩ G)→ A induces an isomorphism 

    ^*_G( A )  ^*_G( A × E(∩ G) )

on equivariant K-groups for every finite G-CW-complex A,
where E(∩ G) is a universal G-space for the family of finite cyclic subgroups
of G.
The Milnor short exact sequence lets us extend this to infinite G-CW-complexes,
so the criterion provided by Proposition <ref> 
for being right induced from the global family  is satisfied.




Our next aim is to establish Bott periodicity for the global K-theory spectrum .


[Inverse Bott class]
The Bott class β∈π_2^e()
was defined in Construction <ref>.subjectBott class!in connective K-theory
We will now define the inverse Bott classsubjectinverse Bott class
λ∈π_-2^e()
and show that it is multiplicatively inverse to the image of β under
the homomorphism of ultra-commutative ring spectra j:→.

We recall that u denotes the underlying euclidean inner product space of .
We let e=[1] and f=[i] be the images in (u)
of the standard orthonormal -basis {1,i} of u.
We denote by q∈(u) the projection
(i.e., self-adjoint idempotent)

    q  =  (1 - i· e f)/2 .

The isomorphism δ_:(u)→_(Λ^*)
defined in (<ref>)
takes the projection q to the orthogonal projection onto Λ^0, 
the constant summand of the exterior algebra, see Example <ref>.

Moreover, we denote by p_0∈_u the orthogonal projection onto
the subspace · 1 in _u=( (u)_).
Then the element q p_0 of (u)_u
is another projection. So the map

    s  → (u)_u ,    
    f  ⟼ f(0)· q p_0

is a /2-graded ∗-homomorphism, i.e., an element in the space

    C^∗_(s,(u)_u)  =  (u)
     =  (^2) .
 
We denote by

    λ ∈ π_-2^e()

the homotopy class represented by this point.




subjectBott periodicity
The relation
j_*(β)·λ =  1
holds in π_0^e().


Step 1: We start with an identification of
() M_2 with (⊕ u),
where M_2 is the C^∗-algebra of 2× 2 complex matrices, 
concentrated in even grading.
We let d=[1]∈() denote the odd unit corresponding to 1∈.
The map

    Ψ(1([ w x; y z ]))
     =  (w+z)/2· 1  +  (z-w)/2· i· e f 
     +  (y-x)/2· d e  -  (x+y)/2· i· d f

identifies the even summand 1 M_2 of 
() M_2 
with the even summand of (⊕ u).
The odd element i· d e f of (⊕ u) 
squares to 1 and commutes with all even elements of (⊕ u).
So we can extend the isomorphism to the odd summands by setting

    Ψ(d([ w x; y z ]))
     =  i· d e f ·Ψ(1([ w x; y z ])) .

The result is an isomorphism of /2-graded C^∗-algebras

    Ψ :  () M_2  (⊕ u)  .

While C^∗-algebras are not required to have multiplicative units, 
and ∗-homo­morphisms are ignorant of units, these two C^∗-algebras are unital
and Ψ happens to preserve multiplicative units.

Step 2: We let j':(^2,S^1)→ C^∗_(s,() M_2)
be the restriction of the map

    j() : ()=((),S^1) → 
    C^∗_(s,()_)=()
 
to those configurations contained in ^2⊂(),
defined by the same formula

    j'[E_1,E_2; v_1,v_2](u)  =   u[v_1] p_E_1 +  u[v_2] p_E_2 .

Our next claim is that the composite

    S^⊕    [] m  S U(2)     (^2,S^1)
        j'  
    C^∗_(s,() M_2)   Ψ_*  
    C^∗_(s,(⊕ u))

is the functional calculus map 
: S^⊕→ C^∗_(s,(⊕ u)),
where m is the homeomorphism (<ref>) defined by the formula

    m(v) =  m(x,z) =  1/|v|^2+1[ |v|^2-1 - i 2 x          2 i z̅;           2 i z |v|^2-1 + i 2 x ] .

The map ∘ m sends (x,z) to the configuration of Cayley transforms
of the eigenvalues of m(x,z),
labeled by their eigenspaces.
So to identify the composite j'∘∘ m we must
calculate these eigenvalues and eigenspaces, and their orthogonal projections.
This is a straightforward exercise in linear algebra:
a direct calculation shows that the matrices

    p_+  =  1/2|v|[  |v|- x      z̅;       z |v| + x ]   and  
     p_-  =  1/2|v|[ |v|+ x    -z̅;    - z |v|- x ]

are projections and orthogonal to each other, i.e., they satisfy the relations

    p_+^2 =  p_+^∗ =  p_+  ,    
     p_-^2 =  p_-^∗ =  p_-    and  
    p_+· p_- =  p_-· p_+  =  0  .

Moreover,

    c(|v|)· p_+  +  c(-|v|)· p_-  =  m(x,z)  ,

so p_+ and p_- are the orthogonal projections onto the eigenspaces
of m(x,z), with eigenvalues 

    c(|v|) =  (|v|+i)(|v|-i)^-1   respectively  
     c(-|v|) =  c(|v|)^-1 =  (|v|-i)(|v|+i)^-1 .

The map j'∘∘ m thus sends v=(x,z) to the ∗-homomorphism

    (j'∘∘ m)(x,z)(u) =  u[|v|] p_+  +  u[-|v|] p_- .

The C^∗-algebra s is generated by the function r(x)=2 i(x-i)^-1.
So to verify that two graded ∗-homomorphisms with source s agree,
it suffices to show that they coincide on the even and odd 
components of the generator r,
which we spelled out explicitly in (<ref>). 
For the even component r_+ the calculation is

    (Ψ_*∘ j'∘∘ m)(x,z)(r_+)    =  Ψ( r_+[|v|] p_+ + r_+[-|v|] p_-) 
       =  Ψ( r_+[|v|] (p_+ +  p_-))  
    =     r_+[|v|]· 1 =     r_+[x,z] .

For the odd component r_- we first observe that

    Ψ( d([ - x  z̅;   z   x ]))     =    
    i· d e f · (  x· i e f  + (z-z̅)/2 · d e
    -(z+z̅ )/2· i· d f  )
       =  
    - x·  d e f e f - Im(z)·  d e f d e +  Re(z) ·  d e f d f
       =  
     x·  d + Im(z)·  f + Re(z)· e =  [x,z]

in (⊕ u). So

    (Ψ_*∘ j'∘∘ m)(x,z)(r_-)    =  Ψ( r_-[|v|] p_+ + r_-[-|v|] p_-) 
       =  Ψ( r_-[|v|] (p_+ -  p_-)) 
       =  Ψ( 2 i [|v|]/|v|^2+11/|v|[ -x z̅;  z  x ]) 
       =  2 i/|v|^2+1·Ψ( d[ -x z̅;  z  x ]) 
       =  2 i· [x,z]/|v|^2+1 =   r_-[x,z] .

This completes the verification that Ψ_*∘ j'∘∘ m
is the functional calculus map : S^⊕→
C^∗_(s,(⊕ u)).


Step 3:
We recall that q=(1 - i e f)/2 in the Clifford algebra (⊕ u), 
which satisfies the relations

    d q =  q d   ,     e q   = (1-q)e
         and     q e q  = 0  .

We consider the even element u of (⊕ u) M_2
defined by

    u  =  q([ 1 0; 0 0 ])
     + q e d([ 0 0; 1 0 ])
     +     e d q([ 0 1; 0 0 ])
     +   (1-q)([ 0 0; 0 1 ])  .

A direct calculation, making repeated use of the identities (<ref>),
shows that this is a unitary element, i.e.,

    u^∗· u =  u·  u^∗ =   1 .

So conjugation by u is a graded ∗-automorphism

    ζ :  (⊕ u) M_2  → (⊕ u) M_2  ,    x  ⟼ u^∗· x· u  .

Now we observe that the following square of graded C^∗-algebras commutes:

    @C=13mm() M_2[d]_-Ψ[r]^-- q -   ()(u) M_2
    [r]^-(<ref>) M_2   (⊕ u) M_2[d]^ζ
    (⊕ u)[rr]_- - ([ 1 0; 0 0 ])      (⊕ u) M_2

This, again, is a direct calculation, using the identities (<ref>).
We claim that the conjugation map ζ
is homotopic, through graded ∗-homomorphisms, to the identity
of (⊕ u) M_2.
To show this we define a continuous path

    u :  [0,π/2]  → (⊕ u) M_2

by the formula

    u(t)  =  (q+sin(t)   (1-q))([ 1 0; 0 0 ])
     + cos(t)· q e d ([ 0 0; 1 0 ])
        +  cos(t)· e d q([ 0 1; 0 0 ])
     +  
    (sin(t)· q + (1-q))([ 0 0; 0 1 ])  .

One more direct calculation shows that all elements in this path are even unitaries.
So conjugation by the elements u(t) is a path of graded ∗-automorphism.
Since u(0)=u, this path starts with the conjugation map ζ;
since u(π/2)=1, the path ends with the identity.
Since ζ is homotopic to the identity, the induced self-map
of C^∗_(s,(⊕ u) M_2)
is based homotopic to the identity.

Step 4: The class λ is represented by the point of (u)
given by the graded ∗-homomorphism

    s   ϵ   · q p_0 (u)_u

associated to the projection q p_0; here ϵ(φ)=φ(0)
is the augmentation. So the multiplication map
-·λ :  π_k+2^e()→π_k^e()
is the effect on homotopy groups of the based continuous map

    (V) =     C^∗_(s,(V)_V)
       - (-· q p_0) 
     C^∗_(s,(V)_V) C^∗_(,(u)_u)  
       ϵ^* C^∗_(s,(V)_V) C^∗_(s,(u)_u)  
       μ^_V,u C^∗_(s,(V⊕ u)_V⊕ u)  =  (V⊕ u) .

The composite sϵ∘Δ is the identity of s, 
so multiplication by λ is the effect of postcomposition
with the graded ∗-homomorphism

    (V)_V    - q - p_0 (V)(u)_V_u
       [    ](<ref>)(<ref>) (V⊕ u)_V⊕ u .



Now we put the pieces together and prove the theorem.
We contemplate the following diagram of continuous based maps:

    @C=5mm@R=8mm 
     S^⊕[d]_∘ m[rr]^-       
    C^∗_(s,( u)) 
    [d]^-(-([ 1 0; 0 0 ]))_*@<8ex>@/^2pc/[dd]^-(.2)(- p_0)_*
    (^2,S^1)[d]_-(i,S^1)[r]_-j'   
    C^∗_(s,() M_2)[r]_-(- q -)_*[ur]^(.4)Ψ_*[d]^-(- i_!)_*   
     C^∗_(s,( u) M_2) [d]^-(- i_!)_*
    ((),S^1)@=[dd]    
    C^∗_(s,()_) @=[dd] [r]_-(- q -)_*    
     C^∗_(s,( u)_) 
    [d]^(- p_0)_*
            C^∗_(s,( u)_⊕ u) @=[d] 
    ()[r]_-j()   ()[r]_-(- q - p_0)_*   (⊕ u)

Here i:^2→() is the identification with the
constant and linear summands of the symmetric algebra used throughout,
and i_!:M_2→_ is the ∗-homomorphism given by extension 
by 0 on the higher symmetric powers.
By Step 2, the upper left part commutes on the nose,
and by Step 3 the upper right triangle commutes up to based homotopy.
The lower left part commutes because j' is the restriction of j(). 
The lower right part commutes as well.
So the whole diagram commutes up to based homotopy.
By Step 4, the composite
through the lower left corner represents the class j_*(β)·λ,
whereas the composite along the right side of the diagram is
the unit map η_⊕:S^⊕→(⊕ u)
of the ring spectrum structure. So the diagram witnesses the desired relation

    j_*(β)·λ =  
    [(- q -  p_0)_*∘ j()∘((),S^1)∘∘ m]
     =  [η_⊕ u]  =  1  .






[Thom isomorphism in equivariant K-theory]
The spectrum  enjoys a much stronger form of periodicity that generalizes
the -graded periodicity manifested by
Theorem <ref>. 
The equivariant Bott class β_G,W associated
to a G-^c-representation W was introduced 
in Construction <ref>,
and β_G,W is in fact an `R O(G)-graded unit' in . 
This kind of equivariant periodicity in K-theory goes back to 
Atiyah <cit.>;subjectBott periodicity!equivariant 
it is formally similar to the periodicity of  and 
for orthogonal representations (compare Theorem <ref> (iii)),
or the periodicity of  and  for unitary representations. 

Even more is true. The equivariant Bott class is a special case of
the Thom class of an equivariant ^c-vector bundle,subjectThom class!in 
and equivariant Bott periodicity is a special case of a Thom isomorphism 
for such bundles.subjectThom isomorphism!for 
This Thom isomorphism for equivariant -theory goes under the name 
`Atiyah-Bott-Shapiro orientation'subjectAtiyah-Bott-Shapiro orientation 
and was established in <cit.>.subjectG-spin c-bundle@(G,^c)-bundlesubjectspin^c groupsubjectspin^c-bundle!equivariant
In <cit.> 
Joachim shows that the Atiyah-Bott-Shapiro orientation
is in fact extremely highly structured, name­ly `globally and ultra-commutative'.
Joachim defines a morphism of ultra-commutative ring spectra
α:𝕄^c→ from a global equivariant version of the
^c-Thom spectrum; his morphism refines the Atiyah-Bott-Shapiro orientation,
in the sense that it takes certain tautological Thom classes 
for ^c-vector bundles in 𝕄^c
to the -theoretic Thom classes of Atiyah-Bott-Shapiro. 


subjectK-theory@K-theory!periodic global|)


subjectK-theory@K-theory!global connective|(

[Global connective K-theory]
Now we define global connective K-theory ,
an ultra-commutative ring spectrum whose associated G-homotopy type,
for compact Lie groups G,
is that of G-equivariant connective K-theory
in the sense of Greenlees <cit.>.
This is not a connective equivariant theory, i.e.,
the equivariant homotopy groups π_*^G (  ) do not
vanish in negative dimensions, as soon as the group G is non-trivial.
Hence the order of the adjectives `global' and `connective' matters, i.e.,
`global connective' K-theory is different from `connective global' K-theory.
Two of the advantages of  are that it is equivariantly 
(and in fact globally) orientable,
and that  satisfies a completion theorem: for every
compact Lie group G, the completion of the graded ring π_*^G(  ) 
at the augmentation ideal of the unitary representation ringsubjectaugmentation ideal!of the unitary representation ring
is the connective -cohomology 
of the classifying space B G, see <cit.>.

Our construction of  is a direct `globalization' of Greenlees' 
definition in <cit.>.
We define symbol - global connective K-theorysubjectK-theory@K-theory!global connective 
as the homotopy pullback in the square of ultra-commutative ring spectra

    [r]^-[d]    
    b()[d]^b j
    [r]_-i_    b()

The morphism j:→ from connective to periodic global K-theory
was defined in Construction <ref>;
the Borel theory functor b and the natural transformation
i:→ b were defined in Construction <ref>.
So more explicitly, we set 

    =  ×_b() b()^[0,1]×_b() b()  .

Since the spectra , b() and b()
are ultra-commutative ring spectra and the two morphisms
i_:→ b() and b j:b()→ b()
are homomorphisms, the homotopy pullback
is canonically an ultra-commutative ring spectrum
and the two morphisms from  to  and b()
are morphisms of ultra-commutative ring spectra.
As a homotopy pullback, the square (<ref>)
does not commute, but the construction comes with
a preferred homotopy between the two composites around the square.

The morphisms j:→, i_:→ b()
and the constant homotopy provide a morphism of ultra-commutative ring spectra

    ι :   →

from connective global K-theory to global connective K-theory.
For finite groups, this morphism induces an isomorphism on homotopy global functors
in non-negative dimensions.
The construction of  endows it with a morphism of ultra-commutative ring spectra
→. As explained by Greenlees 
in <cit.>,
this morphism becomes a global equivalence after inverting
the Bott class v∈π_2^e(  ), 
the image of the Bott class β∈π_2^e()
defined in Construction <ref>.subjectBott class!in connective K-theory
Indeed, for every compact Lie group G, the morphism
j:→ induces a morphism of graded rings

    j^*(B G) :  ^*(B G)  → ^*( B G)

that becomes an isomorphism after inverting the Bott class
by <cit.>.
The natural isomorphisms

    π_-*^G(b())     ^*(B G)    and  π_-*^G(b ())     ^*(B G)

of Proposition <ref>
then show that the map

    _*(b j)  :  _*(b())  → _*(b ())

becomes an isomorphism of graded global functors after inverting the Bott class.
Since the right vertical morphism in the 
defining global homotopy pullback (<ref>)
becomes a global equivalence after inverting the Bott class, the same is true for
the left vertical morphism.


The composite

    ^c  :   → b()   b()  b()

is a morphism of ultra-commutative ring spectra,
where  is the Eilenberg-Mac Lane spectrum of the integers 
(see Construction <ref>)subjectEilenberg-Mac Lane spectrum!of the integers 
and the dimension homomorphism :→
was defined in Example <ref>.subjectdimension homomorphism 
The dimension morphism annihilates the Bott class β∈π_2^e().
We claim that the sequence

    S^2  ṽ   ^c  b()

is a `global homotopy cofiber sequence', i.e., part of a distinguished
triangle in the global stable homotopy category, where ṽ
is the extension of the Bott class v∈π_2^e() 
to a morphism of -module spectra.
Indeed, the sequence 

    S^2   β̃

is a non-equivariant homotopy fiber sequence;
the Borel theory functor b takes this to the global homotopy fiber sequence

    b() S^2  b(β̃)  b()
      b()  b()  .

The spectrum  is Bott periodic, i.e., the image of the Bott class
in π_2^e() is invertible by Theorem <ref>.
Since the morphism → b() is multiplicative, the same goes
for the Borel theory b(). So the morphisms

    β̃ :   S^2  →    and  
     b(β̃) :  b() S^2  → b()

are global equivalences, and their respective homotopy fibers 
are globally stably contractible.
Passing to homotopy pullbacks gives the desired 
global homotopy fiber sequence (<ref>).

The global homotopy fiber sequence (<ref>)
and the isomorphism

    π_k^G( b())  H^-k(B G, )

of Proposition <ref> (ii)
give rise to a long exact sequence of global functors

    ⋯→_k+1 ( b() )   ∂ _k-2()   · v  _k()   ^c_* _k ( b() )
      →⋯ .

This way Greenlees calculates some of the homotopy group global functors
of equivariant connective K-theory in <cit.>.
We review Greenlees' calculations in our language.
The group H^-k(B G,) vanishes for k>0.
So multiplication by the Bott class

    -· v  : _k-2()  → _k()

is an isomorphism for k>0 and a monomorphism for k=0. 
In particular, we conclude that

    _k()      for k≥ 0 and k even, and
      0      for k≥ -1 and k odd.

More precisely, for every m≥ 0, the composite

    [-] _0() _0(ι) _0(^c)  · v^m  _2m(^c)

is an isomorphism of global functors.

The global functor _0(b()) sends G to π_0^G(b())=H^0(B G,); 
so _0(b()) is constant with value  and the morphism 
^c_*:_0() →_0 ( b() )
is isomorphic to the augmentation morphism
:→ of global functors, by Proposition <ref>.
This is an epimorphism, so the sequence of global functors

    0  → _-2()   · v   _0()   ^c_*  _0 ( b() ) → 0

is short exact and

    _-2()   𝐈 𝐔  = (:→)

is the augmentation ideal global functor.subjectaugmentation ideal!of the unitary representation ring
Again since the map  
^c_*:_0() →_0 ( b() )
is surjective, the global functor _-3()
injects into _-1(), which is trivial by the above. 
So we conclude that _-3()  =  0.

This method can be pushed a little further to also determine 
the global functors π_-4() and π_-5();
we refer to <cit.> for the argument.
The result is that

    _-4()   𝐈 𝐒 𝐔(G)  =  {x∈𝐈 𝐔(G) | (x)=0}

and that _-5()=0.
After this point things become less explicit.


subjectK-theory@K-theory!global connective|)
subjectultra-commutative ring spectrum|)




  



CHAPTER: COMPACTLY GENERATED SPACES



In this appendix we recall some background material about compactly
generated spaces, our basic category to work in.
Compactly generated spaces are in particular `k-spaces',
a notion that seems to go back to Kelley's book <cit.>.
Compactly generated spaces were
popularized by Steenrod in his paper <cit.>
as a `convenient category of spaces'; however, in contrast to our usage of
the term, Steenrod includes the Hausdorff property in his definition
of `compactly generated'.
But Steenrod already writes that `(...) The Hausdorff property is
imposed to ensure that compact subsets are closed. (...)';
the weak Hausdorff condition it thus the next logical step,
as it isolates this relevant property.
The weak Hausdorff condition first appears in print in 
McCord's paper <cit.>, who credits the idea to J. C. Moore. 
McCord also was the first to use the terminology
`compactly generated spaces' for weak Hausdorff k-spaces.
The fact that much of the recent literature in equivariant and stable
homotopy theory uses compactly generated spaces
can be taken as evidence that the weak Hausdorff condition 
is even more convenient than the actual Hausdorff separation property.
Section 7.9 of tom Dieck's textbook <cit.>
is a nice summary of compactly generated spaces (where these are called
whk-spaces), and contains most of the material that we discuss here.
Two influential – but unpublished – sources about compactly generated
spaces are the Appendix A of Gaunce Lewis's thesis <cit.>
and Neil Strickland's preprint <cit.>.

I want to emphasize that this appendix does not contain new mathematics
and makes no claim to originality.
I have decided to include it because I found it cumbersome to
collect proofs of all the relevant properties of compactly generated spaces 
from the scattered literature. An additional complication stems
from the fact that the basic 
references <cit.>
all work in slightly different categories; so in the interest of a self-contained
treatment, I felt obliged to fill in arguments 
where a reference confines itself to the statement that 
`(...) the proof is analogous to that of (...)'.





We fix some terminology. A topological space is compactsubjectcompact!topological space
if it is quasi-compact (i.e., every open cover has a finite subcover)
and satisfies the Hausdorff separation property (i.e., every pair of distinct
points can be separated by disjoint open subsets).


 Let X be a topological space.
  
  
  * A subset A of X is compactly closedsubjectcompactly closed 
    if for every compact space K and every continuous map f:K→ X, 
    the inverse image f^-1(A) is closed in K.
  
  * X is a k-spacesubjectk-space@k-space 
    if every compactly closed subset is closed.
  
  * X is weak Hausdorffsubjectweak Hausdorff space
    subjectHausdorff space!weak|seeweak Hausdorff space 
    if for every compact space K and
    every continuous map f:K→ X the image f(K) is closed in X.
  
  * X is a compactly generatedsubjectcompactly generated!topological space space if it is a k-space and weak Hausdorff.



Every closed subset is also compactly closed. One can similarly define
compactly opensubjectcompactly open  subsets of X by demanding 
that for every compact space K and every continuous map f:K→ X, 
the inverse image is open in K. A subset is then compactly open 
if and only if its complement is compactly closed.
Thus k-spaces can equivalently be defined by the property 
that all compactly open subsets are open.

We denote by symbol - category of topological spaces
the category of topological spaces and
continuous maps. We denote by symbol - category of k-spaces 
respectively symbol - category of compactly generated spaces 
the full subcategories of   
consisting of k-spaces respectively compactly generated spaces.
So we have full embeddings

    ⊂  ⊂  .

We will see below that the inclusion ⊂
has a right adjoint `Kelleyfication' k:→
and the inclusion ⊂ has a left adjoint  w:→.




We follow the terminology introduced by McCord in <cit.>.
We warn the reader that the usage of the terms `k-space' and
 `compactly generated' is not consistent throughout the literature. 
For example, some authors define
k-spaces by the property that a subset is closed if and only its intersection
with every compact subspace is closed. For Hausdorff spaces,
and more generally weak Hausdorff spaces, that definition agrees with the
one in Definition <ref> because for such spaces 
the image of a compact space under a continuous map is automatically compact,
by Proposition <ref> (v) below.
Moreover, there are sources (for example <cit.>)
that require a compactly generated space
to be Hausdorff (as opposed to only weak Hausdorff);
on the other hand, several recent references do not include the weak Hausdorff condition
in `compactly generated'.



We recall various useful properties of k-spaces, 
weak Hausdorff spaces and compactly generated spaces.
We start by discussing k-spaces.
A Hausdorff topological space X 
is locally compactsubjectlocally compact!topological space
if the following condition holds: for every open subset U of X
and every point x∈ U, there is an open neighborhood V of x
whose closure V̅ is compact and contained in U.
For example, every compact space is in particular locally compact.
A topological space is first countable
if every point has a countable basis of neighborhoods.

Because the product of two k-spaces need not be a k-space in
the usual product topology, we need notation for two different kinds 
of product topologies.
We shall denote by X×_0 Y the cartesian
product of two spaces X and Y, endowed with the product topology
(which makes it a categorical product in the category  of all topological spaces).
So a basis of the topology X×_0 Y is given by products of open subsets
in the two factors. We reserve the symbol X× Y 
for k(X×_0 Y), the Kelleyfication of the product topology, 
discussed in more detail below. This product X× Y
is a categorical product in the categories 𝐊 and .
Part (vi) of the following proposition says that these two product topologies
coincide if one factor is a k-space and the other factor is locally compact Hausdorff.


  
  
  * Every quotient space of a k-space is a k-space.
  
  * Every closed subset of a k-space is a k-space in the subspace topology.
  
  * Every locally compact Hausdorff space, and hence every compact space,
    is a k-space.
  
  * Every first countable space is a k-space.
  
  * Every metric space is first countable, and hence a k-space.
  
  * If X is a k-space and Y a locally compact Hausdorff space, then
    the product X×_0 Y is a k-space in the product topology.
  


  (i) Let p:X→ Y be a quotient projection and B a compactly closed subset of Y.
  We claim that p^-1(B) is compactly closed in X. Indeed, if f:K→ X
  is a continuous map from a compact space, then p f:K→ Y is continuous, so
  f^-1(p^-1(B))=(p f)^-1(B) is closed because B is compactly closed.
  Since p^-1(B) is compactly closed and X is a k-space, the set p^-1(B) 
  is in fact closed in X. So B is closed in Y by definition of
  the quotient topology.

  (ii) We let Y be a closed subset of a k-space X,
  and A a compactly closed subset of Y with respect to the subspace topology.
  We claim that then A is also compactly closed as a subset of X. 
  Indeed, if f:K→ X is a continuous map from a compact space, then
  L=f^-1(Y) is closed in K, and hence compact. The restriction
  f|_L:L→ Y is then continuous and so (f|_L)^-1(A)=f^-1(A) is
  closed in L because A was assumed to be compactly closed in Y.
  Since L is closed in K, the set f^-1(A) is closed in K.
  This shows that A is compactly closed as a subset of X.
  Since X is a k-space, A is closed in X. But then A is also closed in Y
  in the subspace topology, so this concludes the proof that Y is a k-space.

  (iii) 
  The argument goes all the way back to 
  Kelley <cit.>.
  We let A be a compactly closed subset of a locally compact Hausdorff space X. 
  We let A̅ be the closure of A in X and x∈A̅.
  Since X is locally compact, the point x has a compact neighborhood K.
  Since A is compactly closed and the inclusion K→ X is continuous,
  the set K∩ A is closed inside K. Since K is compact and X is Hausdorff,
  K is closed in X. So K∩ A is closed in X.

  Now we claim that x∈ K∩ A.
  We argue by contraction and suppose that x∉K∩ A.
  Then X-(K∩ A) is an open neighborhood of x, 
  and hence K∩(X-(K∩ A))=K∩(X-A)
  is another neighborhood of x.
  Let U be an open subset of X with x∈ U⊂ K∩(X-A). 
  Then A⊂ X-U, and hence x∈A̅⊂ X-U because X-U is closed.
  But his contradicts the hypothesis x∈ U.
  Altogether this proves the claim that x∈ K∩ A⊂ A.
  Hence A=A̅, and so A is closed.

  (iv) 
  This, too, goes back to Kelley <cit.>.
  We let X be a first countable space and A a compactly closed subset of X.
  We let z∈A̅ be a point in the closure of A. 
  The point z has a countable basis of open neighborhoods {U_n}_n≥ 1,
  which we can moreover take to be nested, i.e.,
  
    U_1  ⊃ U_2  ⊃ ⋯ ⊃ U_n ⊃ ⋯ .

  If the intersection of U_n and A were empty, then
  A̅⊂ X-U_n which contradicts the fact that z∈A̅∩ U_n. 
  So for every n≥ 1 there is a point x_n∈ U_n∩ A.
  We define a map
  
    g :  K =  {0}∪{1/n  |  n≥ 1} → X

  by g(0)=z and g(1/n)=x_n. The hypotheses
  imply that the map g is continuous if we give the source K
  the subspace topology of the interval [0,1].
  In this topology the space K is compact, so g^-1(A) is
  closed since A was assumed to be compactly closed.
  On the other hand, all the points 1/n are contained in g^-1(A),
  and the closure of the set of these point is all of K. 
  So 0∈ g^-1(A), which means that z=g(0)∈ A.
  So A coincides with its closure, i.e., A is closed in X.

  (v) 
  In a metric space the balls of radius 1/n for all n≥ 1
  form a countable neighborhood basis of a given point. 
  So metric spaces are first countable, hence k-spaces by (iv).

  (vi) This argument goes back to D. E. Cohen <cit.>
  who thanks J. H. C. White­head for `suggesting the subject (...) and for 
  valuable help'.
  We let A be a compactly closed subset of X×_0 Y.
  We let (x_0,y_0)∈ (X× Y)-A be a point in the complement.
  We let
  
    A_0  =  { y∈ Y |  (x_0,y)∈ A}

  be the `slice' of A through x_0∈ X.
  We let N be a compact neighborhood of y_0 in Y.
  Then A_0∩ N is closed in N because A is compactly closed.
  Since Y is Hausdorff, N is closed in Y, and hence
  A_0∩ N is closed in Y. 
  Since y_0∉A_0, the set Y-A_0 is an open neighborhood of y_0.
  Since Y is locally compact Hausdorff, there is a compact neighborhood K 
  of y_0 with K⊂ Y-A_0.
  We let
  
    B  =  {x ∈ X  |  ({x}× K)∩ A ∅}

  be the projection of (X× K)∩ A to X.
  The condition K⊂ Y-A_0 is then equivalent to x_0∉B.
  
  We let f:C→ X be a continuous map from a compact space. Then C× K is compact,
  and so (f× K)^-1((X× K)∩ A) is closed in C× K 
  since A is compactly closed.
  Hence (f× K)^-1((X× K)∩ A) is compact in the subspace topology
  inherited from C× K.
  Since 
  
    f^-1(B)  =  {c ∈ C  |  ({f(c)}× K)∩ A ∅}

  is the projection of (f× K)^-1((X× K)∩ A) onto C, 
  the set f^-1(B) is closed in C.
  Altogether this shows that the set B is compactly closed.
  Since X is a k-space, B must be closed in X.
  Since x_0∉B, the set (X-B)× K is a neighborhood of (x_0,y_0).
  Moreover, (X-B)× K is disjoint from A by definition of the set B.
  This shows that the complement of the original set A is open,
  hence A is closed. This completes the proof that 
  X×_0 Y is a k-space in the product topology.



The analog of Proposition <ref> (ii) 
is also true for open subsets, i.e., every open subset 
of a k-space is a k-space with respect to the subspace topology.
We will not use this, 
so we refer the reader to <cit.>.



If X is any topological space we let k X be the space which has the
same underlying set as X, but such that the closed subsets of k X
are the compactly closed subsets of X. This indeed defines a topology 
which makes k X into a k-space
and such that the identity :k X→ X is continuous.
Moreover, any continuous map Y→ X whose source Y is a k-space 
is also continuous when viewed as a map to k X. 
In more fancy language, the assignment X↦ k X extends to a functor
k:→𝐊 that is right adjoint
to the inclusion of the full subcategory of k-spaces.
Since the inclusion 𝐊→ has a right adjoint,
the category 𝐊 of k-spaces has small limits and colimits. 
Colimits can be calculated in the ambient category of all topological spaces;
equivalently, any colimit of k-spaces is again a k-space.
To construct limits, we can first take a limit 
in the ambient category of all topological spaces; this ambient limit
need not be a k-space, but applying the Kelleyfication functor  
k:→𝐊 yields a limit in 𝐊. Since k
does not change the underlying set, the categories 𝐊 and
 share the property that the forgetful functor to sets
preserves all limits and colimits. More loosely speaking, the underlying
set of a limit or colimit in 𝐊 is what one first thinks of.

The discussion about limits above applies in particular to products,
and the product of two k-spaces need not be a k-space in
the usual product topology.
As already mentioned, we denote by X×_0 Y the cartesian
product of two spaces X and Y, endowed with the product topology
(which makes it a categorical product in the category  of all topological spaces).
We denote by X× Y=k(X×_0 Y) the Kelleyfication 
of the product topology; if X and Y are k-spaces, then X× Y
is a categorical product in the category 𝐊.
Proposition <ref> (vi)
shows that Kelleyfication is unnecessary if one
of the factors is locally compact Hausdorff.

An important example where products in  and  
can differ is the product of two `sufficiently large' CW-complexes X and Y. 
Every CW-complex is a k-space,
for example by <cit.>.
The product X×_0 Y with the usual product topology 
comes with a filtration (X×_0 Y)_(n)=∪_p+q=nX_(p)×_0 Y_(q),
where X_(p) is the p-skeleton of the CW-structure on X.
If X or Y is locally compact, then the product topology
is a k-space by Proposition <ref> (vi),
and then the above filtration makes X×_0 Y into a CW-complex,
as was already noted 
by J. H. C. Whitehead <cit.>.
In general, however, 
X×_0 Y may not be a k-space, and hence cannot have a CW-structure.
The first example of this phenomenon 
was given by Dowker <cit.>,
namely where X and Y are countably respectively uncountably
infinite wedges of circles.
This is also an example where the topology on X× Y=k(X×_0 Y)
is strictly finer than the product topology.
The product in the category 𝐊, i.e., the space X× Y=k(X×_0 Y),
is always compactly generated and a CW-complex via the above filtration,
see for example <cit.>.




A surjective continuous 
map p:X→ Y is a proclusionsubjectproclusion
if whenever O⊂ Y is such that p^-1(O)
is open in X, then O is already open in Y.
So a proclusion is a continuous map that is homeomorphic, under X,
to the projection onto a quotient space.
The following proposition is one of the key properties of the category of k-spaces,
and is stated without proof as Proposition 2.2 of <cit.>.
The argument goes back, at least, to Steenrod,
who states it in <cit.>
for Hausdorff k-spaces. The analog of the following proposition
does not hold for general topological spaces in the usual product topology,
and the examples in <cit.>
illustrate what can go wrong.



Let X and Z be k-spaces and p:X→ Y a proclusion.
Then the map p× Z:X× Z→ Y× Z is a proclusion.


 We adapt Steenrod's argument from <cit.>
to the more general context, i.e., for k-spaces
without any additional separation hypothesis.
We start with the special case when Z is compact. 
In that case X× Z=X×_0 Z and
Y× Z=Y×_0 Z by Proposition <ref> (vi),
i.e., the usual product topologies coincide with their Kelleyfications.
In this formulation, the special case goes back 
to J. H. C. Whitehead <cit.>,
and the proof can also be found in many topology textbooks,
for example <cit.>.

Now we reduce the general case to the special case.
We consider a subset A⊂ Y× Z
such that (p× Z)^-1(A) is closed in X× Z.  
We let

    f  =  (f_1,f_2)  :  K → Y× Z

be a continuous map from a compact space.
Then 

    (p× K)^-1((Y× f_2)^-1(A)) =  
     (X× f_2)^-1((p× Z)^-1(A))

is closed in X× K.
The map p× K:X× K→ Y× K
is a proclusion by the special case, so 
the set (Y× f_2)^-1(A) is closed in Y× K.
So

    f^-1(A) =  (f_1,_K)^-1( (Y× f_2)^-1(A))

is closed in K.
This shows that A is compactly closed, and hence closed in
the k-space topology of Y× Z.


Now we turn to weak Hausdorff spaces, and collect various
useful properties in the following proposition.



  
    
  * If i:A→ X is a continuous injection and X is weak Hausdorff, then
    A is also weak Hausdorff. In particular, every subspace 
of a weak Hausdorff space is again weak Hausdorff.
  
  * Every Hausdorff space is also weak Hausdorff.
  
  * Every finite subset of a weak Hausdorff space is closed.
  
  * Every continuous bijection from a compact space to weak Hausdorff space 
    is a homeomorphism.
  
  * Let f:K→ X be a continuous map from a compact space to a weak Hausdorff
    space. Then the image f(K) is compact in the subspace topology.
  
  * Every disjoint union of weak Hausdorff spaces is weak Hausdorff.
  
  * Every limit in  of a functor with values in 
    weak Hausdorff spaces is weak Hausdorff.
  
  * If X is a weak Hausdorff space, then its Kelleyfication
    k X is again weak Hausdorff.
  


  (i) Let f:K→ A be a continuous map from a compact space.
  Then i f:K→ X is also continuous, hence (i f)(K) is closed in X.
  Since i is injective we have f(K)=i^-1( (i f)(K)), which is thus
  closed in A. This shows that A is a weak Hausdorff space.

  (ii) If K is compact and f:K→ X continuous, then
  the image f(K) is always quasi-compact; if X is Hausdorff, then
  any quasi-compact subset such as f(K) is closed. 
  So X is weak Hausdorff.

  (iii) Every one point space is compact, so  every point of any space is the continuous 
 image of a compact space. So in weak Hausdorff spaces, all points and 
 thus all finite subsets are closed.

 (iv) Let f:X→ Y be a continuous bijection from a compact space 
 to a weak Hausdorff space. Every closed subset A of X is compact
 in the subspace topology, so f(A) is closed in Y by the weak Hausdorff
 property. This shows that f is also a closed map, hence a homeomorphism.
 
 (v) The property of being quasi-compact is automatically inherited under
 continuous surjections, so the main issue is the Hausdorff property of f(K),
 for which we reproduce the argument from <cit.>.
 We let x,y∈ f(K) be two distinct points. The sets {x}
 and {y} are closed in X by part (iii), so
 f^-1(x) and f^-1(y) are disjoint closed subsets of K.
 Since compact spaces are normal, there are disjoint open subsets U and V
 of K with f^-1(x)⊂ U and f^-1(y)⊂ V.
 Since X is weak Hausdorff, the sets f(K-U) and f(K-V)
 are closed in X, and hence also in f(K).
 But then f(K)-f(K-U) and f(K)-f(K-V) are disjoint open subsets of f(K) 
 that separate x and y.

 (vi) We let {Y_i}_i∈ I be a family of weak Hausdorff spaces and
 f:K→∐_i∈ I Y_i a continuous map from a compact space to
 the disjoint union. Since K is the disjoint union of the closed subspaces
 K_i=f^-1(Y_i), each K_i is compact in the subspace topology. 
 The image of the restriction f|_K_i:K_i→ Y_i is closed
 since Y_i is weak Hausdorff. So f(K)=∐_i∈ If(K_i)
 is closed in the disjoint union.

 (vii) In a first step we show that a product of any family 
 {Y_i}_i∈ I of weak Hausdorff spaces is weak Hausdorff.
 We let f=(f_i)_i∈ I:K→∏^0_i∈ IY_i be a continuous map from a compact space,
 with respect to the product topology on the target. 
 Because Y_i is weak Hausdorff, the subset f_i(K) is closed in Y_i, 
 and a Hausdorff space in the subspace topology by part (v).
 So the subset ∏_i∈ I f_i(K) is closed in ∏_i∈ I^0 Y_i
 and Hausdorff in the subspace topology. 
 Since K is compact  and the image of f is contained in
 the Hausdorff space ∏_i∈ I f_i(K), 
 the image f(K) is closed in the subspace topology of ∏_i∈ I f_i(K).
 Since ∏_i∈ I f_i(K) is closed in ∏_i∈ I^0 Y_i,
 the image f(K) is also closed in ∏^0_i∈ IY_i.
 This proves that the product ∏^0_i∈ IY_i
 is weak Hausdorff.

 A limit in  of a functor F:I→ is a subspace
 of the product of the values F(i) for all object i∈ I.
 Since a subspace of a weak Hausdorff space is weak Hausdorff by part (i),
 this proves the claim.

 (viii) The identity is a continuous injection k X→ X, so part (i) proves the claim.


Now we turn to compactly generated spaces. 
By combining Propositions <ref>
and <ref>, we get some immediate corollaries:



  
  
  * Every closed subset of a compactly generated space
    is compactly generated in the subspace topology.
  
  * Every locally compact Hausdorff space, and hence every compact space,
    is compactly generated.
  
  * Every metric space is compactly generated.
  
  * Every disjoint union of compactly generated spaces is compactly generated.
    
  * If X is compactly generated and Y locally compact Hausdorff, then
      X×_0 Y is compactly generated in the product topology.
  



There is a useful criterion, due to McCord <cit.>,
for when a k-space is weak Hausdorff (and hence compactly generated).

 
A k-space X is weak Hausdorff if and only if 
the diagonal is closed in X× X=k(X×_0 X). 


We suppose first that the  diagonal is closed in X× X. 
We let f:K→ X be a continuous map from a compact space.
Then the map

    f× X  :  K× X  → X× X

is continuous, and so (f× X)^-1(Δ_X) is closed in
K× X. Since K is compact, the topology on K× X
is the usual product topology by Proposition <ref> (vi).
Moreover,  the projection p:K× X→ X away from K is a closed map,
also by compactness.
Hence

    f(K) =  p((f× X)^-1)(Δ_X)

is closed in X. This shows that X is weak Hausdorff.

For the converse we assume that X is weak Hausdorff.
We show that the diagonal Δ_X is compactly closed in X×_0 X,
and hence closed in X× X. 
We let f=(f_1,f_2):K→ X×_0 X be a continuous map from a compact space.
Then the map

    f_1+f_2  :  K⨿ K  → X

is continuous.
Since X is weak Hausdorff, the subset A=(f_1+f_2)(K⨿ K)=f_1(K)∪ f_2(K)
of X is closed and a Hausdorff space in the subspace topology, by
Proposition <ref> (v). 
So the diagonal Δ_A is closed in A×_0 A.
Since moreover, f(K) is contained in A×_0 A,
the set
f^-1(Δ_X)  =  f^-1(Δ_A)
is closed in K. This completes the proof that Δ_X is compactly closed.


Proposition <ref> now leads to the following useful criterion
for when a quotient space of a k-space is weak Hausdorff (and hence compactly generated).

  Let X be a k-space and E⊂ X× X an equivalence relation.
  Then the quotient space X/E is compactly generated if and only if E is
  closed in the k-topology of X× X.


We let p:X→ X/E denote the projection.
Any quotient space of a k-space is automatically a k-space
by Proposition <ref> (i).
If X/E is also weak Hausdorff, then Δ_X/E is closed in 
X/E× X/E by Proposition <ref>. 
Hence 

    E  =  (p× p)^-1(Δ_X/E)

is closed in X× X.
Conversely, suppose that E is closed in X× X.
Since X and X/E are k-spaces, the map

    p× p  :  X× X  → (X/E)×(X/E)

is a proclusion by two applications of Proposition <ref>.
Since E is closed by hypothesis, the relation (<ref>) 
shows that Δ_X/E is closed in (X/E)×(X/E).
So X/E is weak Hausdorff by Proposition <ref>.





Let X be a compactly generated space and A a closed subset of X.
Then the quotient topology on X/A is again compactly generated.


Since A is closed in X, A× A is closed in X× X.
Since X is weak Hausdorff, the diagonal Δ_X is closed X× X.
So the equivalence relation E=(A× A)∪Δ_X is closed in X× X,
and the quotient space X/A is compactly generated 
by Proposition <ref>.



Proposition <ref> 
also suggests how to construct a `maximal weak Hausdorff quotient'
of a k-space:



Let X be a k-space. Let E_min⊂ X× X be the intersection of
all equivalence relations on X that are closed
in the k-topology of X× X.
Then X/E_min with the quotient topology is a compactly generated space
and the quotient map X→ X/E_min is initial among
continuous maps from X to a compactly generated space.


The intersection E_min is again an equivalence relation,
and E_min is closed in  X× X as an intersection of closed subsets; 
so the quotient topology is weak Hausdorff by Proposition <ref>.

Now we let f:X→ Y be a continuous map to a compactly generated space.
The equivalence relation

    E  =  { (x,x')∈ X× X  |  f(x)=f(x') }

is the preimage of the diagonal under the continuous map

    f× f  :  X× X → Y× Y  .

Since the diagonal is closed in Y× Y by Proposition <ref>, 
the set E is closed in X× X, 
and so E_min⊆ E. So f factors uniquely over
a continuous map from X/E_min, by the universal property of the
quotient topology.



The previous proposition implies that the assignment

    X ⟼ X/E_min =  w(X)

extends canonically to a functor

    w  :   →

that is left adjoint to the inclusion of compactly generated spaces into k-spaces. 
Moreover, if X is already compactly generated, then the 
diagonal is closed in X× X by Proposition <ref>; 
every equivalence relation contains the diagonal, 
so E_min=Δ_X whenever X is 
compactly generated. In this situation the quotient map X→ X/E_min=w(X)
is a homeomorphism.

It follows formally from the existence of a left adjoint 
to the inclusion 𝐊⊂ that the category of compactly generated spaces 
has small limits and colimits; 
limits can be calculated in the category 𝐊 of k-spaces.subjectlimit!in 
To construct a colimit of a diagram in , we can first take a colimit 
in the category 𝐊 of k-spaces (or equivalently in );
while a k-space, this colimit need not be weak Hausdorff,
but applying the left adjoint w:𝐊→ yields a colimit in .
subjectcolimit!in  



If the colimit, taken in the category  of k-spaces,
of a diagram of compactly generated spaces is not already weak Hausdorff,
then the minimal closed equivalence relation on it is strictly larger than the
diagonal, so the `maximal weak Hausdorff quotient' 
identifies at least two distinct points and thus changes the underlying set.
So one has to be especially careful with general colimits
in : unlike for  of 𝐊,
the forgetful functor from  to sets need not preserve colimits.
More loosely speaking, the underlying set  of colimit in  
may be smaller than one first thinks.



The fact that colimits in  may be hard to identify 
may seem like a problem at first. However, the issue is largely irrelevant
for purposes of homotopy theory because we don't expect to have
homotopical control over arbitrary colimits anyhow.
The colimits that we do care about turn out to be `as expected';
in particular,
for pushouts along closed embeddings (see Proposition <ref>),  
filtered colimits along closed embeddings 
(see Proposition <ref>), 
wedges (see Proposition <ref>), 
and orbits by actions of compact topological groups 
(see Proposition <ref>),
it makes no difference if we calculate 
the colimit in the category  or in  respectively .


We call a continuous map i:A→ X between topological spaces
a closed embeddingsubjectclosed embedding if i is injective
and a closed map; equivalently, the image i(A) is closed in X 
and i is a homeomorphism onto its image.
By Proposition <ref> below,
the cobase change, in ,  or , of a closed embedding is again 
a closed embedding.



There is an ambiguity with the meaning of `embedding' in general, 
due to the fact that a general subset of a k-space, endowed with the
subspace topology, need not be a k-space,
and so one may or may not want to apply `Kelleyfication' k:→
to the subspace topology. However, closed subsets of k-spaces
are again k-spaces in the usual subspace topology,
so there is no such ambiguity with the notion of `closed embedding'.




Let i:A→ X and j:B→ Y be closed embeddings between topological spaces.
Then the product maps i×_0 j:A×_0 B→ X×_0 Y
and i× j:A× B→ X× Y are closed embeddings.


The map i× j is clearly injective, so must show that it is a closed
map with respect to the product topologies, and with respect to their Kelleyfications.
We may assume that i is the inclusion of a closed subset A⊂ X
and j is the inclusion of a closed subset B⊂ Y.
The subspace topology on A× B induced from X×_0 Y
agrees with the product topology A×_0 B,
which is the claim for i×_0 j, i.e., the usual product topologies.

For the second claim we must show that every compactly closed subset C
of A×_0 B is also compactly closed in X×_0 Y.
So we let f:K→ X×_0 Y be a continuous map from a compact space.
Since A×_0 B is closed inside X×_0 Y,
the subset L=f^-1(A×_0 B) is closed in K, and hence compact
in the subspace topology. 
The relation

    f^-1(C) =  (f|_L)^-1(C)

and the hypothesis that C is compactly closed inside A×_0 B
then show that f^-1(C) is closed in K. 
So C is compactly closed in X×_0 Y.




This following proposition is <cit.>. 



Let i:X→ Y be a continuous map between compactly generated spaces
that admits a continuous retraction.
Then i is a closed embedding.


Let r:Y→ X be a continuous retraction, i.e., such that r i=_X.
Then the map

    (_Y,i r)  :  Y → Y× Y

is continuous and for every subset A⊂ X we have

    i(A)  =   {y∈ Y |  y=i (r(y))  and  r(y)∈ A} =  (_Y,r i)^-1(Δ_Y) ∩  r^-1(A) .

Since Y is compactly generated, the diagonal Δ_Y is closed
in Y× Y by Proposition <ref>;
hence the set (_Y,r i)^-1(Δ_Y) is closed in Y.
If A is closed, then so is r^-1(A), and hence also i(A).
This proves that i is a closed embedding.





    Given a pushout square in the category  of topological spaces
    
    X [r]^-f [d]_i     Z [d]^j 
    
          Y [r]_-g     P

       such that i is a closed embedding, then j is also a closed
    embedding. If moreover Y and Z are compactly generated, then
    so is P, and hence the square is a pushout in .


We adapt the argument given in <cit.>.
The map j is injective because i is. Indeed, we can choose
a set-theoretic retraction r:Y→ X to i (not necessarily continuous),
and then (f r)∪: P→ Z is a set-theoretic retraction to j.

For the other claims we first treat the special case where the map g is a proclusion.
We let A⊂ Z be a closed subset; then f^-1(A) is closed in X. 
Since the map i is injective, the relation

    g^-1(j(A))  =  i(f^-1(A))

holds as subsets of Y. So i(f^-1(A)), and hence g^-1(j(A)),
is closed in Y because i is a closed embedding.
Since g is a proclusion, this shows that j(A) is closed, 
and hence j is a closed map.
Altogether this proves that j is a closed embedding.

Now we suppose that Y and Z are compactly generated.
Then P is a k-space by Proposition <ref> (i)
because g:Y→ P is a proclusion.
Moreover, the diagonals of Y and Z
are closed in Y× Y respectively Z× Z by 
Proposition <ref>. 
Hence (f× f)^-1(Δ_Z) is closed in X× X.
Since i is a closed embedding, so is i× i:X× X→ Y× Y,
by Proposition <ref>.
Because

    (g× g)^-1(Δ_P) =  Δ_Y∪ (i× i)( (f× f)^-1(Δ_Z))

we conclude that (g× g)^-1(Δ_P) is closed in Y× Y.
Since g is a proclusion, so is g× g, by two applications of
Proposition <ref>.
So Δ_P is closed in P× P, 
and P is weak Hausdorff by the criterion of Proposition <ref>. 

Now we treat the general case.
The pushout P can be constructed as a quotient space of the disjoint union
Y⨿ Z, which means that the lower horizontal map 
in the commutative square

    @C=12mm X⨿ Z [r]^-f+[d]_i⨿    Z [d]^j 
    
      Y ⨿ Z[r]_-g+j    P

is a proclusion.
Since the original square is a pushout, so is this new square.
The spaces Y⨿ Z and Z are compactly generated by hypothesis
and Proposition <ref> (iv),
and the left vertical map is a closed embedding. 
So we can apply the special case above to this new square, 
and conclude that P is compactly generated.


The following example of Lewis <cit.> 
shows that a pushout of compactly generated spaces calculated 
in the ambient category  need not be compactly generated, and that 
a cobase change in  of a continuous injection need 
not be injective. We consider the diagram

    {-1,1}      [-1,0) ∪ (0,1]  inclusion [-1,1]

where all three spaces have the subspace topology of , and the left map 
takes [-1,0) to -1 and it takes (0,1] to 1.
All three spaces are compactly generated, and the pushout P 
in the categories  or  has three points, only one of which is closed.
Any continuous map from P to a weak Hausdorff space must be constant, 
so the space w(P) (which is a pushout in ) has only one point.





We recall that a poset (P,≤) 
is filteredsubjectfiltered  poset if 
for all elements i,j∈ P there is a k∈ P such that i≤ k and j≤ k.
The associated category has object set P 
and a unique morphism (i,j):i→ j for every pair of elements such that i≤ j.
The filtered poset we mostly care about is (,≤), the set of natural numbers
under it usual ordering; a functor F:→ from the associated poset category
is determined by the sequence of morphisms

    F(0) F(0,1) F(1) F(1,2) F(2) F(2,3) … F(n-1,n) F(n)      …

So colimits of such functors are sequential colimits.
Other filtered posets that come up are the set of finite subsets of an infinite set,
and the set of finite-dimensional vector subspaces of an infinite dimensional
vector spaces, both ordered by inclusion.

  
  Let P be a filtered poset and F:P→ a
  functor from the associated poset category to the category of topological spaces. 
  Let F_∞ be a colimit of F in the category 
  with respect to continuous maps κ_i:F(i)→ F_∞.
  
  
  * If the map F(i,j):F(i)→ F(j) is a closed embedding
    for every comparable pair of elements of P, then 
    all the maps κ_i:F(i)→ F_∞ are closed embeddings.
  
  * If the space F(i) is compactly generated for every element i∈ P
    and the map F(i,j):F(i)→ F(j) in injective 
    for every comparable pair of elements,
    then the colimit F_∞ is compactly generated.
    Hence F_∞ is also a colimit of the functor F in the category .
  


(i) Colimits in  are created on underlying sets, so the map
κ_i:F(i)→ F_∞  is injective because all the maps F(i,j) are injective.
Now we let A⊂ F(i) be closed subset. If j∈ P is
another element, we can choose k∈ P such that i≤ k and j≤ k.
Then

    κ_k^-1(κ_k(F(i,k)(A)))  =  F(i,k)(A)

because the map κ_k is injective.
Hence

    κ_j^-1(κ_i(A)) = F(j,k)^-1(κ_k^-1(κ_k(F(i,k)(A)))) = 
    F(j,k)^-1(F(i,k)(A))  .

Since F(i,k) is a closed map, this shows that
κ_j^-1(κ_i(A)) is closed in F(j) for all j∈ P.
The map ∐_j∈ PF(j)→ F_∞
given by κ_j on F(j) is a proclusion, 
so we conclude that κ_i(A) is closed in F_∞.
So κ_i is a closed embedding.

(ii) The map κ:∐_j∈ PF(j)→ F_∞
given by κ_j on F(j) is a proclusion, and the source is compactly generated by 
Proposition <ref> (iv).
We show that the equivalence relation 

    E  =  (κ×κ)^-1(Δ_F_∞)  ⊂ (⨿_j∈ P  F(j))×(⨿_j∈ P  F(j))

that gives rise to F_∞ is closed; the criterion
of Proposition <ref> then shows that F_∞
is compactly generated.
The product of ∐_j∈ PF(j) with itself
is the disjoint union of the subspaces F(i)× F(j) for all i,j∈ P,
so it suffices to show that the intersection of E with all
F(i)× F(j) is closed. We choose k∈ P such that i≤ k and j≤ k.
Then

    E∩ (F(i)× F(j)) =  (F(i,k)× F(j,k))^-1(Δ_F(k))

because the map κ_k:F(k)→ F_∞ is injective.
The diagonal is closed in F(k)× F(k) since F(k) is compactly generated
(by Proposition <ref>).
So E∩ (F(i)× F(j)) is closed in F(i)× F(j).
This verifies the criterion of Proposition <ref> 
and concludes the proof that the colimit F_∞ is compactly generated.




Many functors F:P→ out of filtered posets that arise in practice 
have the property that a continuous map f:K→ F_∞
from a compact space factors through F(i) for some i∈ P.
Such filtered colimits then tend to preserve weak equivalences.
We now formulate a precise version of this property for certain lattices.
The most common special case is the poset (,≤) 
of natural numbers under the usual linear ordering.

We recall that a poset (P,≤) is a latticesubjectlattice
if every pair of elements p,q∈ P has a join p∨ q (least upper bound) 
and a meet p∧ q (greatest lower bound).
Every lattice is in particular a filtered poset.

 
Let (P,≤) be a lattice with the following property:
for every element q∈ P and every countable chain

    p_1 ≤ p_2 ≤ … ≤ p_n ≤ …

in P with p_n≤ q for all n≥ 1, 
the sequence is eventually constant.


  * Let F:P→ be a functor to the category of compactly generated spaces
with the following properties:


  * the map F(i,j):F(i)→ F(j) is a closed embedding for all comparable 
pairs i≤ j of elements of P, and

  * for all elements p,q∈ P the following square is a pullback:

    F( p∧ q) [r][d]     F(q)[d]
    
    F(p)[r]     F(p∨ q)

Let F_∞ be a colimit of F with respect to the continuous maps
κ_i:F(i)→ F_∞.
Then for every  continuous map α:K→ F_∞ from a compact space K
there exists an element i∈ P and a  continuous map α':K→ F(i)
such that α=κ_i∘α'.

  * Let F':P→ be another functor satisfying conditions (a) 
and (b) and ψ:F→ F' a natural transformation.
Suppose that the map ψ(i):F(i)→ F'(i) is m-connected for all i∈ P,
for some m≥ 0.
Then the map ψ_∞:F_∞→ F'_∞ induced on colimits is m-connected.

  * Let F:P→ satisfy conditions (a) 
and (b) and suppose that the map F(i,j):F(i)→ F(j) is a m-connected
for all pairs of comparable elements.
Then the map  κ_i:F(i)→ F_∞ is m-connected for every i∈ P.



(i) The canonical maps κ_i:F(i)→ F_∞ are
closed embeddings by Proposition <ref> (i). 
Hence we can, and will, pretend that F(i) is a closed subspace
of F_∞ and the maps F(i,j):F(i)→ F(j) are inclusions.

The image of the continuous map f:K→ F_∞ is compact in the subspace topology
by Proposition <ref> (v). So we may replace
K by its image and pretend that f is the inclusion of a compact subspace
of F_∞. We then have to show that K⊂ F(i) for some i∈ P.

We argue by contradiction and assume that K is not contained in F(i)
for any i∈ P.
We can then choose comparable elements

    p_1 ≤ p_2  ≤ … ≤ p_n  ≤ …

of P and elements x_i∈ K with x_i∈ F(p_i) F(p_i-1).
Indeed, we start with any choice x_1∈ F(p_1) and continue inductively:
since K is not contained in F(p_n), there is an element x_n+1∈ K F(p_n).
We must have x_n+1∈ F(q) for some q∈ P, and then p_n+1=p_n∨ q
can serve for the inductive step.

We set C =  {x_1,x_2,x_3,…}, a countably infinite subset of K.
Now we claim that for every q∈ P the 
intersection C∩ F(q) is finite.
If this were not the case, then there would be infinitely many indices
i_1<i_2<i_3<… such that x_i_j∈ F(q) for all j≥ 1.
The commutative square

    F( p_i_j∧ q) [r][d]     F(q)[d]
     F(p_i_j) [r]    
    F(p_i_j∨ q)

is a pullback by hypothesis,
and so x_i_j∈ F(p_i_j∧ q) for all j≥ 1.
Also by hypothesis there is an N≥ 1 such that

    p_N∧ q =   p_N+1∧ q =   p_N+2∧ q =  ⋯ .

So F( p_N∧ q), and hence also F(p_N), contains infinitely
many of the elements of the set C. This contradicts the construction of C,
and we can conclude that the intersection C∩ F(q) is finite.

Now we let T be any subset of C.
Since T∩ F(q) is finite, it is a closed subset of F(q)
by Proposition <ref> (iii).
The map

    ∐κ_q  :  ∐_q∈ P F(q) → F_∞

is a proclusion, i.e., the space F_∞ carries 
the quotient topology. Since T∩ F(q) is closed for all q∈ P,
the set T is closed in F_∞. Since T was any subset of C,
the subset C is discrete in the subspace topology from F_∞.
So altogether C is an infinite discrete subset of the compact space K,
which is a contradiction. This proves that K is contained in F(i)
for some i∈ P.

(ii) We consider a commutative square of continuous maps for k≤ m:

    ∂ D^k[r]^α[d]_incl    F_∞[d]^ψ_∞
    
    D^k[r]_β    F'_∞

Part (i) provides i,j∈ P and continuous maps α':∂ D^k→ F(i)
and β':D^k→ F(j) such that 
κ_i∘α'=α and κ'_j∘β'=β.
By replacing i and j by their join we may suppose that i=j.
Then 

    κ'_i∘ψ(i)∘α' = ψ_∞∘κ_i∘α' = ψ_∞∘α = β|_∂ D^k =  κ'_i∘β'|_∂ D^k .

Since κ'_i is injective this shows that ψ(i)∘α'=β'|_∂ D^k.
Since ψ(i) is m-connected, there is 
a continuous map λ:D^k→ F(i) that restricts to α'
on ∂ D^k and such that ψ(i)∘λ is homotopic, 
relative ∂ D^k,
to β'. Then κ_i∘λ:D^k→ F_∞
solves the original lifting problem. So the map F_∞ is m-connected.

(iii) We consider a commutative square of continuous maps for k≤ m:

    ∂ D^k[r]^α[d]_incl    F(i)  [d]^κ_i
    
    D^k[r]_β    F_∞

Part (i) provides a j∈ P and a continuous map β':D^k→ F(j) such that 
κ_j∘β'=β.
By replacing j by the join i∨ j 
and β' by the map F(i,i∨ j)∘β':D^k→ F(i∨ j),
we can assume without loss of generality that i≤ j.
Then 

    κ_j∘ F(i,j)∘α = κ_i∘α = β|_∂ D^k =  κ_j∘β'|_∂ D^k .

Since κ_j is injective this shows that F(i,j)∘α=β'|_∂ D^k.
Since F(i,j) is m-connected, there is 
a continuous map λ:D^k→ F(i) that restricts to α
on ∂ D^k and such that F(i,j)∘λ is homotopic, 
relative ∂ D^k, to β'. Then λ also
solves the original lifting problem. So the map κ_i is a m-connected.




We let (P,≤) be a linear ordersubjectlinear order,
i.e., a poset in which every two elements are comparable.
Such a poset is in particular a lattice. 
We observe that for every linear order, 
condition (b) of Proposition <ref> (i)
is trivially satisfied for every functor.
Indeed, any two elements p,q∈ P are comparable,
and we suppose that p≤ q, the other case being analogous. 
Then p∧ q=p and p∨ q=q, and
the two vertical maps in the commutative square of condition (b) are
identity maps. Hence the square is a pullback.  




The most familiar special case of Proposition <ref>
is the poset (,≤) of natural numbers under the usual ordering.
This poset is a linear order, so as we explained in Example <ref>,
condition (b) of Proposition <ref> (i)
is automatically satisfied for every functor.
For (,≤), part (i) of Proposition <ref>
can be found in many topology textbooks, for example <cit.>.
For easier reference we spell out the content of part (ii) and (iii) 
of Proposition <ref> for the poset (,≤)
and for m=∞, i.e., for weak equivalences.


  
   
  * Let e_n:X_n→ X_n+1 and f_n:Y_n→ Y_n+1 be closed embeddings
    between compactly generated spaces, for n≥ 0. 
    Let ψ_n:X_n→ Y_n be weak equivalences
    that satisfy ψ_n+1∘ e_n=f_n∘ψ_n for all n≥ 0.
    Then the induced map ψ_∞:X_∞→ Y_∞ 
    between the colimits of the sequences is a weak equivalence.
  
  * Let f_n:Y_n→ Y_n+1 be a closed embedding of compactly generated spaces
   that is also a weak equivalence, for n≥ 0. 
   Then the canonical map Y_0→ Y_∞ to the colimit of the sequence
   is a weak equivalence.
  


Besides the linear order (,≤), 
further examples of relevant posets that satisfy the hypotheses of 
Proposition <ref> are 
the set of finite subsets of a given set 
(which features in the following proposition),
and the set of finite-dimensional vector subspaces of a given vector space.


 
Let {X_i}_i∈ I be a family of based compactly generated
spaces. Then the wedge (one-point union) ⋁_i∈ IX_i
is compactly generated, and thus a coproduct in the category _* 
of based compactly generated spaces. 
Moreover, for every compact space K and every continuous map
f:K→⋁_i∈ IX_i there is a finite subset J of I such that
f factors through the subwedge ⋁_j∈ J X_j.


The disjoint union ∐_i∈ IX_i is compactly generated
by Proposition <ref> (iv).
Moreover, the equivalence relation that identifies all the basepoints is closed
in (∐_i∈ IX_i)×(∐_i∈ IX_i).
So the quotient space ⋁_i∈ IX_i is compactly generated by 
Proposition <ref>.

We let P be the poset of finite subsets of I, ordered by inclusion;
this is a lattice satisfying the hypotheses of 
Proposition <ref>.
We consider the functor F:P→ sending a finite subset J⊂ I
to the finite wedge ⋁_j∈ JX_j.
For J'⊂ J⊂ I, the map ⋁_j∈ J'X_j→⋁_j∈ JX_j
is a closed embedding 
(by direct inspection, or by Proposition <ref>),
and property (b) of Proposition <ref> (i)
is satisfied.
Proposition <ref> (i) 
thus provides the desired factorization through ⋁_j∈ JX_j 
for some finite subset J of I.




Now we let X and Y be compactly generated topological spaces 
equipped with basepoints x_0∈ X respectively y_0∈ Y. 
We define the smash product assubjectsmash product!of based spaces

    X Y  =   X× Y / ( X×{y_0}∪{x_0}× Y)  ,

the quotient of the product (with the k-topology)
by the subspace X×{y_0}∪{x_0}× Y.
We write x y for the class of  (x,y)∈ X× Y in the quotient space.
Since points in compactly generated spaces are closed
(Proposition <ref> (iii)), the map 

    X∨ Y → X× Y

from the coproduct to the product in _*
is a closed embedding with image the subspace 
X×{y_0}∪{x_0}× Y;
so X Y is also a cokernel of this map.

The next proposition is another important reason for working in 
the category  of compactly generated spaces. 
Indeed, in the larger category  of all topological spaces,
the smash product is not generally associative:
if smash products are defined as quotients of the
usual product topology, then the canonical set-theoretic bijection from 
(X Y) Z to X (Y Z) need not be continuous.
An explicit example is mentioned without proof in <cit.>,
namely that () and () 
are not homeomorphic in , where  has the subspace topology of ,
and  has the discrete topology.
A proof can be found in <cit.>.

 
Let X, Y, and Z be based compactly generated spaces.
  
  
  * The space X Y is compactly generated in the quotient topology
    of X× Y.
  
  * The maps
    
    X Y  → Y X ,     x y ⟼ y x

    and
    
    (X Y) Z  → X (Y Z) ,     
        (x y) z ⟼ x (y z)

    are homeomorphisms.
  


(i) In weak Hausdorff spaces all points are closed 
(Proposition <ref> (iii)), hence
X×{y_0}∪{x_0}× Y is a closed subspace of X× Y.
So the quotient topology is compactly generated by Corollary <ref>. 

The first claim of part (ii) is straightforward from the fact that the twist homeomorphism
X× Y→ Y× X sending (x,y) to (y,x) takes the subspace
X×{y_0}∪{x_0}× Y to the subspace
Y×{x_0}∪{y_0}× X and hence descends to a homeomorphism
on quotient spaces.

Since the projection p_X,Y:X× Y→ X Y is a proclusion,
so is p_X,Y× Z: X× Y× Z→ (X Y)× Z,
by Proposition <ref>. Hence the map

    p_X Y,Z∘(p_X,Y× Z)  : X× Y× Z → (X Y) Z  ,    
    (x,y,z) ⟼ (x y) z
 
is a proclusion as the composite of two proclusions.
The composite

    X× Y× Z p_X Y,Z∘(p_X,Y× Z) 
    (X Y) Z  (x y) z ↦ x(y z) X (Y Z)

is the map p_X,Y Z∘(X× p_Y,Z), and hence continuous.
Since  p_X Y,Z∘(p_X,Y× Z) is a proclusion, the
second map (X Y) Z → X (Y Z) is continuous.
By the same reasoning the inverse map 
X (Y Z)→ (X Y) Z is continuous. Hence the two maps
are mutually inverse homeomorphisms.




Another key advantage of the categories  and 
over the category of all topological spaces is that they are cartesian closed,
i.e., categorical product with a fixed object is a left adjoint.
The discovery of these facts was predated 
by similar result in closely related categories;
for example, Brown <cit.>
discusses internal function spaces in the category of Hausdorff k-spaces,
and this was popularized by Steenrod in <cit.>.
The first reference to cartesian closedness for  and  that I am aware of is
the appendix of Lewis' thesis <cit.>;
we will closely follow Lewis' exposition.
Internal function spaces in  and  are given by the set of all continuous maps 
endowed with the Kelleyfication 
of a slight modification of the compact-open topology.
For weak Hausdorff spaces, the `modified' compact-open topology actually coincides with
the compact-open topology.


For topological spaces X and Y, we let C(X,Y)
denote the set of continuous maps from X to Y. 
For a continuous map h:K→ X from a compact space K and an open subset U of Y
we define a subset of C(X,Y) by

    N(h,U) =  {φ∈ C(X,Y) |  φ(h(K))⊂ U} .

We endow the set C(X,Y) of continuous maps with the topology
generated by the subbasis consisting of all these sets N(h,U).
This topology is covariantly functorial in Y and contravariantly functorial in X.
Indeed, for every continuous map f:Y'→ Y we have

    C(X,f)^-1(N(h,U)) =  N(h,f^-1(U)) ,

which is open in C(X,Y'), and so C(X,f):C(X,Y')→ C(X,Y) is continuous.
Similarly, for a continuous maps g:X→ X' and h:K→ X,
with K a compact space, we have

    C(g,Y)^-1(N(h,U)) =  N(g h,U) ,

which is open in C(X',Y), and so C(g,Y):C(X',Y)→ C(X,Y) is continuous.

Now we let X and Y be k-spaces.
Then C(X,Y) need not be a k-space. We fix this by Kelleyfication,
i.e., we define the internal mapping space in  assymbol(Y,Z) - unbased mapping space 

    (X,Y) =  k C(X,Y)  .




[Relation to the compact-open topology]
We recall that a subbasis for the compact-open topologysubjectcompact-open topology 
on C(X,Y) is given by all sets

    W(K,U) =  {φ∈ C(X,Y) |  φ(K)⊂ U} ,

where K is compact subset of X and U is an open subset of Y.
Because W(K,U)=N(incl:K→ X,U),
the topology generated by the sets N(h,U) is finer than the compact-open topology.
In general, the compact-open topology may be strictly coarser.

For weak Hausdorff spaces X, the topology on C(X,Y) 
coincides with the compact-open topology.
Indeed, if X is weak Hausdorff, then the image of every continuous map h:K→ X
is compact in the subspace topology (Proposition <ref> (v)). 
So in this case N(h,U)=W(h(K),U), and the two subbases coincide.



We recall that a category 
is cartesian closedsubjectcartesian closed category
if it has finite products and product with a fix object is a left adjoint.
The proof of the following theorem can be found 
in <cit.>.

 
For all k-spaces X,Y and Z, the natural map

    Ψ :  (X× Y,Z)  → (X,(Y,Z))   ,   Ψ(f)(x)(y)  =  f(x,y)

is a homeomorphism. 
In particular, the category of k-spaces is cartesian closed.


We reproduce Lewis' arguments.
We start by checking that the map

    η_X  :  X  → C(Y,X× Y) ,   η_X(x)(y) = (x,y)

is continuous. 
Since X is a k-space, it suffices to check that the composite
η_X∘ f is continuous for every continuous map f:L→ X 
from a compact space. Since η_X∘ f=C(Y,f× Y)∘η_L
and C(Y,f× Y) is continuous,
we can assume without loss of generality that X is compact.
If X is compact, then X× Y=X×_0 Y 
by Proposition <ref> (vi), i.e.,
X× Y is already a k-space in the usual product topology.

We let h:K→ Y be a continuous map from a compact space and U an open subset 
of X× Y=X×_0 Y.
Then

    η_X^-1(N(h,U))
      =  {x∈ X | {x}× K ⊂ (X× h)^-1(U)} .

For every x∈η_X^-1(N(h,U)), the `tube lemma' 
(see for example <cit.>)
provides an open neighborhood O of x in X 
such that O× K⊂ (X× h)^-1(U), 
i.e., such that O⊂η_X^-1(N(h,U)).
So the set η_X^-1(N(h,U)) contains a neighborhood of each of its points,
and is thus open. Altogether this shows that inverse images of
the subbasis of the topology on C(Y,X× Y) are open; 
hence η_X is continuous.
Since X is a k-space, the map η_X is also continuous
with respect to the Kelleyfied topology on the target,
i.e., when considered as a map to (Y,X× Y).

We denote the evaluation map by 

    ϵ_Z  :  (Y,Z)× Y =  
    k( C(Y,Z)×_0 Y)  → Z ,   ϵ(f,y) = f(y) .

We show next that for every continuous map
h:K→ Y from a compact space the composite
ϵ_Z∘(C(Y,Z)×_0 h):C(Y,Z)×_0 K→ Z is continuous. 
We let U be an open subset of Z and consider 
a point (φ,k)∈ C(Y,Z)×_0 K in (ϵ_Z∘(C(Y,Z)×_0 h))^-1(U);
this simply means that k∈ (φ∘ h)^-1(U).
Since K is compact and (φ∘ h)^-1(U) is open in K,
there is an open neighborhood V of k in K whose closure V̅
is contained in (φ∘ h)^-1(U).
This closure V̅ is compact in the subspace topology, 
so the set N(h|_V̅:V̅→ Y, U) is open in C(Y,Z),
and it contains the map φ.
Hence N(h|_V̅:V̅→ Y, U)× V is open in C(Y,Z)×_0 K,
and contains the point (φ,k).
Since this open set is also contained in (ϵ_Z∘(C(Y,Z)×_0 h))^-1(U),
this latter set is a neighborhood of each of its points, and hence open 
in C(Y,Z)×_0 K. This completes the proof that the map 
ϵ_Z∘ (C(Y,Z)×_0 h) is continuous. 

Now we show that the evaluation map ϵ_Z itself is continuous. 
At this point the Kelleyfication of the source is crucial, 
as ϵ_Z need not be continuous 
with respect to the topology C(Y,Z)×_0 Y.
So it suffices to show that the composite ϵ_Z∘(g,h):K→ Z
is continuous for every continuous map (g,h):K→ C(Y,Z)×_0 Y 
from a compact space.
The map ϵ_Z∘ (g,h) coincides with the composite

    K  (g,_K) C(Y,Z)×_0 K  C(Y,Z)×_0 h 
    C(Y,Z)×_0 Y   ϵ_Z   Z  .

The composite of the last two maps is continuous by the previous paragraph,
so the whole composite ϵ_Z∘ (g,h) is continuous.

At this point we know that the two maps

    η_X  :  X  → (Y,X× Y)   and  ϵ_Z  :  (Y,Z)× Y  → Z

are continuous.
The rest of the argument is formal.
Since ϵ_X× Y∘(η_X× Y) is the identity of X× Y
and (Y,ϵ_Z)∘η_(Y,Z) is the identity of (Y,Z),
the natural transformations η and ϵ are the unit respectively counit
of an adjunction (-× Y,(Y,-)).
In particular, the map

    Ψ :  C(X× Y,Z)  → C(X,(Y,Z))   ,   Ψ(f)(x)(y)  =  f(x,y)

is bijective.
We let T be another k-space. Replacing X by T× X shows that 

    Ψ :  C(T× X× Y, Z)  → C(T×  X,(Y,Z))

is bijective. The adjunction just established translates this into the fact that 

    C(T,Ψ) :  C(T,(X× Y,Z))  → C(T,(X,(Y,Z)))

is bijective. Since T is an arbitrary k-space, Ψ is a homeomorphism.


Now we turn to compactly generated spaces.

 
  
  
  * If Y is a weak Hausdorff space and X is any topological space,
    then the spaces C(X,Y) and (X,Y) are weak Hausdorff.   
    In particular, if X and Y are compactly generated, then so is (X,Y).
  
  * For all compactly generated spaces X,Y and Z, the natural map
    
    Ψ :  (X× Y,Z)  → (X,(Y,Z))   ,   Ψ(f)(x)(y)  =  f(x,y)

    is a homeomorphism. 
    In particular, the category of compactly generated spaces is cartesian closed.
  


(i) For every point x∈ X the evaluation map

    _x  :  C(X,Y) → Y  ,    f ⟼ f(x)

is continuous. Indeed, for an open set U of Y we have
_x^-1(U)=N(incl:{x}→ X,U) 
which is open because a one-point space is compact.

The evaluation map

    C(X,Y) → ∏^0_x∈ X Y  ,    f  ⟼ {f(x)}_x∈ X

is injective, and it is continuous by the above.
Any product, with the usual product topology, of copies of Y is again weak Hausdorff
by Proposition <ref> (vii).
So C(X,Y) is weak Hausdorff by Proposition <ref> (i).
The space (X,Y)=k C(X,Y) is then still weak Hausdorff
by Proposition <ref> (viii),
and hence compactly generated.

(ii)  Part (i) shows that the category of compactly generated
spaces is closed under the internal function objects in the category 
of k-spaces. Since products in  are also formed in the ambient category ,
part (ii) becomes a special case of 
Theorem <ref>.



[Based mapping spaces]
Since we work a lot with based spaces, we record that the based
version of the mapping space is right adjoint to the smash product in _*.
In combination with the commutativity and associativity property of
the smash product (Proposition <ref>),
and the natural homeomorphisms X S^0 X S^0 X,
this shows that the smash product provides a closed symmetric monoidal structure
on the category _* of based compactly generated spaces.

The based adjunction is a straightforward consequence of the unbased
version in Theorem <ref>.
We let (X,x_0), (Y,y_0) and (Z,z_0) be based compactly generated spaces.
We denote by _*(Y,Z) the subspace of (Y,Z)symbol_*(Y,Z) - based mapping space 
consisting of all based continuous maps f, i.e., such that f(y_0)=z_0;
the basepoint is the constant map with value z_0. 
Since points are closed in compactly generated spaces and
_*(Y,Z) is the preimage of {z_0} under the continuous evaluation map

    (Y,Z) → Z  ,    f ⟼ f(y_0) ,

we see that _*(Y,Z) is a closed subset of (Y,Z).
So _*(Y,Z) is compactly generated in the subspace topology 
(Proposition <ref> (i)).
The composite

    X   η_X  (Y,X× Y)  (Y,proj)(Y,X Y)

takes values in the subspace _*(Y,X Y), so it restricts to a continuous map

    η'_X  :  X → _*(Y,X Y) ,

which is moreover based.
The composite

    _*(Y,Z)× Y  incl× Y (Y,Z)× Y   ϵ_Z  Z

takes the subspace {const}× Y∪_*(Y,Z)×{y_0}
to the basepoint of Z, so it factors over the quotient space
though a continuous based map

    ϵ_Z'  :  _*(Y,Z) Y  → Z  .

By direct verification, the composites

    ϵ'_X Y∘(η'_X Y)
       and  (Y,ϵ'_Z)∘η'__*(Y,Z)

are the identity of X Y respectively _*(Y,Z).
So η' and ϵ' are the unit respectively counit of an 
adjunction for the pair of endofunctors (- Y,_*(Y,-))
on the category _*.



Now we specialize to the space of continuous homomorphisms
between Lie groups, with compact source. 
If (Y,d) is a metric space and K a compact topological space,
then the supremum metric

    d(f,g) =  sup_x∈ K { d(f(x),g(x))}

is a metric on the set C(K,Y) of continuous maps.
The topology induced by the supremum metric 
agrees with the compact-open topology, see for example <cit.>,
and hence also with  the topology of Construction <ref>.
Since the topology on C(K,Y) is metrizable, the space C(X,Y) 
is in particular compactly generated,
by Proposition <ref> (iii).
Hence (X,Y)=C(X,Y), with topology induced by the supremum metric.

The topology of every second countable smooth manifold is metrizable,
so the previous discussion applies in particular when K is a compact Lie group and 
G is a Lie group.
So in that case the compact-open topology
on the space C(K,G) of continuous maps is metrizable,
and hence compactly generated. Hence C(K,G)=(K,G),
i.e., the compact-open topology is the topology of the internal mapping
space in the category  of compactly generated spaces. 

We let (K,G) denote the set of continuous homomorphisms
with the subspace topology of (K,G).
Since (K,G) is metrizable, so is its subspace (K,G),
which is then compactly generated by Proposition <ref> (iii).
The next proposition in particular shows that the space (K,G)
is the topological disjoint union of the orbits, under conjugation,
of the identity component group G^∘.
The key input is a theorem of 
Montgomery and Zippin <cit.>
from 1942 which roughly says that in a Lie group `nearby compact subgroups are conjugate'.
The following consequence of Montgomery and Zippin's result
appears in <cit.>.




Let K and G be Lie groups, and suppose that K is compact.
Then every orbit of the conjugation action by G^∘ on (K,G)
is an open subset of the space (K,G).
In particular, the connected components of the space (K,G) coincide
with its path components, and with the G^∘-orbits under the conjugation action.


We reproduce the argument from <cit.>, in somewhat
expanded form. 
We start by showing that the G^∘-orbits are open subsets of (K,G).
Let α:K→ G be a continuous homomorphism. 
Its graph Γ_α={(k,α(k)) | k∈ K} is then a compact subgroup
of the Lie group K× G. 
By <cit.>,
there is an open subset O of K× G containing Γ_α
with the following property:
for every closed subgroup Δ of K× G with Δ⊆ O
there is an element (k,g)∈ K^∘× G^∘ such that
(k,g)^-1·Δ·(k,g)⊆Γ_α.
We set

    U  =  {β∈(K,G) |  Γ_β⊆ O } ;

then α∈ U, and U is contained in the G^∘-orbit of α because

    (k,g)^-1·Γ_β·(k,g) =  Γ_c_g∘ c_β(k)^-1∘β .

We claim that U is an open subset of (K,G).
To see this, we let β:K→ G be a continuous homomorphism
such that Γ_β⊂ O,
and k∈ K. Since O is open and (k,β(k))∈ O,
there are open subsets U_k⊂ K and V_k⊂ G with

    (k,β(k)) ⊆ U_k× V_k  ⊆ O  .

Since β is continuous, the set β^-1(V_k) is open in K,
hence so is U_k∩β^-1(V_k); moreover, k∈ U_k∩β^-1(V_k).
Since K is compact, there is a compact neighborhood L_k of k 
that is contained in U_k∩β^-1 (V_k).
Since L_k is a neighborhood of k and K is compact, there
are finitely many points k_1,…,k_n such that
K=L_k_1∪…∪ L_k_n.
Since β(L_k_i)⊂ V_k_i for all 1≤ i≤ n, the 
homomorphism β is contained in the set

    W =  ⋂_i=1,…,n  N(incl:L_k_i→ K,V_k_i)

which is an open subset of (K,G).
So we are done if we can show that every β∈ W∩(K,G)
satisfies Γ_β⊂ O.
Given k∈ K, there is an i∈{1,…,n} with k∈ L_k_i.
Thus β(k)∈ V_k_i by hypothesis on β, and so

    (k,β(k)) ∈ L_k_i× V_k_i ⊆ O  .

Since k was arbitrary, this proves that Γ_β⊆ O.
Since the set U is open in (K,G) and contained in the G^∘-orbit 
of α, the G^∘-orbit of α is open in (K,G).

Now we show that every G^∘-orbit is path connected.
For every continuous homomorphism α:K→ G, the map 

    G × K  → G  ,   
    (g,k) ⟼ g^-1·α(k)· g

is continuous since it can be written as a composite of a diagonal map,
an evaluation map and data from the group structure.
The adjoint 

    G  → (K,G)  ,     g ⟼ c_g∘α

is thus continuous as well.
Since the component group G^∘ is path connected and conjugation is continuous,
every G^∘-orbit is path connected, so in particular connected. 
The G^∘-orbits are open by the previous paragraph. Since the complement
of an orbit is a union of other orbits, the G^∘-orbits are also closed.
So the G^∘-orbits are open, closed and path connected; 
hence they coincide with the path components and the connected components.



For a closed subgroup H of a compact Lie group G,
we let C_G H and N_G H denote the centralizer respectively
normalizer of H in G, and we write W_G H=(N_G H)/H for the Weyl group.


For every closed subgroup H of a compact Lie group G,
the composite

    (C_G H)^∘  incl 
    (N_G H)^∘  proj ( W_G H)^∘

is surjective.


We consider a continuous path ω:[0,1]→ W_G H
starting at ω(0)= e H.
The projection N_G H→ (N_G H)/H=W_G H is a locally trivial fiber bundle,
so the path can be lifted to a continuous path
ω̅:[0,1]→ N_G H with ω̅(0)=e.
Proposition <ref> shows that the
image of the map

    H^∘ → (H,H) ,    h  ⟼ c_h

is the path component of the identity.
Since H^∘ is compact and (H,H) is a Hausdorff space, 
this map factors over a homeomorphism from
H^∘ /(H^∘∩ Z(H)) onto the path component of the identity,
where Z(H) is the center of H.
In particular, the map (<ref>)
is a locally trivial fiber bundle over the path component of the identity.
The path

    [0,1] → (H,H)  ,    t  ⟼ c_ω̅(t)

can thus be lifted to a continuous path ν:[0,1]→ H^∘ satisfying

    ν(0)=e      and     c_ω̅(t) =  c_ν(t)

for all t∈ [0,1].
The second relation means that ω̅(t)·ν(t)^-1 centralizes H,
so 

    ω̅·ν^-1 :  [0,1] → C_G H

is a path in the centralizer of H that starts at the identity element.
Moreover,

    ω̅(1)·ν^-1(1)· H  = ω̅(1)· H  =  ω(1) .




The h-cofibrations are the morphisms with
the homotopy extension property. We will use this concept in various categories,
for example in the category of G-spaces, orthogonal spaces 
and orthogonal spectra. So we recall some basic properties of
h-cofibrations in the context of categories enriched over the category of spaces. 
The arguments are all standard and well known, and we include them
for completeness and convenience.

For the discussion of h-cofibrations we work in a cocomplete category  
that is tensored and cotensored over the category  of compactly generated spaces.
We write `×' for the pairing
and X^K for the cotensor of an object X with a compactly generated space K.
A homotopy is then a morphism H:A× [0,1]→ X
defined on the pairing of a -object with the unit interval.
For a homotopy and any t∈ [0,1] we denote by H_t:A→ X the
composite morphism

    A   A×{t} A×incl A× [0,1]  H   X  .




Let  be a category tensored over the category  of spaces. 
A -morphism f:A→ B is an 
h-cofibrationsubjecth-cofibration 
if it has the homotopy extension property, 
i.e., given a morphism φ:B→ X and a homotopy
H:A× [0,1]→ X such that H_0=φ f,
there is a homotopy H̅:B× [0,1]→ X such that
H̅∘(f× [0,1])=H and H̅_0=φ.


There is a universal test case for the homotopy extension problem, 
namely when X is the pushout:

    A [d]_f [r]^-(-,0)    A× [0,1] [d]^H 
    
    B [r]_-φ    B∪_f  ( A× [0,1] )

So a morphism f:A→ B is an h-cofibration if and only if the canonical morphism

    B ∪_f ( A×  [0,1] ) → B× [0,1]

has a retraction. Also, the adjunction between -× [0,1] and (-)^[0,1]
lets us rewrite any homotopy extension data (φ,H) in adjoint form
as a commutative square:

    A[d]_f [r]^-Ĥ    X^[0,1][d]^_0
    
    B [r]_-φ    X

A solution to the homotopy extension problem is adjoint to a lifting, 
i.e., a morphism λ:B→ X^[0,1] such that 
λ f=Ĥ and _0∘λ=φ.
So a morphism f:A→ B is an h-cofibration if and only if it 
has the left lifting propertysubjectlifting property!left
with respect to the morphisms _0:X^[0,1]→ X for all
objects in .

The three equivalent characterizations of h-cofibrations quickly imply
various closure properties.




 
Let  be a cocomplete category tensored and cotensored over the category 
 of compactly generated spaces.


  * The class of h-cofibrations in  is closed under
retracts, cobase change, coproducts and sequential compositions.

  * Let ' be another category tensored over the category ,
and F:→' a continuous functor that commutes with colimits and
tensors with [0,1]. Then F takes h-cofibrations
in  to  h-cofibrations in '.

  * If  is a topological model category in which every object is fibrant,
  then every cofibration is an h-cofibration.



(i) Every class of morphisms that can be characterized by the left lifting property 
with respect to some other class has the closure properties listed.

(ii) Let f:A→ B be a cofibration in  and 
r:B× [0,1]→ B ∪_f (A× [0,1]) a retraction to the canonical morphism.
The composite

    F B× [0,1]       F(B× [0,1])   Fr 
       F(B∪_f ( A× [0,1]) )     F B ∪_Ff ( F A× [0,1] )

is then a retraction to the canonical morphism for Ff:F A→ F B.
So Ff is an h-cofibration.

(iii)
Since the model structure is topological, for every cofibration
f:A→ B the canonical morphism (<ref>)
is an acyclic cofibration.
Since every object is fibrant, this morphism has a retraction, 
and so f is an h-cofibration.


We turn to a key technical property of h-cofibrations in the category 
of compactly generated spaces, namely that these are always closed embeddings.
The same conclusion also holds for h-cofibrations in the category _*
of based compactly generated spaces. 

The forgetful functor _*→ does not preserve the tensors with
unbased spaces, and it does not preserve h-cofibrations.
Indeed, the `based tensor' of a based space (A,a_0) with an unbased space K is 
A K_+, whereas the unbased tensor of the underlying space is simply A× K.
For every based space (A,a_0) the inclusion 
{a_0}→ A is an h-cofibration in the based sense, but not generally 
in the unbased sense. 
By definition, (A,a_0) is well-pointedsubjectwell-pointed
if this inclusion is an h-cofibration of unbased spaces.



The following proposition is Lemma 8.2 in <cit.>.


  Every h-cofibration between compactly generated spaces is a closed embedding.
  Every based h-cofibration between compactly generated based spaces 
  is a closed embedding.


Let f:A→ B be an h-cofibration between
compactly generated spaces.
Since the map (-,0):A→ A×[0,1] is a closed embedding, 
Proposition <ref> says that the
pushout B∪_f ( A×[0,1]) in the category  can 
in fact be calculated in the ambient category .
So the map

    (-,1) :  A → B∪_f (A×[0,1])  ,    a  ⟼ (a,1)

is a closed embedding by direct inspection of the topology on the pushout.

The universal example of the homotopy extension property provides 
a continuous retraction to the canonical map 

    i =   (-,0)∪ (f× [0,1]) : B∪_f (A×[0,1]) → B× [0,1] .

So i is a closed embedding by Proposition <ref>.
In the commutative square

    @C=12mm 
     A[r]^-(-,1)[d]_f     B∪_f (A×[0,1]) [d]^i 
    
     B[r]_-(-,1)     B×[0,1]

the two horizontal and the right vertical maps are thus closed embeddings. 
So the map f is a closed embedding as well.

The based case, albeit logically independent, proceeds along the same lines.
We let f:(A,a_0)→ (B.b_0) 
be a based h-cofibration between compactly generated based spaces.
Since points in weak Hausdorff spaces are closed
(Proposition <ref> (iii)), 
the set {a_0}× [0,1] is closed in A× [0,1], and so the map

    (-,t) : A→ A [0,1]_+ ,    a ⟼ a t

is a closed embedding for every t∈ [0,1].
Any pushout of based spaces can be calculated in the ambient category
of unbased spaces; so Proposition <ref> says that the
pushout B∪_f ( A [0,1]_+) in the category _* can 
in fact be calculated in the ambient category .
So the map

    (-,1) :  A → B∪_f (A [0,1]_+)  ,    a  ⟼ a 1

is a closed embedding by direct inspection.

The universal example of the homotopy extension property provides 
a continuous retraction to the canonical based map 

    i =   (-,0)∪ (f [0,1]_+) : B∪_f (A [0,1]_+) → B [0,1]_+ .

So i is a closed embedding by Proposition <ref>.
In the commutative square

    @C=15mm 
     A[r]^-(-,1)[d]_f     B∪_f (A[0,1]_+) [d]^i 
    
     B[r]_-(-,1)     B[0,1]_+

the two horizontal and the right vertical maps are thus closed embeddings. 
So the map f is a closed embedding as well.


Now we turn to geometric realization of simplicial spaces,
which is a frequent tool to construct interesting homotopy types.




We recall the geometric realization of a simplicial space.
Geometric realization was originally introduced 
by Milnor <cit.> for simplicial sets 
(which were called `semi-simplicial complexes' at that time);
the version for simplicial spaces is a straightforward generalization.
We let  denote
the simplicial indexing category, with objects the finite
totally ordered sets [n]={0≤ 1≤…≤ n}
for n≥ 0. Morphisms in  are all weakly monotone maps.
We let

    Δ^n  =  { (t_1,…,t_n)∈ [0,1]^n  | t_1≤ t_2≤…≤ t_n }

be the topological n-simplex.
As n varies, these topological simplices assemble into a covariant functor

    Δ^∙ :   → ,    [n] ⟼ Δ^n ;

the coface maps are given by

    (d_i)_*(t_1,…,t_n)  = {[         (0,t_1,…,t_n)                      ; (t_1,…,t_i,t_i,…,t_n)                      ;         (t_1,…,t_n,1)                       ].

The codegeneracy map (s_i)_*:Δ^n→Δ^n-1 drops the entry t_i+1.

A simplicial spacesubjectsimplicial space
is functor X:^→, i.e., a contravariant functor from 
to the category of topological spaces.
We use the customary notation X_n=X([n]) for the value of a simplicial space
at [n]. 
The geometric realizationsubjectgeometric realization!of a simplicial space
of X is

    |X| =  ( ∐_n≥ 0  X_n×_0 Δ^n ) /  ∼ ,

the quotient space of the disjoint union by the equivalence relation generated by

    (x,α_*(t)) ∼ (α^*(x),t)

for all morphisms α:[n]→ [m] in  
and all (x,t)∈ X_m×_0 Δ^n.
A more categorical way to say this is that |X| is a coend of the functor

    ^× →  ,    
    ([m],[n]) ⟼ X_m×_0Δ^n  .



We recall that the equivalence relation generated by (<ref>)
is well understood in terms of `minimal representatives':


  * every equivalence class has a unique 
representative (x,t)∈ X_l×Δ^l of minimal dimension l;

  * an element (x,t)∈ X_l×Δ^l is the minimal representative in
its equivalence class if and only if x is non-degenerate 
(i.e., not in the image of any degeneracy map s_i^*:X_l-1→ X_l)
and t belongs to the interior of Δ^l;

  * if (x,t)∈ X_l×Δ^l is a minimal representative 
and (y,s)∈ X_m×Δ^m is equivalent to (x,t), 
then there is a unique surjective morphism σ:[k]→[l], 
a unique injective morphism δ:[k]→[m]
and a unique u∈Δ^k such that

    δ^*(y) =  σ^*(x) ,    
    s =  δ_*(u)    and   t =σ_*(u)  .


These facts go back all the way to Milnor <cit.>.

The next proposition shows that compactly generated spaces are 
`closed under geometric realization';
more precisely, we consider a simplicial space X_∙ 
such that X_n is compactly generated for every n≥ 0.
The geometric realization, formed in the ambient category , 
is an enriched coend, hence automatically a k-space.
Since colimits of compactly generated spaces are not, in general,
formed in the ambient category, it is not completely obvious, though,
that the ambient realization is again a weak Hausdorff space.
I learned about this fact as a parenthetical remark 
in the introduction of the paper <cit.>.




  Let X:^→ be a simplicial k-space.
  
  
  * The the geometric realization |X| is a k-space,
    and hence also a coend internal to the category ,
    of the functor (<ref>).
  
  * If X_n is compactly generated for every n≥ 0, 
    then the geometric realization |X| is compactly generated,
    and hence also a coend internal to the category ,
    of the functor (<ref>).
  
  * Let X_n be compactly generated for all n≥ 0.
    Let Y be a simplicial subspace of X such that
    Y_n is closed in X_n for all n≥ 0. 
    Then the inclusion induces a closed embedding |Y|→ |X|.
  


(i) Since Δ^n is compact and X_n is a k-space,
X_n×_0Δ^n=X×Δ^n is a k-space in the product topology 
by Proposition <ref> (vi),
and so the disjoint union ∐_n≥ 0  X_n×Δ^n
is a k-space.
As a quotient space, the geometric realization |X| is then 
a k-space by Proposition <ref> (i).

(ii) As we argued in part (i),
the space ∐_n≥ 0 X_n×_0 Δ^n=∐_n≥ 0 X_n×Δ^n
is a k-space.
We can thus use the criterion given by Proposition <ref>
to show that the quotient space |X| is compactly generated.
We let E⊂ (∐_n≥ 0 X_n×Δ^n)^2
be the equivalence relation generated by (<ref>),
which we had simply denoted `∼' above. We will show that
E is closed in the k-topology of (∐_n≥ 0 X_n×Δ^n)^2.
Since products distribute over disjoint unions, we may show that

    E_m,n =  E∩ (X_m×Δ^m× X_n×Δ^n)

is closed in X_m×Δ^m× X_n×Δ^n for all m,n≥ 0.
If (y,s,y̅,s̅)∈ E_m,n, then (y,s) and (y̅,s̅)
have the same minimal representative (x,t)∈ X_l×Δ^l.
So there are surjective morphisms
σ:[k]→[l] and σ̅:[k̅]→[l], 
injective morphisms δ:[k]→[m] and δ̅:[k̅]→[n],
and u∈Δ^k, u̅∈Δ^k̅ such that

    δ^*(y) =  σ^*(x) ,    
    s =  δ_*(u)    and   t =σ_*(u)

and

    δ̅^*(y̅) =  σ̅^*(x) ,   s̅ =  δ̅_*(u̅)    and   t =σ̅_*(u̅)  .

Hence E_m,n is the union,
indexed over l,σ,σ̅,δ,δ̅ as above,
of the finite number of sets

    (δ^*×Δ^m×δ̅^*×Δ^n)^-1(
     (σ^*×δ_*×σ̅^*×δ̅_*)
    ((X_l×σ_*× X_l×σ̅_*)^-1
    ( Δ_X_l×Δ^l)))  .

The diagonal Δ_X_l×Δ^l is closed in (X_l×Δ^l)^2
because X_l, and hence X_l×Δ^l, is compactly generated.
So its inverse image under the continuous map
X_l×σ_*× X_l×σ̅_*
is closed in X_l×Δ^k× X_l×Δ^k̅.
Every surjective morphism in  has a section,
so σ^*:X_l→ X_k and σ̅^*:X_l→ X_k̅ 
have continuous retractions.
The maps δ_*:Δ^k→Δ^m and δ̅_*:Δ^k̅→Δ^n 
also have continuous retractions, hence so does
σ^*×δ_*×σ̅^*×δ̅_*.
This map is then a closed embedding by Proposition <ref>.
So the set 

    (σ^*×δ_*×σ̅^*×δ̅_*)
    ((X_l×σ_*× X_l×σ̅_*)^-1
    ( Δ_X_l×Δ^l))

is closed in  X_k×Δ^m× X_k̅×Δ^n.
Since E_m,n is the inverse image of this latter closed set
under a continuous map, this show the claim that E_m,n
is a closed subset of X_m×Δ^m× X_n×Δ^n.

  (iii) We write ι:Y→ X for the inclusion. 
  Our first claim is that the induced map |ι|:|Y|→ |X| is injective.
  We consider a point (y,s)∈ Y_m×Δ^m 
  and we let (x,t)∈ X_l×Δ^l be the minimal representative 
  of the equivalence class of (y,s) 
  in the ambient space ∐_n≥ 0X_n×Δ^n.
  Then there is a surjective morphism
  σ:[k]→[l], an  injective morphism δ:[k]→[m] 
  and u∈Δ^k such that
  
    δ^*(y) =  σ^*(x) ,    
      s =  δ_*(u)    and   t =σ_*(u)  .

  We let δ̅:[l]→ [k] be a morphism in  
  such that σδ̅=_[l]. 
  Then δ̅^*(δ^*(y))=δ̅^*(σ^*(x))=x.
  Since Y is a simplicial subspace of X, this shows that x belongs to Y_l.
  If (y̅,s̅)∈ Y_n×Δ^n is another pair that
  represents the same point in |X| as (y,s), then
  by the above, the minimal representative of the equivalence class belongs
  to ∐_n≥ 0 Y_n×Δ^n, and so (y,s) and (y̅,s̅)
  already represent the same point in |Y|. So the map |ι| is injective. 

  It remains to show that the continuous injection |ι| is a closed map.
  We consider the commutative square
  
    @C=15mm∐_n≥ 0 Y_n×Δ^n[r]^-∐ι_n×Δ^n[d]_p   ∐_n≥ 0 X_n×Δ^n[d]^q
    
        |Y|[r]_-|ι|    |X|

  where the vertical maps are the quotient maps.
  We let (y,s)∈ X_m×Δ^m be a point whose equivalence class
  lies in the image of |ι|:|Y|→|X|.
  As we showed in the previous paragraph, the representative 
  (x,t)∈ X_l×Δ^l 
  of minimal dimension in the equivalence class of (y,s) must then
  lie in the simplicial subspace Y, i.e., we must have x∈ Y_l.
  Moreover, there is a surjective morphism
  σ:[k]→[l], an  injective morphism δ:[k]→[m] 
  and u∈Δ^k such that
  
    δ^*(y) =  σ^*(x) ,    
      s =  δ_*(u)      and     t =σ_*(u)  .

  So for every subset A⊂ |Y|, we have
   
    ( X_m×Δ^m) ∩   q^-1( |ι|(A) )
     
      =  ⋃_σ,δ      (δ^*×Δ^m)^-1(( σ^*×δ_*)
       (  ( X_l×σ_*)^-1(( Y_l×Δ^l)∩ p^-1(A) ) ) ) .

  The union is over the finite set of pairs consisting of a surjective 
  morphism σ:[k]→ [l] and an injective morphism δ:[k]→ [m].

  Now we assume that A is closed inside |Y|. Because p is continuous
  and Y_l is closed in X_l, the set
  (Y_l×Δ^l)∩ p^-1(A) is closed inside X_l×Δ^l.
  So
  (X_l×σ_*)^-1(( Y_l×Δ^l)∩ p^-1(A)) 
  is a closed subset of X_l×Δ^k.
  Since the map σ^*×δ_*:X_l×Δ^k→ X_k×Δ^m 
  has a continuous retraction, it is a 
  closed embedding by Proposition <ref>.
  So the set (σ^*×δ_*) 
  (  ( X_l×σ_*)^-1((Y_l×Δ^l)∩ p^-1(A) ))
  is closed in X_k×Δ^m.
  As the inverse image under a continuous map, the set
  
    (δ^*×Δ^m)^-1 ( (σ^*×δ_*) 
      (  ( X_l×σ_*)^-1(( Y_l×Δ^l)∩ p^-1(A) ) ) )

  is then closed in X_m×Δ^m.

  So each set in the finite union above is closed inside X_m×Δ^m.
  We conclude that ( X_m×Δ^m) ∩ q^-1( |ι|(A) )
  is closed in X_m×Δ^m for every m≥ 0, 
  hence the set q^-1(|ι|(A)) is closed.
  Since q is a quotient map, this shows that |ι|(A) is closed in |X|.



The next proposition is about the interaction of geometric realization
and products for simplicial k-spaces and simplicial compactly generated spaces.
The essential input here is that the categories  and 
are cartesian closed, hence product with a fixed object
is a left adjoint and commutes with colimits and coends.
A direct consequence is that geometric realization 
commutes with colimits and products with a fixed space,
see part (i) and (ii) of Proposition <ref> below.


We also recall that different ways to realize a multi-simplicial space are homeomorphic.
To this end we consider a 
bisimplicial k-space,subjectbisimplicial space i.e., a functor
Z:^×^→.
The diagonal simplicial k-space Z is the composite 
with the diagonal functor ^→^×^.
On the other hand, for every fixed `horizontal' dimension m 
we can define the simplicial space ∂_m Z as the composite 

    ^  (_[m],-)   ^×^  Z   ,

so that (∂_m Z)_n=Z_m,n. For varying m, these define a simplicial object
in the category of simplicial k-spaces; so the assignment

    [m] ⟼ | ∂_m Z|

becomes a simplicial k-space. We can geometrically realize this simplicial space
and arrive at the iterated realizationsubjectiterated realization!of a bisimplicial space that we denote

    |Z|^it =  |[m]↦ |∂_m Z| |  .

The diagonal realization and the iterated realization are related
by a natural continuous map, see (<ref>) below.
We show in Proposition <ref> (iii)
that this preferred natural map is a homeomorphism.

I am not sure who deserves credit for parts (ii) and (iii)
of the following proposition.
For preservation of products, Milnor <cit.> 
already treats the special case of
simplicial sets (which we can view as discrete simplicial spaces),
where he imposes size conditions to ensure that the ordinary product 
of the two realizations is a k-space.
The earliest reference in the present generality I know of 
is <cit.>, 
where Segal states the result without proof.
May <cit.> gives a proof
in the category of Hausdorff k-spaces (which he calls 
`compactly generated Hausdorff spaces'), but the proof does not use 
the Hausdorff property. 


  
  
  * For every simplicial k-space Y:^→ and every k-space K,
    the canonical map
    
    |K× Y| → K× |Y|  ,    [k,y,t ] ⟼ (k,[y,t])

    is a homeomorphism.
  
  * The realization functors  
    
    |-| : ^ →      and    
        |-| : ^ →

    for simplicial k-spaces respectively 
    simplicial compactly generated spaces 
    preserves colimits and finite products.
  
  * For every bisimplicial k-space Z:^×^→, 
    the map 
    
    δ :  | Z|  → |Z|^ it   given by  δ[z,t]  =  [ [z,t],t]

    is a homeomorphism.
  


(i)
The category of k-spaces is cartesian closed
(Theorem <ref>), so product with K 
is a left adjoint and commutes with coends. Hence the canonical map

    |K× Y | =  ∫^[n]∈( K× Y_n×Δ^n)
     → K×( ∫^[n]∈ Y_n×Δ^n)  =  K× |Y|

is a homeomorphism. 

(ii) We start by showing that the realization functors preserve colimits.
The argument is entirely formal, and literally the same for the
categories  and , so we only treat the former case.
We let I be a small category and F:I→^ a functor.
The category of k-spaces is cartesian closed
(Theorem <ref>), so product with Δ^n 
is a left adjoint and commutes with coends. 
Moreover, colimits commute with coends, so the canonical maps

    _I    ( |-|∘ F) =  _I  ( ∫^[n]∈ F_n×Δ^n)   ∫^[n]∈_I (F_n ×Δ^n) 
          ∫^[n]∈ ( _I F_n )×Δ^n= ∫^[n]∈ ( _I F)_n ×Δ^n =  |_I F|

are homeomorphisms. 

We postpone the proof that the realization functors preserve finite products
until later.

(iii) 
In any presheaf category with values in , every functor is canonically
a colimit of representable functors times a fixed space.
For the category ×, 
this means that every bisimplicial k-space can be expressed as a colimit of
bisimplicial spaces of the form (×)(-, ([m],[n]))× K
for m,n≥ 0 and k-spaces K.
Passage to the diagonal clearly commutes with colimits and products,
so source and target of the map δ commute with colimits in  
and products with a fixed k-space, by parts (i) and (ii).
This reduces the claim to the special case 
Z=Δ[m,n]=(×)(-, ([m],[n])),
the bisimplicial discrete space represented by the object ([m],[n]).
The diagonal of this bisimplicial space is Δ[m]×Δ[n],
the product of the represented simplicial sets Δ[m] and Δ[n].

On the other hand, for fixed k≥ 0, the simplicial discrete space 
∂_k Δ[m,n] is isomorphic to ([k],[m])×Δ[n],
and so its realization is homeomorphic to ([k],[m])× |Δ[n]|.
The simplicial space |∂_∙Δ[m,n] | is thus isomorphic
to Δ[m]× |Δ[n]|. Since product with |Δ[n]|
commutes with realization by part (i), we conclude that
the iterated realization |Δ[m,n]|^it is homeomorphic to the 
product |Δ[m]|× |Δ[n]|.
Moreover, under these identifications, the map δ for Z=Δ[m,n] specializes
to the canonical map |Δ[m]×Δ[n]|→|Δ[m]|×|Δ[n]|.
It is a classical fact,
already observed by Milnor <cit.>,
that this canonical map is a homeomorphism;
other references are <cit.>
and <cit.>.

We still need to show that the realization functors preserve finite products.
The category  of compactly generated space is closed under products
in the ambient category  of k-spaces (because the inclusion →
is a left adjoint), and closed under realization 
by Proposition <ref> (ii).
So it suffices to treat realization of simplicial k-spaces.

We consider the bisimplicial k-space X×̅Y defined as the composite

    ^×^ X× Y ×  ×  ;

then X× Y=(X×̅Y) is its diagonal.
The canonical map

    |∂_m (X×̅Y) |  =  
    | X_m× Y |
     → X_m×( ∫^[n]∈ Y_n×Δ^n)
     =   X_m× |Y|

is a homeomorphism by part (i). 
Taking realizations over varying m and using part (i) again 
gives two homeomorphisms

    |X×̅Y|^it =  
    |[m]↦ | X_m× Y | |
        |X × |Y| |     
    |X |× |Y|  .

Under these identifications, the canonical map
(|p_X|,|p_Y|): |X× Y| → |X|×  |Y|
becomes the map δ:|(X×̅Y)|→ |X×̅Y|^it
of part (iii). Since δ is a homeomorphism, this shows the claim.


For general simplicial topological spaces, i.e., 
functors X,Y:^→ geometric realization need not
commute with products in . Examples already arise when X and Y 
are simplicial sets (considered as simplicial discrete spaces), 
in which case |X× Y| is compactly generated. 
However, if X and Y are sufficiently large 
(i.e., neither countable nor locally finite), then the product topology
need not make |X|×_0 |Y| into a k-space. A specific example
is given by taking both X and Y as wedges of the simplicial 1-sphere, 
where X has countably infinitely many copies,
and Y has uncountably many copies, 
compare <cit.>.




Now we discuss the latching spacessubjectlatching space!of a simplicial space 
of a simplicial space X, of which there are competing definitions in the literature.
The definition of the n-th latching object that we adopt is 
as a colimit of a certain functor whose values are the subspaces X_i for i<n.
A possible point of confusion is that different references work
in slightly different categories (such as general topological spaces, 
k-spaces, Hausdorff k-spaces or weak Hausdorff k-spaces), 
and the presence of a Hausdorff or weak Hausdorff condition
affects the meaning of `colimit'.
Some authors define the n-th latching space of X as a subspace of X_n,
namely the union of the subspaces s_i^*(X_n-1) for i=0,…,n-1,
where s_i^*:X_n-1→ X_n is the i-th degeneracy map.
I suspect that the different definitions are not generally homeomorphic 
in the context of simplicial topological spaces, 
i.e., functors X:^→.


As we will discuss now, all these definitions in fact coincide for
simplicial compactly generated spaces, i.e., functors ^→.
We already explained in Proposition <ref> (ii)
that then the geometric realization (formed in the ambient
category  of all topological spaces) is automatically compactly generated,
and hence also a coend internal to the category .
We will now show that also the latching objects, 
defined in the ambient category ,
are automatically compactly generated, and hence also a colimit in the category .
Moreover, the latching map l_n:L_n X→ X_n 
is a closed embedding, with the expected image.



For n≥ 0 we let (n) denote category with objects the weakly 
monotone surjections σ:[n]→ [k];
a morphism from σ:[n]→ [k] to σ':[n]→ [k']
is a morphism α:[k]→ [k'] in  with α∘σ=σ'.
Such a morphism α, if it exists, is necessarily unique and also surjective.
We let (n)_∘ denote the full subcategory with all objects
except the identity of [n].

A simplicial topological space X:^→ 
can be restricted along the forgetful functor

    (n)_∘^  u   ^ ,   
    (σ:[n]→ [k]) ⟼ [k] ,   α ⟼ α .

The n-th latching space of X is the colimit of the composite
functor X∘ u:(n)_∘^→.
As σ ranges over the objects of (n)^_∘, the maps

    σ^*  :  X_k → X_n

assemble into a continuous map l_n:L_n X→ X,
the n-th latching map.


 
The category (n)^ is isomorphic to the poset category (n) 
of subsets of the set {1,…,n}.
Indeed, an isomorphism κ:(n)^→(n) is given on objects by

    κ(σ:[n]→ [k])  =  { i∈{1,…, n} :  σ(i) > σ(i-1)} .

In the other direction, a subset U⊂{1,…,n} 
is taken to  the monotone surjection κ^-1(U):[n]→[|U|]
defined by

    κ^-1(U)(i) =  |U∩{1,…,i}| .

Since κ(_[n])={1,…,n}, 
the subcategory (n)^_∘ is taken
to the poset (n)^∘ of proper subsets of {1,…,n}.

The maximal elements of the poset (n)^∘ correspond to the morphisms

    s_i :  [n] → [n-1]  ,    
      s_i(j) =   j    for 0≤ j≤ i, and
    
    j-1    for i+1≤ j≤ n.

So the latching space L_n X can also be presented 
as a coequalizer of two continuous maps

    ∐_0≤ i < j≤ n-1  X_n-2  @<.4ex>[r]^-α@<-.4ex>[r]_-β     ∐_0≤ i≤  n-1  X_n-1 .

The map α sends the (i,j)-th summand to the i-summand by the degeneracy
map s_j-1^*:X_n-2→ X_n-1.
The map β sends the (i,j)-th summand to the j-summand by the degeneracy
map s_i^*:X_n-2→ X_n-1.
For example, the category (0)_∘ is empty, so L_0(X) is empty.
The category (1)_∘ has a unique object s_0:[1]→ [0],
so L_1(X) = X_0 and the latching map is given by s_0^*:X_0→ X_1.
The category (2)_∘ has three objects and two non-identity morphisms,
and L_2(X) is a pushout of the diagram

    X_1   s_0^*   X_0   s_0^*  X_1  .

 




Let X:^→ be a simplicial topological space and n≥ 0.  


  * The continuous latching map l_n:L_n X→ X_n is injective
and its image is the union of the sets s_i^*(X_n-1) for i=0,…,n-1.

  * Suppose that the degeneracy map s_i^*:X_n-1→ X_n is
a closed embedding for all 0≤ i≤ n-1.
Then the latching map l_n:L_n X→ X_n is a closed embedding.

  * Suppose that the space X_n is compactly generated for every n≥ 0.
Then the n-latching space L_n X formed in the ambient category 
of topological spaces is compactly generated, and hence a latching object
of X internal to the category .
Moreover, the latching map l_n:L_n X→ X_n is a closed embedding.



(i) This argument is purely combinatorial and does not use the topology in any way.
For easier book keeping we label the different summands in the source
and target of the coequalizer diagram (<ref>)
as X_n-2^[i,j] respectively X_n-1^[i].
We let q:∐_0≤ i≤ n-1X_n-1^[i]→ L_n X denote the quotient map.
We consider x∈ X_n-1^[i] and y∈ X_n-1^[j] such that
l_n(q(x))=l_n(q(y)) in X_n. This means that s_i^*(x)=s_j^*(y).
We assume without loss of generality that i≤ j.
We write the elements as degeneracies of non-degenerate elements, 
i.e., x=σ^*(z) and y=σ̅^*(z̅)
for surjective morphisms σ:[n-1]→[k],σ̅:[n-1]→[k̅] 
and non-degenerate simplices z and z̅. Then

    (σ s_i)^*(z) =    s_i^*(σ^*(z)) =  s_i^*(x) = s_j^*(y)  =   
    s_j^*(σ̅^*(z̅)) =  (σ̅s_j)^*(z̅) .

By the `Eilenberg-Zilber lemma' (<cit.>, 
see also <cit.>),
the representation of a simplex as a degeneracy of a non-degenerate
element is unique, so k=k̅, z=z̅ 
and σ s_i=σ̅s_j.
Because i≤ j, the second relation means that

    σ̅(a) =  σ(a)    for 0≤ a≤ i,
    σ(a-1)    for i+1≤ a≤ j, and
     σ(a)    for j+1≤ a≤ n-1.

We define τ:[n-2]→[k] by

    τ(a) =  σ̅(a)    for 0≤ a≤ i,
    σ̅(a+1)    for i+1≤ a≤ n-2.

Then

    σ =  τ s_j-1     and    σ̅ =  τ s_i  .

Setting w=τ^*(z) then yields

    x  =  σ^*(z) =  s_j-1^*(w)     and    
    y =  σ̅^*(z) =  s_i^*(w) .

This means that x and y are equivalent, and hence q(x)=q(y).
So the latching map l_n:L_n X→ X_n is injective.

(ii) The latching map is continuous and injective by part (i),
so it remains to show that l_n is also a closed map.
The composite

    ∐_0≤ i≤ n-1 X_n-1^[i]  q  L_n X   l_n  X_n

of the quotient map and the latching map is the disjoint union
of the degeneracy maps s_i^*:X_n-1→ X_n.
So for every subset A of L_n X, we have

    l_n(A)  =  ⋃_0≤ i≤ n-1  s_i^*(q^-1(A)∩ X_n-1^[i]) .

Now we suppose that A is closed in L_n X.
Since each s_i^* is a closed embedding,
l_n(A) is a finite union of closed subsets of X_n, and hence itself closed.

(iii)
Since the morphism s_i:[n]→[n-1] has a retraction, 
the degeneracy map s_i^*:X_n-1→ X_n has a continuous retraction.
Since X_n-1 and X_n are compactly generated, 
s_i^*:X_n-1→ X_n is a closed embedding 
by Proposition <ref>.
So part (ii) applies and shows that l_n:L_n X→ X_n is a closed embedding.
Since X_n is compactly generated, so is  L_n X,
by Proposition <ref> (i).


The proof of part (iii) of the previous proposition
makes critical use of the fact that all degeneracy maps
in a simplicial compactly generated space are closed embeddings.
This is not the case more generally for simplicial k-spaces 
or simplicial topological spaces; so while a latching map
is always a continuous injection, it need not be a homeomorphism onto its image.
So if latching spaces are ever considered in this generality,
one has to beware the possible difference between L_n X defined as the colimit
or as the subspace s_0^*(X_n-1)∪…∪ s_n-1^*(X_n-1) of X_n.



Now we recall Reedy cofibrancy, which will be our main condition to ensure
good homotopical behavior of geometric realization.


A simplicial topological space X:^→ 
is Reedy cofibrantsubjectReedy cofibrant!simplicial space
is the latching morphism l_n:L_n X→ X_n 
is a cofibration in the Quillen model structure on the category of topological spaces.


The next proposition is the key reason why we care about Reedy cofibrancy.


  Let f:X→ Y be a morphism between Reedy cofibrant simplicial topological spaces.
  If f_n:X_n→ Y_n is a weak equivalence for every n≥ 0, then
  the geometric realization |f|:|X|→ |Y| is a weak equivalence.


We use the Reedy model structure on
the category of simplicial topological spaces.
We endow the category  of topological space
with the Quillen model structure, see <cit.>,
which makes it into a simplicial model category.
Geometric realization thus takes levelwise weak equivalences between
Reedy cofibrant objects to weak equivalences by <cit.>.




By the precious proposition, the condition that a simplicial space is Reedy cofibrant
ensures that its geometric realization is `homotopically invariant'.
There are two other widely used conditions for ensuring that 
a geometric realization is well-behaved, the notions of a 
good and proper simplicial spaces.
A simplicial space X is proper
in the sense of May <cit.> if
the latching map ν_n:L_n X→ X_n is a closed h-cofibration
for every n≥ 0. 
A simplicial space X is good 
in the sense of Segal <cit.>
if the degeneracy map s_i^*:X_n-1→ X_n is a closed h-cofibration
for all 0≤ i≤ n-1. 
One should beware that in the generality of arbitrary topological spaces,
the homotopy extension property does not imply that a map is a closed
embedding, hence `closed h-cofibration' is a stronger condition.
For compactly generated spaces, however, a continuous map with 
the homotopy extension property is automatically a closed embedding,
by Proposition <ref>.

For proper simplicial spaces, homotopical invariance of geometric realization 
is proved in <cit.>.
The case of good simplicial spaces then follows from the fact that
`goodness' implies `properness'; this is implicit in the proof 
of <cit.>, and stated explicitly in
the proof of Lewis' <cit.>. 
The argument makes essential use of Lillig's `union theorem' <cit.>,
so it is not of a formal, model category theoretical nature.

Since cofibrations of compactly generated spaces are in particular 
closed h-cofibrations, 
every Reedy cofibrant simplicial compactly generated space 
is in particular proper and good.
Whenever we want to appeal to homotopy invariance of geometric realization,
our simplicial spaces are Reedy cofibrant; so we do not use the more
general conditions `good' and `proper', and we don't elaborate on these any further.




We recall that the singular complex (K) of a topological space K
is the simplicial set defined as the composite

    ^ (Δ^∙)^ ^ (-,K) (sets) .

In particular, the set of n-simplices (K)_n is the set
of continuous maps from Δ^n to K.
Now we consider a simplicial topological space X.
Applying the singular complex functor levelwise yields a simplicial simplicial set
(i.e., a bisimplicial set) ∘ X, 
sending [n] to the simplicial set (X_n).
We apply geometric realization to the simplicial sets (X_n)
for all n≥ 0, and end up with a new simplicial space  |-|∘∘ X.

The singular complex functor is right adjoint the geometric realization
(restricted from simplicial spaces to simplicial sets),
and the adjunction counit is the continuous map

    ϵ_K :  |(K)|  → K  ,    [f,t] ⟼ f(t) .

As was already noticed Milnor when he introduced the geometric
realization of simplicial sets <cit.>, 
the map ϵ_X is a weak equivalence.
Since ϵ_K is natural, these maps assemble into a
morphism of simplicial spaces

    ϵ_X :   |-|∘∘ X  → X  .

We observe that ||-|∘∘ X|=|∘ X|^it
is the iterated realization (<ref>) 
of the bisimplicial set ∘ X.


Let X:^→ be a Reedy cofibrant simplicial topological space.


  * The map | ϵ_X |:|∘ X|^ it→ |X|
is a weak equivalence.

  * If the spaces X_0,…, X_n consist only of a single point each,
then the realization |X| is n-connected.



(i) We claim that this simplicial space |-|∘∘ X
sending [m] to the realization of the simplicial set (X_m)
is automatically Reedy cofibrant.
Indeed, for every simplicial set A the latching maps
L_n A→ A_n are automatically injective
(a special case of Proposition <ref> (i)). 
Hence for fixed n≥ 0 and varying [m]∈^, the maps 

    L_n^(  (X_m) ) → (X_m)_n

(with latching object taken in the direction of the singular complex)
are a monomorphism of simplicial sets.
Geometric realization takes monomorphisms of simplicial sets to
relative CW-inclusions, and hence to cofibrations of spaces, 
and it commutes with colimits.
So the simplicial space |-|∘∘ X is Reedy cofibrant.
Since we assumed that X is also Reedy cofibrant, 
the map |ϵ_X| is a weak equivalence 
by Proposition <ref>.


(ii) By part (i) we may show that the space
| ∘ X|^it is n-connected.
The iterated realization is homeomorphic to the diagonal realization, 
by Proposition <ref> (iii);
so we may show that the space |( ∘ X)| is n-connected.
But the simplicial set ( ∘ X) has only a single k-simplex
for 0≤ k≤ n, by hypothesis. So its geometric realization 
admits a CW-structure with one 0-cell and no cells in dimension 1 through n.
Hence the space |( ∘ X)| is n-connected.



In <cit.>, Bousfield and 
Friedlander establish a fairly general criterion to ensure that a dimensionwise
homotopy cartesian square remains homotopy cartesian after realization.
The criterion involves the `π_*-Kan condition' introduced 
in <cit.>. 
We now translate the condition into the context of simplicial spaces;
it roughly says that the simplicial sets obtained by taking t-th homotopy groups
dimensionwise satisfy the Kan extension condition.
The precise definition is more elaborate because one needs to properly 
take care of varying basepoints for homotopy groups.


Let X:^→ be a simplicial topological space, 
m,t≥ 1 and a∈ X_m. 
Then X satisfies the π_t-Kan condition at a 
if the following condition holds:
for all tuples of elements x_i∈π_t(X_m-1,d_i^*(a))
for i∈{0,1,…,k-1,k+1,…,m} satisfying 

    d_i^*(x_j) =  d_j-1^*(x_i)    in  π_t(X_m-2,(d_j d_i)^*(a))

for all 0≤ i<j≤ m with i k, j k, there is an element
y∈π_t(X_m,a) such that

    d_i^*(y)  =  x_i

for all 0≤ i≤ m with i k.

The simplicial space X satisfies the 
π_*-Kan conditionsubjectpi star Kan condition@π_*-Kan condition
if it satisfies the π_t-Kan condition for all m,t≥ 1 and all a∈ X_m.




As Bousfield and Friedlander already remark,
the π_t-Kan condition is automatically satisfied at a∈ X_m
for all t≥ 1 whenever a=σ_m^*(b) 
is the degeneracy of some point b∈ X_0,
where σ_m:[m]→[0] is the unique morphism.
Indeed, in that special case, the collection of 
homotopy groups π_t(X_n,σ_n^*(b)) forms a simplicial group
as n varies over the objects of ; 
the simplicial sets underlying simplicial groups always 
satisfy the Kan extension condition, an observation due to Moore,
see for example <cit.>.


Bousfield and Friedlander work with simplicial sets,
whereas we need their result for topological spaces.
The next theorem does the straightforward translation 
using the singular complex functor.
The reader may observe that <cit.>
has no Reedy cofibrancy condition; this should not be surprising 
because latching morphisms of simplicial objects in simplicial sets 
are automatically injective, and hence cofibrations. 
Put differently, bisimplicial spaces are automatically
Reedy cofibrant when considered as simplicial objects in simplicial sets.

A homotopy fiber sequencesubjecthomotopy fiber sequence
is a pair of composable continuous maps

    A   i   X   f  Y

whose composite f i:A→ Y is constant with value y_0,
and such that the induced map

    A     →  F(f) =  {(λ,x)∈ Y^[0,1]× X |  λ(0)=y_0, λ(1)=f(x)}
    
    a     ⟼     (const_y_0, i(a))

from A to the homotopy fiber of f is a weak equivalence.


Let

    A   i   X   f  Y

be morphisms of Reedy cofibrant simplicial topological spaces
such that the composite f∘ i is constant.
Suppose that the following conditions hold:


  * For every n≥ 0 the sequence (i_n,f_n) is
a homotopy fiber sequence.

  * The simplicial spaces X and Y satisfy the π_*-Kan condition.

  * The morphism of simplicial sets π_0(f):π_0(X)→π_0(Y)
is a Kan fibration.

Then the sequence (|i|,|f|) is a homotopy fiber sequence.


We transfer the question into the context of simplicial sets by use of the
singular complex functor, and then quote the theorem of Bousfield and Friedlander.
In the commutative diagram

    @C=13mm 
    |∘ A|^it[r]^-|∘ i|^it[d]_ϵ_A    
    |∘ X|^it[d]^ϵ_X[r]^-|∘ f|^it   
    |∘ Y|^it[d]^ϵ_Y
    
    |A| [r]_|i|    |X| [r]_-|f|     |Y|

all vertical maps are weak equivalences by Proposition <ref>,
because we assumed that the simplicial spaces A, X and Y are
Reedy cofibrant.
The property of being a homotopy fiber sequence is invariant under 
pointwise weak equivalences, so we may show that the upper row
is a homotopy fiber sequence.
This sequence arises from a commutative square of bisimplicial sets

    @C=12mm∘ A [r]^-∘ i[d]    ∘ X[d]^∘ f
    ∗[r]    ∘ Y

by taking iterated realizations. 
In every fixed simplicial dimension, the square of simplicial sets
is a homotopy fiber square in the sense of <cit.>,
by hypothesis (i).
Moreover, the bisimplicial sets ∘ X and ∘ Y
satisfy the π_*-Kan condition by assumption (ii),
and the morphism of simplicial sets 
π_0^v(∘ f):π_0^v(∘ X)→π_0^v(∘ Y)
is a Kan fibration by assumption (iii).
So <cit.> shows that the
square of diagonal simplicial sets

    @C=15mm(∘ A) [r]^-(∘ i)[d]    (∘ X)[d]^(∘ f)
    ∗[r]    (∘ Y)

is homotopy cartesian.
After geometric realization we can identify the diagonal realization 
with the iterated realization 
as in Proposition <ref> (iii),
so this proves the claim.




At some later stage we will need to know that the product of two
Reedy cofibrant simplicial spaces is again Reedy cofibrant.
The proof of this fact is rather formal and works in a much broader context,
as the following proposition shows.


Let  be a model category that is equipped with a monoidal product ⊠
satisfying the pushout product property for cofibrations.
Then for all Reedy cofibrant simplicial objects X,Y:^→,
the objectwise monoidal product X⊠ Y:^→
is Reedy cofibrant.


I learned this proof from Cary Malkiewich.
We let (m) denote the power set of the set {1,…,m}, 
i.e., the set of subsets.
We also write (m) for the associated poset category, under inclusion.
An m-cube in  is simply a functor F:(m)→.
We call such an m-cube cofibrant if for every subset B of {1,…,m}
the canonical morphism

    l_B  :  _A⊊ B  F(A) →  F(B)

is a cofibration.

We claim that for every cofibrant m-cube F and all
sets ⊂⊂(m) that are both closed under passage to subsets,
the canonical morphism

    _A∈  F(A) → _A∈  F(A)

is a cofibration. 
We start with the special case where =∪{B} 
for some subset B of {1,…,m} that does not belong to .
Since  is closed under taking subsets, 
every proper subset of B belongs to , and hence to .
Then the square

    @C=10mm_A⊊ B  F(A) [d][r]^-l_B     F(B)[d]
    _A∈  F(A)[r]    _A∈  F(A)

is a  pushout in .
The upper horizontal latching morphism is a cofibration by hypothesis;
so the lower horizontal morphism is a cofibration.  

In the general case we choose a chain of intermediate subsets

    =  _0  ⊂ _1  ⊂ … ⊂_k  =

such that each _i is closed under taking subsets 
and _i has exactly one element more than _i-1. 
The claim then holds for each pair (_i,_i-1).
Since the composite of two cofibrations 
is a cofibration, this proves the general case.

Now we turn to the proof of the proposition.
In Remark <ref> we specified an isomorphism of categories
κ:(m)^(m).
Since X is Reedy cofibrant, the m-cube

    (m) [] κ^-1 (m)^  u   ^   X

is cofibrant for every m≥ 0,
and the same is true for Y. We observe that the (m+n)-cube

    X Y :  (m+n) →  , 
       U+V ⟼ X_|U|⊠ Y_|V|

is then again cofibrant, where `+' denotes disjoint union of sets.
Indeed, for B∈(m) and B'∈(n),
the latching object

    _A+A'⊊ B+B' X_|A|⊠ Y_|A'|

is isomorphic to the pushout

    (L_B X)⊠ Y_|B'|∪_(L_B X)⊠ (L_B'Y) X_|B|⊠ (L_B'Y)

because ⊠ preserves colimits in both variables.
Under this isomorphism, the latching morphism

    l_B+B' :  _A+A'⊊ B+B' X_|A|⊠ Y_|A'| → X_|B|⊠ Y_|B'|

becomes the pushout product l_B^X l_B'^Y
of the latching morphisms for X respectively Y.
This latter morphism is a cofibration by the pushout product property.

Now we take m=n and let  be the subposet of (n+n) 
of proper diagonal elements, i.e., the sets U+U 
for a proper subset U of {1,…,n}.
The latching object L_n(X⊠ Y) is then a colimit
of the functor X Y over the poset .
We let  be the subposet of (n+n)
consisting of those sets U+V such that U∪ V{1,…,n}.
The latching map for X⊠ Y factors as the composite

    L_n(X⊠ Y) =     _U+U∈  X_|U|⊠ Y_|U|
    →    _U+V∈  X_|U|⊠ Y_|V| → X_n⊠ Y_n  .

We observe that the inclusion → is final, i.e., 
for every U∈ the comma category U↓
is non-empty and connected. 
So the first morphism in (<ref>) is an isomorphism.
We set =(n+n);  since  has a terminal object,
the object X_n⊠ Y_n is a colimit 
of the functor X Y over .
Both  and  are closed under passage to subsets, 
so the first paragraph shows that 
the second morphism in (<ref>) is a cofibration.
This proves the claim.  







CHAPTER: EQUIVARIANT SPACES





In this appendix we collect basic results about the equivariant homotopy
theory of G-spaces. Initially G can be any topological group, but we will
eventually specialize to compact Lie groups.
We start out by checking that taking fixed points commutes with certain kinds of
colimits, namely pushouts and sequential colimits along closed embeddings,
smash products, geometric realization and latching objects,
see Proposition <ref>.
Proposition <ref>
provides a self-contained proof of the standard model structure 
on the category of G-spaces, relative to a set of closed subgroups.
In the realm of compact Lie groups, 
various change of group functors preserve cofibrations,
namely fixed points (see Proposition <ref>),
restriction along a continuous homomorphism, induction,
and orbits (see Proposition <ref>). 
Proposition <ref>
is a useful decomposition result for (G X)^K, 
the K-fixed points of the G-orbits of a G-free 
(K× G)-space.

Then we turn to equivariant -spaces.
We start with the observation that compactly generated spaces are 
`closed under prolongation of -spaces'.subjectGamma-space@-space
More precisely, if F:→ is a -k-space
such that F(n_+) is compactly generated for every n≥ 0,
then it is not completely obvious whether the prolongation F(K)
to a general compactly generated space K, defined as a certain quotient space of
∐_n≥ 0 F(n_+)× K^n, is weak Hausdorff.
We show in Proposition <ref>
that this is automatically the case.
Because no `weak Hausdorffication' is necessary, 
there are no unexpected identifications, and the underlying set of F(K) 
is what one first thinks of.

For us, the main purpose of -spaces is to provide spectra
by evaluation on spheres. We want to do this equivariantly, 
in the presence of an action of a compact Lie group.
Proposition <ref> gives a way to calculate 
the fixed points of a prolonged -G-space
under the action of a connected compact Lie group.
Since G-fixed points can be obtained by first taking fixed points
of the identity component G^∘, and then fixed points 
of the finite group π_0 G=G/G^∘, this formula effectively reduces
questions about fixed points to the case of finite groups.

In order to analyze prolonged -G-spaces homotopically,
we need a cofibrancy condition that we introduce 
in Definition <ref>.
This kind of cofibrancy is stable under passage to closed subgroups
and taking fixed points (see Proposition <ref>)
and guarantees that evaluation on spheres produces
a G-spectrum that is equivariantly connective (see Proposition <ref>).
Moreover, for G-co­fibrant -G-spaces,
prolongation is homotopical, i.e., strict equivalences prolong
to G-weak equivalences on finite G-CW-complexes
(see Proposition <ref>).

We conclude this appendix with a reformulation and generalization 
of the Segal-Shima­kawa delooping formalism
for equivariant -spaces, see Theorems <ref> 
and <ref>. 
When restricted to finite groups, these theorems show that
evaluation of a G-cofibrant -G-space F  on spheres provides
a positive G-Ω-spectrum if F is `special' 
(compare Definition <ref>),
and a full fledged G-Ω-spectrum if F is `very special'
(compare Definition <ref>).
As we explain in Remark <ref>, there is no hope
to obtain a G-Ω-spectrum
for compact Lie groups of positive dimension.
However, we do have partial delooping results for non-finite compact Lie groups:
Theorem <ref> 
effectively says that evaluating a G-cofibrant special -G-space
on spheres yields a `G^∘-trivial positive G-Ω-spectrum',
where G^∘ is the identity component of G.
In other words, evaluating on spheres provides equivariant deloopings with respect to 
all those G-representations on which G^∘ acts trivially.
If G is not finite, this is of course a very restricted class of
representations; but for trivial representations we can at least conclude
that evaluation on spheres gives a `naive' G-Ω-spectrum.



We let G be a topological group, which we take to mean a group
object in the category  of compactly generated spaces.
So a topological group is a compactly generated
space equipped with an associative and unital multiplication

    μ :   G × G  → G

that is continuous with respect to the 
compactly generated product topology, and such that the shearing map

    G × G  →  G × G  ,   (g,h) ⟼ (g, g h)

is a homeomorphism (again for the compactly generated product topology). 
This implies in particular that inverses exist in G,
and that the inverse map g↦ g^-1 is continuous.
A G-space is then a compactly generated space X 
equipped with an associative and unital action 

    α :  G  × X  → X

that is continuous with respect to the compactly generated product topology.
We write GsymbolG - category of G-spaces
for the category of G-spaces and continuous G-maps.

The forgetful functor from G-spaces to compactly generated spaces 
has both a left and a right adjoint,
and hence limits and colimits of G-spaces are created in the underlying
category . This has nothing to do with topology, and is entirely formal,
using only that the underlying category  is cartesian closed, 
complete and cocomplete. 




Since colimits of G-spaces are created in the underlying category 
of compactly generated spaces, the earlier caveat applies as well.
A colimit in G-spaces is calculated by first forming a colimit 
in the category  of all topological spaces (or equivalently,
in the full subcategory  of k-spaces); the result is a k-space,
but it need not be weak Hausdorff. In that case an application of
the functor w:→ left adjoint to the inclusion produces 
a colimit in . This colimit comes with a preferred G-action making
it a colimit in the category of G-spaces. 
One has to beware that whenever the colimit in  is not weak Hausdorff,
the functor w changes the underlying set; in particular, the forgetful functor to sets
does not preserve such colimits.





Now we consider a closed subgroup H of a topological group G.
Then H is compactly generated in the subspace topology by
Proposition <ref> (i), and
hence a topological group (internal to the category ) in its own right.
For a G-space X we denote by

    X^H  =  { x∈ X |  h x = x  for all h∈ H}

the subspace of H-fixed points. For an individual element h∈ H
the h-fixed subspace {x∈ X |  h x = x }
is the preimage of the diagonal under the continuous map
(,h·-):X→ X× X, so it is a closed subspace of X
by Proposition <ref>.
As an intersection of closed subsets, X^H is then closed in X, and
hence compactly generated in the subspace topology,
by Proposition <ref> (i).



The following proposition records that fixed points preserve certain kinds of colimits.
In part (iv) we consider a simplicial G-space X:^→ G,
and we write X^G for the simplicial space consisting of the fixed points of X,
i.e., the composite functor

    ^  X   G (-)^G  .



Let G be a topological group.


  * For every pushout square of G-spaces on the left

    A [d]_-i [r]^-f     C [d]^-j        
    A^G [d]_-i^G[r]^f^G    C^G [d]^-j^G
    
    B[r]_-h     D        B^G[r]_-h^G    D^G

in which the map i is a closed embedding,
the square of fixed point spaces on the right is a pushout.

  * Taking G-fixed points commutes with filtered colimits
along continuous G-maps that are closed embeddings.

  * For all based G-spaces X and Y 
 the canonical map X^G Y^G→ (X Y)^G  is a homeomorphism.  

  * For every simplicial G-space X:^→ G and every n≥ 0,
the canonical maps 

    |X^G|→ |X|^G     and     L_n(X^G)→ (L_n X)^G

are homeomorphisms.



(i) Pushouts in G-spaces are formed on underlying compactly generated spaces.
Since i is a closed embedding, so is its restriction to
fixed points i^G. So by Proposition <ref> 
both pushouts are `as expected', i.e., formed in the ambient category
of all topological spaces. In particular, D is the set-theoretic
disjoint union of the images of B-i(A) and C, which are both G-invariant.
So D^G is the set-theoretic disjoint union of the images of
(B-i(A))^G=B^G-(i^G)(A^G) and C^G. The canonical map
B^G∪_A^GC^G→ D^G is thus a continuous bijection.

Now we show that the canonical map is closed.
We consider the commutative square:

    @C=12mm 
    B^G⨿ C^G [r]^-incl[d]_p     B⨿ C[d]^h+j
    
    B^G∪_A^GC^G[r]_-h^G∪ j^G    D

If O is a closed subset of B^G∪_A^GC^G,
then p^-1(O) is closed in B^G⨿ C^G, and hence in B⨿ C
(since fixed points are closed in the ambient space).
The relation

    p^-1(O) =  (h+j)^-1( (h^G∪ j^G)(O))

holds in B⨿ C; since the right vertical map h+j 
is a proclusion, (h^G∪ j^G)(O) is closed in D, and hence also in D^G.

(ii) We let P be a filtered poset and F:P→ G a functor
to the category of G-spaces. 
We must show that the canonical continuous map

    κ :  _P   F^G  → (_P F)^G

is a homeomorphism, where F^G=(-)^G∘ F.
Proposition <ref> ensures that both colimits
are formed in the ambient category of sets, so the map is bijective.
To see that the canonical map is also closed, we consider the commutative square

    ∐_j∈ P F(j)^G [r]^-incl[d]_p    ∐_j∈ P F(j) [d]^q
    _P   F^G [r]_-κ   _P F

in which the vertical maps are proclusions.
The canonical maps κ_j:F(j)→_P F are injective,
so every point of F(j) that becomes G-fixed in the colimit
is already G-fixed in F(j), i.e., the square is a pullback.
If A is a closed subset of _P F^G, then p^-1(A)
is closed in the coproduct of the F(j)^G's. Since F(j)^G
is closed in F(j), the set q^-1(κ(A)) = p^-1(A) 
is closed in the coproduct of the F(j)'s. Since q is a proclusion,
the set κ(A) is closed in _P F, and hence also in (_P F)^G.
This shows that κ is a closed map.

(iii)
Points of compactly generated spaces are closed 
(Proposition <ref> (iii)),
so the subspace X×{y_0}∪{x_0}× Y is closed in X× Y.
The claim then follows by applying part (i) to the pushout of the diagram

    ∗ ⟵ X×{y_0}∪{x_0}× Y  →  X× Y  .


(iv) Since X_n^G is closed inside X_n, the inclusion X^G⊂ X 
induces a closed embedding
|X^G|→ |X| by Proposition <ref> (iii).
The image of this map is clearly G-fixed. So it remains to show 
that every G-fixed point of |X| is in the image of |X^G|.
We let (x,t)∈ X_l×Δ^l be the  minimal representative
of a given G-fixed point in |X|^G. Then for every g∈ G 
the point (g x,t) is equivalent to (x,t).
Since the minimal representative is unique, this forces g x=x.
Hence x∈ X_l^G, which proves the claim.

The latching map l_n:L_n X→ X_n
is a closed embedding by Proposition <ref> (iii).
Hence the restriction to fixed points (l_n)^G:(L_n X)^G→ (X_n)^G 
is also a closed embedding.
The composite

    L_n(X^G) →  (L_n X)^G (l_n)^G (X_n)^G

is the latching map for the simplicial compactly generated space X^G,
hence also a closed embedding, 
again by Proposition <ref> (iii).
So the map in question is a closed embedding.

It remains to show that every G-fixed point of L_n X 
is the image of a point in L_n(X^G).
By Proposition <ref>,
the latching map identifies L_n X with the union
of the subspaces s_i^*(X_n-1)  for i=0,…,n-1.
So we may show that the composite

    L_n(X^G) →  (L_n X)^G [] (l_n)^G  (⋃_i=0,…,n-1 s_i^*(X_n-1) )^G

is surjective.
If  y∈ X_n-1 is such that x=s_i^*(y) is G-fixed, then y is
G-fixed because every degeneracy map is injective and G-equivariant.
So y∈ (X_n-1)^G represents a point in L_n(X^G) that maps to x.



 
  Let H be a closed subgroup of a topological group G.
  
  
  * The orbit space G/H is compactly generated in the quotient topology,
    and hence a G-space.  
  
  * Let
    
    X_0  → X_1  → … → X_n  → …

    be a sequence of closed embeddings of G-spaces
    and X_∞ a colimit of the sequence in the category of G-spaces.
    Then for every compact space K,
    every continuous G-map from G/H× K to X_∞ 
    factors through X_n for some n, and that factorization is
    a continuous G-map.
  


(i) The equivalence relation E⊂ G× G
that gives rise to G/H is the inverse image of H under the continuous map

    G× G  → G  ,    (g,g̅) ⟼ g^-1·g̅ .

Because H is closed, so is the equivalence relation, and hence
the quotient topology is compactly generated by 
the criterion given by Proposition <ref>.

(ii)
We let f : G/H × K → X_∞ be a continuous G-map. 
Colimits of G-spaces are created in
the underlying category  of compactly generated spaces;
since the maps in question are closed embeddings, 
the sequential colimit is created in the ambient category 
of all topological spaces, by Proposition <ref>.  
The composite 

    K  (e H,-) G/H × K   f  X_∞

factors through a continuous map f̅:K→ X_n for some n≥ 0,
by Proposition <ref> (i).
The canonical map X_n→  X_∞ is injective
by Proposition <ref>.
Since the image of the composite (<ref>)
is contained in the H-fixed points of X_∞,
the image of the factorization f̅:K→ X_n is contained
in the H-fixed points of X_n.
So the composite

    G× K  G×f̅ G× X_n  act X_n

factors through a well-defined set theoretic G-equivariant map

    f'  :   G/H× K  → X_n  .

The map proj× K:G× K→ G/H× K is a
proclusion by Proposition <ref>,
so the continuity of (<ref>) implies that f' is also continuous,
and hence the desired factorization of the original morphism f.





An argument that we need several times in this book proves
that a certain model structure is topological.
To avoid repeating the same kind of argument, we axiomatize it.
The `classical' model structure on the category of all topological spaces
was established by Quillen in <cit.>.
We use the straightforward adaptation of this model structure to
the category of compactly generated spaces,
which is described for example in <cit.>.
In this model structure on the category , the weak equivalences 
are the weak homotopy equivalences and fibrations
are the Serre fibrations. The cofibrations are the retracts of 
generalized CW-complexes, i.e., relative cell complexes in which cells
can be attached in any order and not necessarily to cells of lower dimensions.

We consider a model category  that is also enriched, tensored and cotensored
over the category  of compactly generated spaces.
We denote the tensor by ×.
Given a morphism f:X→ Y in   and a continuous map of spaces g:A→ B,
we denote by f g the 
pushout product morphismsubjectpushout product defined as

    f g = (f× B)∪(Y× g)  :  
    X× B∪_X× AY× A  → Y× B .

We recall that the model structure is called topologicalsubjecttopological model structuresubjectmodel structure!topological|seetopological model structure
if the following two conditions hold:


  * if f is a cofibration in  and g is a cofibration of spaces, 
  then the pushout product morphism f g is a cofibration;

  * if in addition f or g is a weak equivalence, then so is the
  pushout product morphism  f g.

The pushout product condition can also be stated in two different, but equivalent,
adjoint forms, compare <cit.>.
In the next proposition, we denote by

    i_k  :  ∂ D^k  → D^k      and    
     j_k  :   D^k×{0} → D^k×[0,1]

the inclusions. Then {i_k}_k≥ 0 is the standard set
of generating cofibrations for the Quillen model structure on the category
of spaces, and  {j_k}_k≥ 0 is the standard set
of generating acyclic cofibrations, compare <cit.>.



Let  be a model category that is also enriched, tensored and cotensored
over the category  of spaces.
Suppose that there is a set of cofibrant objects  
and a set of acyclic cofibrations  of  with the following properties:


  * The acyclic fibrations are characterized by the right lifting property
with respect to the morphisms K× i_k for all K∈ and k≥ 0.

  * The fibrations are characterized by the right lifting property
with respect to the union of the morphisms K× j_k for all K∈ and k≥ 0
and the pushout products c i_k for all c∈ and k≥ 0.

Then the model structure is topological.


Since the tensor bifunctor × has an adjoint in each variable, 
it preserves colimits in each variable.
So it suffices to check the pushout product properties
when the maps f and g are from the sets of generating (acyclic) cofibrations,
compare <cit.>.
The set of inclusions of spheres into discs is closed under pushout product,
in the sense that i_k i_l is homeomorphic to i_k+l.
So pushout product with i_l preserves the set of
generating cofibrations {K× i_k}_K∈,k≥ 0 (up to isomorphism). 
This takes care of the part
of the pushout product property that involves the cofibrations only.

Similarly, the pushout product of j_k with i_l is isomorphic to j_k+l.
So pushout product with i_l preserves the set  of
generating acyclic cofibrations {K× j_k}_K∈,k≥ 0, and
it preserves the set of generating acyclic cofibrations  {c× i_k}_c∈,k≥ 0.
This shows that the pushout product of an acyclic cofibration in 
with a cofibration of spaces is again an acyclic cofibration.

Finally, pushout product with j_l takes the generating cofibrations 
K× i_k to morphisms of the form
K× j_m with  K∈ and m≥ 0, which are acyclic cofibrations.
This shows that the pushout product of a cofibration in 
with an acyclic cofibration of spaces is again an acyclic cofibration.



Now we let  be a set of closed subgroups of a topological group G.
We call a morphism f:X→ Y of G-spaces a -equivalence
(respectively -fibration)subjectC-fibration@-fibration 
if the restriction f^H:X^H→ Y^H to H-fixed points is a
weak equivalence (respectively Serre fibration)
of spaces for all subgroups H of G in .
A -cofibrationsubjectC-cofibration@-cofibration 
is a morphism with the left lifting property 
with respect to all morphisms that are simultaneously
-equivalences and -fibrations.




  Let G be a topological group and  a set 
  of closed subgroups of G. Consider  a commutative diagram of G-spaces 
  
    C  [d]_γ    A [l]_-g [d]^α[r]^-f     B [d]^β
    C̅   A̅[l]^-g̅[r]_-f̅   B̅

  such that f and f̅ are h-cofibrations of G-spaces.
  Suppose that the maps α,β and γ are -equivalences.
  Then the induced map of pushouts
    
    γ∪β :  C∪_A B  → C̅∪_A̅B̅

    is a -equivalence.


  We let H be a closed subgroup from the set ,
  and we contemplate the commutative diagram of fixed points:
  
    @C=10mm
        C^H  [d]_γ^H    A^H [l]_-g^H[d]^α^H[r]^-f^H    
        B^H [d]^β^H
    C̅^H     A̅^H [l]^-g̅^H[r]_-f̅^H   B̅^H

  Since f and f̅ are h-cofibrations of G-spaces,
  f^H and f̅^H are h-cofibrations of spaces. 
  The three vertical maps are weak equivalences by hypothesis.
  The gluing lemma for weak equivalences and pushouts along 
  h-cofibrations then shows that the induced map on horizontal pushouts 
    
    γ^H∪β^H :  C^H ∪_A^H B^H  → C̅^H∪_A̅^HB̅^H

  is a weak equivalence,
  see for example <cit.>.
  Since f and f̅ are h-cofibrations of G-spaces, they are in particular 
  h-cofibrations of underlying spaces, and hence closed embeddings
  (Proposition <ref> (ii)).
  So taking H-fixed points commutes with the horizontal pushout
  (by Proposition <ref> (i)), and we conclude that
  also the map
    
    (γ∪β)^H :  (C∪_A B)^H  → (C̅∪_A̅B̅)^H

    is a weak equivalence. This proves the claim.



The following -projective model structure
is well known and fairly standard, and mentioned in various places in the literature,
for example in <cit.> and <cit.>.
However, I do not know a reference that is both self-contained and complete, 
so I provide the proof.


  Let G be a topological group and  a set 
  of closed subgroups of G. 
  Then the -equivalences, -cofibrations and
  -fibrations form a model structure,
  the -projective model structure
  on the category of G-spaces.
  This model structure is proper, cofibrantly generated and topological.


We number the model category axioms as in <cit.>.
The category of G-spaces is complete and cocomplete and all
limits and colimits are created in the underlying category of compactly generated spaces.
Model category axioms MC2 (2-out-of-3)
and MC3 (closure under retracts) are clear. One half of MC4 
(lifting properties) holds by the definition of -cofibrations.
The proof of the remaining axioms uses Quillen'ssubjectsmall object argument
small object argument, originally given in <cit.>, 
and later axiomatized in various places,
for example in <cit.> or <cit.>.
In the category of (non-equivariant) spaces, 
the set  {i_k: ∂ D^k → D^k}_k≥ 0 of inclusions of spheres into discs 
detects Serre fibrations that are simultaneously weak equivalences. 
By adjointness, the set

    I_ =  { G/H× i_k  :  G/H ×∂ D^k → G/H ×  D^k }_k≥ 0, H∈

then detects acyclic fibrations in the -projective 
model structure on G-spaces.
Similarly, the set of inclusions {j_k: D^k×{0}→ D^k× [0,1]}_k≥ 0 
detects Serre fibrations; so by adjointness, the set

    J_ =  { G/H × j_k }_k≥ 0, H∈

detects fibrations in the -projective  model structure on G-spaces.

All morphisms in I_ and J_ are closed embeddings,
and this property is preserved by coproducts, cobase change and
sequential colimits in the category of G-spaces.
Proposition <ref> guarantees that sources and targets
of all  morphisms in I_ and J_ 
are finite (sometimes called `finitely presented') 
with respect to sequences of closed embeddings of G-spaces.
In particular, the sources of all these morphisms
are finite with respect to sequences of I_-cell complexes and
J_-cell complexes.

Now we can prove the factorization axiom MC5.
Every morphism in I_ and J_ is a -cofibration by adjointness.  
Hence every I_-cofibration or J_-cofibration is a -cofibration of
G-spaces.
The small object argument applied to the set  
I_ gives a factorization of any morphism of G-spaces as
a -co­fibra­tion followed by a morphism with the right lifting property
with respect to I_. Since I_ detects the -acyclic -fibrations,
this provides the factorizations as cofibrations followed by acyclic fibrations.

For the other half of the factorization axiom MC5
we apply the small object argument to the set J_; 
we obtain a factorization of any morphism of G-spaces as
a J_-cell complex followed by a morphism with the right lifting property
with respect to J_. Since J_ detects the -fibrations,
it remains to show that every  J_-cell complex 
is a -equivalence.
To this end we observe that the morphisms in J_ are inclusions of deformation
retracts internal to the category of G-spaces.
This property is inherited by coproducts and cobase changes, 
so every morphism obtained by cobase changes of coproducts
of morphisms in J_ is a homotopy equivalence of G-spaces,
hence also a -equivalence. We also need to pass to sequential colimits,
which is fine because J_-cell complexes are closed embeddings,
and taking H-fixed points commutes with sequential colimits over
closed embeddings (Proposition <ref> (ii)).

It remains to prove the other half of MC4, i.e., that every -acyclic
-cofibration f:A→ B has the left lifting property
with respect to -fibrations.
The small object argument provides a factorization 

    A   j  W  q  B

as a J_-cell complex followed by a -fibration.
In addition, q is a -equivalence since f and j are.
Since f is a -cofibration, a lifting in 

    A [r]^-j [d]_f     W [d]^q_(.6)∼
    
    B @=[r] @..>[ur]     B

exists. Thus f is a retract of the morphism j that has the left lifting
property for -fibrations. 
So f itself has the left lifting property for -fibrations.

The model structure is topological by Proposition <ref>.
Right properness of the model structure is a straightforward consequence
of right properness of the model structure on spaces,
since the H-fixed point functor
preserves pullbacks and takes -fibrations to Serre fibrations.
Since the projective -model structure is topological
and all objects are fibrant, every cofibration is an h-cofibration
by Corollary <ref> (iii). 
So left properness follows from the gluing lemma for -equivalences
(Proposition <ref>).





Let G be a topological group.
A universal spacesubjectuniversal space!for a set of subgroups
for a set  of closed subgroups of G
is a -cofibrant G-space E such that
for every subgroup H∈ the fixed point space E^H 
is weakly contractible.



Any cofibrant replacement of the one-point G-space 
in the -projective model structure is a universal space for the set ,
so universal spaces exist for any set of closed subgroups.
Moreover, any two universal spaces for the same subgroups
are G-equivariantly homotopy equivalent:



Let G be a topological group,  a set 
of closed subgroups of G, and E a universal G-space for the set .


  * Every -cofibrant G-space admits a continuous G-map to E,
and any two such maps are homotopic as G-maps.

  * If E' is another universal G-space for , then every continuous G-map
from E' to E is a G-equivariant homotopy equivalence.



(i) This all follows from the existence of the -projective model
structure described in Proposition <ref>.
We let A be a -cofibrant G-space.
The unique map E→∗ to a one-point G-space is a -equivalence and
a -fibration. So the unique morphism from A
to the one-point G-space lifts to a continuous G-map A→ E.
The -projective model structure is topological, so 
the inclusion A×{0,1}→ A× [0,1] is a -cofibration,
and thus has the left lifting property with respect to E→∗.
Given two G-maps f,f':A→ E, any solution to the lifting problem

    A×{0,1}[r]^-f+f'[d]     E[d]
    
    A×[0,1][r]    ∗

is a G-homotopy from f to f'.
Since universal G-spaces are -cofibrant, part (ii) is a consequence of (i). 



A morphism of G-spaces is a G-cofibrationsubjectG-cofibration@G-cofibration 
if it has the left lifting property with respect 
to all morphisms that are simultaneously weak equivalences and Serre
fibrations on the fixed points for all closed subgroups of G.
Equivalently, G-cofibrations are the -cofibrations
in the sense of Proposition <ref>
for the maximal set of all closed subgroups of G.
Relative G-CW-complexes are special kinds of I_-cell complexes
(with I_ defined in (<ref>)),
namely those where all equivariant cells of the same
dimension are attachable at once, and in order of increasing dimensions.
Thus we have the following implications between the various kinds of
`nice equivariant embeddings':

    relative G-CW-complex ⟹G-cofibration ⟹h-cofibration of G-spaces

The second implication is Corollary <ref>,
applied to the -projective model structure.
Both implications are strict. 


  Let N be a closed normal subgroup of a topological group G. 
  Then for every G-cofibration i:A→ B the map
  
    i^N :  A^N →  B^N

  is a G/N-cofibration, and the map
  
    i∪ incl :  A∪_A^N B^N  → B

  is a G-cofibration.


  The class of G-cofibrations for which the claim holds is clearly closed under
  coproducts and retracts. We consider a pushout square of G-spaces on the left
  
    A [r]^-i[d]   B[d]       
        A^N[r]^-i^N[d]   B^N[d]
      
        C [r]_j    D           C^N [r]_j^N    D^N

  such that i is a G-cofibration for which the claim holds. 
  Then i is a closed embedding,
  and the square on the right is also a pushout of G-spaces
  (by Proposition <ref> (i)).
  In particular, j^N is a G-cofibration whenever i^N is.
  This means that C∪_C^N D^N is also a pushout
  of C and B^N over A^N, and hence the square
  
    @C=12mm 
        A∪_A^N B^N[r]^-i∪incl[d]     B [d]
    
        C∪_C^N D^N[r]_-j∪incl     D

  is another pushout of G-spaces. The upper horizontal map is a G-cofibration
  by hypothesis, hence so is the lower horizontal map.
  So the class of G-cofibrations satisfying the claim is closed under cobase change.
  Because N-fixed points preserve sequential colimits along closed embeddings
  (Proposition <ref> (ii)),
  the class of G-cofibrations satisfying the claim is also closed under
  sequential composites.
  
  Given these closure properties, it suffices to
  verify the claim for the generating G-cofibrations of the form
  
    G/H× i_k  :  G/H×∂ D^k  → G/H× D^k .

  Because (G/H)^N is either empty (whenever N is not contained in H)
  or all of G/H (whenever N≤ H),
  the map in question is either the map G/H× i_k
  or the identity of G/H× D^k; in either case it is a G-cofibration.
  This proves the claim for the generating cofibrations, and thus concludes the proof. 






Now we specialize from general topological groups to 
the class of compact topological groups. The first useful
special feature is that in this restricted context,
orbit spaces are always compactly generated without
any need to force the weak Hausdorff condition.
The arguments involved in the following proposition are well-known,
see for example <cit.> or <cit.>
for closely related statements.
Since the references I know of work in slightly different categories,
I spell out the proof.



  Let G be a compact topological group and X a G-space.
  
  
  * The orbit space G X is compactly generated in the quotient topology.
  
  * The projection Π_X:X→ G X is both open and closed.
  
  * For every G-invariant closed subset Y of X, the tautological map
    G Y → G X is a closed embedding.
  


(i)
Proposition <ref> reduces the claim to showing that
the orbit equivalence relation

    E  =  {  (g x, x) |  g∈ G,  x∈ X }

is closed in X× X.
Since G is compact, projection away from G is a closed map;
so the composite

    G× X× X  (g,x,y)↦ (g,g x,y) G× X× X   proj  X× X

is a closed map since the first map is a homeomorphism.
Since X is weak Hausdorff, the diagonal Δ_X 
is closed in X× X. So G×Δ_X is closed in
G× X× X.
The orbit relation E is the image of G×Δ_X under the above composite,
so E is closed in X× X.

(ii)
If O is an open subset of X, then g O is open for every g∈ G, 
since left translation by g is a homeomorphism. So 

    Π_X^-1(Π_X(O)) =  ⋃_g∈ G  g O

is open as a union of open subsets. Hence Π_X(O) is open in the quotient topology.
The shearing map

    χ :  G× X  → G× X  ,    (g,x) ⟼ (g, g x)

is a homeomorphism, and the composite

    G× X   χ  G× X  proj X

is the action map α:G× X→ X. 
Since G is compact, projection away from G is a closed map; 
so the action map is also a closed map.
If A is closed in X, then G× A is closed in G× X.
So the set

    Π_X^-1(Π_X(A)) =  α(G× A)

is closed in X. Hence Π_X(A) is closed in the quotient topology.

(iii) The tautological map G Y→ G X is continuous 
and injective. We show that the map is also closed.
We denote by Π_Y:Y→ G Y the quotient map.
We let A⊂ G Y be any closed subset. Then Π_Y^-1(A) is closed in Y,
hence also in X.
Since Π_X is a closed map by part (ii), Π_X(A) is closed in G X.






Now we specialize our discussion to compact Lie groups.
If K is a closed subgroup of a compact Lie group G of smaller dimension,
then the underlying K-space of a G-CW-complex 
need not admit a K-CW-structure – 
an example is given by Illman in <cit.>.
Nevertheless, the underlying K-space of a G-CW-complex 
is always K-homotopy equivalent to a K-CW-complex, which can be chosen
to be compact if the original G-space is,
see <cit.>.

The next proposition shows that with respect to restriction of group actions,
the class of cofibrant equivariant spaces
is better behaved than equivariant CW-complexes.
An additional advantage of cofibrant spaces over CW-complexes
is that `cofibrant' is a property, whereas CW-structures are additional data.


 
Let G be a compact Lie group.


  * For every compact Lie group K and every continuous homomorphism α:K→ G
  the restriction functor α^*:G→ K 
  takes G-cofibrations to K-cofibrations.

  * For every closed subgroup H of G 
  the induction functor G×_H -:H→ G
  takes H-cofibrations to G-cofibrations.

  * For every closed normal subgroup N of G 
  the orbit functor N -:G→ (G/N)
  takes G-cofibrations to G/N-cofibrations.



(i) The restriction functor α^* preserves colimits,
so we may show that it takes the generating G-cofibrations
G/H× i_k:G/H×∂ D^k→ G/H× D^k to K-cofibrations, 
for any closed subgroup H of G. 
The K-action on the smooth compact manifold G/H is by
left translation through α; 
continuous homomorphisms between Lie groups are automatically smooth,
compare <cit.>, so the K-action on G/H is smooth.
Illman's theorem <cit.> provides a finite K-CW-structure on G/H,
so in particular G/H is cofibrant as a K-space. Since the projective
model structure on K-spaces (for the set of all closed subgroups) is topological,
the map G/H× i_k is a K-cofibration.

(ii) Since G×_H - preserves colimits,
it suffices to show that it takes the generating H-cofibrations
H/J× i_k:H/J ×∂ D^k→ H/J × D^k to G-cofibrations, 
for any closed subgroup J of H.
This in turn is clear since G×_H (H/J) is G-homeomorphic to G/J. 

(iii)  Since N - preserves colimits,
it suffices to show that it takes the generating G-cofibrations
G/H× i_k:G/H ×∂ D^k→ G/H × D^k to G/N-cofibrations, 
for any closed subgroup H of G.
This in turn is clear since N (G/H) is (G/N)-homeomorphic to (G/N)/(H/H∩ N). 




Now we show that taking cartesian product preserves equivariant cofibrations.
There are two related questions, 
namely `external products' of a G-space and a K-space, 
and `internal products' of two G-spaces with diagonal action.


  Let G and K be topological groups.
  
  
  * The pushout product of a G-cofibration with a K-cofibration is
    a (G× K)-cofibration.
  
  * If G is a compact Lie group, then the pushout product of two G-cofibrations
    is a G-cofibration with respect to the diagonal G-action.
  


(i) The product functor 

    × :  G× K → (G× K)

preserves colimits in each variable, so it suffices to check the 
pushout product of a generating G-cofibration G/H× i_k with
a generating K-cofibration K/L× i_m, where i_k:∂ D^k→ D^k is
the inclusion. The pushout product of these is isomorphic to

    (G× K)/(H× L)× i_k+m

and hence a cofibration of (G× K)-spaces.
  
(ii) By (i) the pushout product of two G-cofibrations
is a (G× G)-cofibration. Since G is compact Lie,
restriction along the diagonal embedding G→ G× G
preserves cofibrations by Proposition <ref> (i). 






Now we prove a decomposition result for certain kinds of fixed points.
We let G and K be topological groups and X a (K× G)-space. 
We want to describe
the K-fixed points (G X)^K of the G-orbit space G X.
For  a continuous homomorphism α:K→ G we set

    X^α =  {x∈ X | (k,α(k)) x = x for all k∈ K} .

Equivalently, X^α is the fixed point space of the graph of α,
which is a closed subgroup of K× G.
The subspace X^α of X is invariant under the action of the centralizer 
of the image of α, i.e., the group

    C(α)  =  {g∈ G | gα(k)=α(k)g  for all k∈ K} .

The inclusion X^α→ X then passes to a continuous map

    α^♭ :   C(α)   X^α →  G X

on orbit spaces. For x∈ X^α and k∈ K the relation

    k(G x) =  G (k x)  =  G (α(k)^-1 x)  =   G x

shows that α^♭ takes values in the K-fixed points (G X)^K.
Moreover,  for g∈ G we have

    g· X^α =  X^c_g∘α

as subspaces of X. So the maps α^♭ and (c_g∘α)^♭
arising from conjugate homomorphisms 
have the same image in the orbit space G X.
It is relatively straightforward to see that the coproduct of
all the maps α^♭ is bijective if the G-action is free; 
some pointset topology is involved in
showing that this continuous bijection is in fact a homeomorphism
if in addition G and K are compact Lie groups.


Let G and K be compact Lie groups.
Let X be a (K× G)-space such that G-action is free.
Then the map

    ∐α^♭ :  ∐_α  C(α)  X^α →   (G X)^K

is a homeomorphism.
Here the disjoint union runs over conjugacy classes of continuous homomorphisms 
α:K→ G and C(α) is the centralizer, in G, of the image of α.


We let Π:X→ G X denote the quotient map.  
We set

    X̅ =  Π^-1((G X)^K)  .

Since X is compactly generated, so is G X 
by Proposition <ref> (i).
Since (G X)^K a closed subset of the orbit space, 
X̅ is a (K× G)-invariant closed subspace of X. 
In particular, X̅ is compactly generated in the subspace topology.

For a given continuous homomorphism α:K→ G, we set

    X^(α) =  G· X^α ,

the smallest G-subspace of X̅ containing X^α. 
The relation (<ref>) shows that X^(α)
is the union of the subsets X^α' as α' runs 
over all conjugates of α.
We factor the map in question as a composite

    ∐_α  C(α)  X^α → ∐_α  G  X^(α) → GX̅ →  (G X)^K  ,

induced by the inclusions X^α→ X^(α)→X̅→ X;
we show that each of the three maps is a homeomorphism.
The third map GX̅→ (G X)^K is a homeomorphism
by Proposition <ref> (iii).

For every x∈X̅ and every k∈ K we have k· (G x)=G(k x)= G x,
so there exists a g∈ G such that k x= g^-1 x.
Since the G-action is free, the element g is uniquely determined
by this property. So we can define a map
β_x:K→ G by the property k x=β_x(k)^-1 x;
equivalently, the characterizing condition for β_x is that
(k,β_x(k))· x = x. 
It is straightforward to see that β_x is a group homomorphism. 

By definition, the stabilizer group of x∈X̅ inside K× G
is precisely the graph of the homomorphism β_x:K→ G.
Since X is compactly generated, this stabilizer group is a closed
subset of K× G, which means that the homomorphism β_x
is continuous.  
Our next claim is that the assignment

    β :  X̅ → (K,G)  ,    x  ⟼ β_x

is continuous.
Since (K,G) has the subspace topology of (K,G), 
it suffices to show that the adjoint map

    β̅ :  X̅× K  → G  ,    (x,k) ⟼ β_x(k)

is continuous. This, in turn, is equivalent to the claim that the graph of β̅
is closed as a subset of X̅× K× G.
The graph of β̅ is the inverse image of the diagonal under the 
continuous map

    X̅× K× G  → X̅×X̅ ,   
    (x,k,g) ⟼ (k x, g x) .

So the graph of β̅ is closed, hence β̅ and β are
continuous.

By Proposition <ref> the space (K,G) 
is the topological disjoint union of the G^∘-orbits under the conjugation action.
In particular, every G^∘-conjugacy class is open and closed.
A G-conjugacy class is a finite union of G^∘-conjugacy classes,
so every G-conjugacy class in (K,G) is also open and closed.
Since β:X̅→(K,G) is continuous, 
X̅ is the topological disjoint union of the subsets X^(α)
as α ranges over all conjugacy classes in (K,G). 
Taking orbits commutes with disjoint unions, so the canonical map
from ∐_(α)  G X^(α) to GX̅ is a homeomorphism.

The final step is to show that for every continuous homomorphism α:K→ G
the canonical continuous map

    C(α) X^α → G X^(α)

is a homeomorphism. The map is surjective because X^(α)=G· X^α.
The map is also injective; if x,y∈ X^α satisfy G x=G y,
then x=g y for some g∈ G, and so x∈ X^α∩ X^c_g∘α.
This implies that α=c_g∘α, and so g∈ C(α).
Hence C(α) x= C(α) g y =C(α) y.
Finally, the map (<ref>) is closed:
if O⊂ C(α) X^α is a closed subset,
then Π_X^α^-1(O) is closed in X^α.
Since X^α is closed in X, hence also in X^(α),
the set Π_X^α^-1(O) is also closed in X^(α).
Since the projection X^(α)→ G X^(α) is a closed map,
the image of O in G X^(α) is closed. This completes the proof
that the map (<ref>) is a homeomorphism.




Let G be a compact Lie group and A a free cofibrant G-space.
Then the functor A×_G - takes G-maps that are
non-equivariant weak equivalences to weak equivalences.


We let f:X→ Y be a continuous G-map
that is a non-equivariant weak equivalence.
We start with the special case where A is a G-CW-complex with skeleton filtration

    ∅ = A^-1 ⊂ A^0  ⊂ A^1  ⊂ … ⊂ A^n 
     ⊂ … .

We show by induction over n that the map A^n×_G f:A^n×_G X→ A^n×_G Y
is a weak equivalence. The induction start with n=-1, where there is nothing to show.
Now we consider n≥ 0 and assume the claim for smaller values of n.
Since G acts freely on A, only free G-cells occur in the equivariant CW-structure.
So there is an index set I and a pushout square of G-spaces:

    G× I ×∂ D^n [r]^-incl[d]_α    
    G× I × D^n [d]
    
    A^n-1[r]    A^n

The map A^n×_G f can thus be obtained by passing to pushouts 
in the horizontal direction of the commutative diagram

    @C=12mm 
    A^n-1×_G X[d]_A^n-1×_G f    
    I×∂ D^n× X[d]^I×∂ D^n× f[l]_-α×_G X[r]^-incl   
    I× D^n× X[d]^I× D^n× f
    
    A^n-1×_G Y    I×∂ D^n× Y [l]^α×_G Y[r]_-incl   
    I× D^n× Y

The left vertical map is a weak equivalence by the inductive hypothesis,
and the middle and right vertical maps are weak equivalences because f is.
The upper and lower right horizontal maps are h-cofibrations,
so the gluing lemma for weak equivalences and pushouts along h-cofibrations 
(see for example <cit.>)
shows that then the map on pushouts A^n×_G f is also a weak equivalence.
Instead of the gluing lemma, we can alternatively quote <cit.>.

The space A×_G X is the sequential colimit, along h-cofibrations,
of the spaces A^n×_G X, and similarly for A×_G Y.
In the category , all h-cofibrations are closed embeddings 
(Proposition <ref>),
and sequential colimits over sequences of closed embeddings in 
preserve weak equivalences 
(Proposition <ref> (i)).
So the map A×_G f is a weak equivalence.
This completes the proof in the special case when A admits a G-CW-structure.
A general cofibrant G-space is G-homotopy equivalent to a 
G-CW-complex, and the functor -×_G X takes
G-homotopy equivalences to non-equivariant homotopy equivalences.
This reduces the general case to the case of G-CW-complexes.




Now we turn to -spaces and study the categorical and homotopical
properties of prolongation of a -space to a continuous
functor defined on all based spaces.
In the body of this book, we always work in the category  of compactly
generated spaces. However, this full subcategory is not closed under
quotient spaces nor coends inside the ambient category
of all topological spaces; since the construction of the prolongation
involves a coend (quotient space), some care needs to be taken. 
So while we are mostly interested  -spaces with values 
in the full subcategory  of compactly generated spaces, 
we define and discuss prolongation for -k-spaces.


We let  denote the category whose objects are the based sets
n_+={0,1,…,n}, with basepoint 0, and with morphisms all based maps.
A -k-space is a functor from  
to the category of k-spaces that is reduced,
i.e., the value at 0_+ is a one-point space.

A -k-space F:→ 
can be `evaluated' at a based k-space K as follows.
We set

    F(K)  =  (∐_n≥ 0    F(n_+)×  K^n) /∼ .
 
In more detail, F(K) is the quotient space of the disjoint union of the spaces
F(n_+)×  K^n by the equivalence relation generated by

    (F(α)(x); k_1,…,k_n) ∼ (x;  k_α(1),…,k_α(m))

for all x∈ F(m_+), all (k_1,…,k_n) in K^n,  and all 
morphisms α:m_+→ n_+ in .
Here k_α(i) is to be interpreted as the basepoint of K
whenever α(i)=0.  
We emphasize that we use the Kelleyfied product topology 
on F(n^+)× K^n, in order to remain inside the category of k-spaces.
We mostly care about the special case when K is compact, and then
this coincides with the ordinary product topology 
F(m^+)×_0 K^×_0 n by Proposition <ref> (vi).
The class of k-spaces is closed under disjoint unions and
passage quotient spaces (by Proposition <ref> (i));
so F(K) is indeed a k-space.
We refer to the extension as the 
prolongationsubjectprolongation!of a -space
of the -space F.
A more categorical way to describe F(K) is as a coend of the functor

    ×^ →  ,    
      (m^+,n^+) ⟼ F(m^+)× K^n  .



 
  We want to justify the abuse of notation of not distinguishing 
  between the original -space and its prolongation:
  the value of the prolongation on m_+ is canonically homeomorphic 
  to the original value.   
  Slightly more is true, namely that the coend description 
  exhibits the prolongation as the left Kan extension of F 
  along the inclusion →_*; we won't use this, however.

  Only for this argument we denote the prolongation of F by F̂. 
  The continuous maps
  
    F(n_+)× (m_+)^n  → F(m_+) ,   
      (x;  i_1,…,i_n) ⟼ F(i)(x)

  are compatible with the equivalence relation defining F̂(m_+);
  here i:n_+→ m_+ is the based map with i(j)=i_j.
  So these maps assemble into a continuous map F̂(m_+)→ F(m_+)
  that is inverse to the continuous map
  
    F(m_+) → F̂(m_+) ,    x  ⟼ [x; 1,2,…,m] .

  
  


We have to analyze the equivalence relation
generated by (<ref>) more closely.
This part of the argument has nothing to do with topology, is purely combinatorial,
and of a very similar flavor as the analysis of the 
equivalence relation defining the geometric realization of a simplicial set.
We call an element x∈ F(m_+) degenerate
if it is in the image of the map F(α):F((m-1)_+)→ F(m_+)
for some morphism α:(m-1)_+→ m_+ in .
We call x non-degenerate if it is not degenerate.
We let C_m(K)⊂ K^m be the set of those tuples
(k_1,…,k_m) whose coordinates k_i are pairwise distinct and all different 
from the basepoint of K.
Part (i) of the following proposition 
is a special case of <cit.>;
part (ii) also ought to be well known, but I am not aware of a reference.
The next proposition in particular implies that
the reduction map (<ref>) defined in the proof of part (ii)
is a bijection from F(K) to the set-theoretic disjoint union, 
for m≥ 0, of the sets F(m_+)^nd×_Σ_mC_m(K),
where F(m_+)^nd is the set of non-generate elements of F(m_+).




Let F:→ (sets) be a functor such that F(0_+) has one element.


  * For every element y∈ F(m_+) there exists an injective morphism
δ:l_+→ m_+ and a non-degenerate element x∈ F(l_+)^ nd
such that y=F(δ)(x).
If moreover y=F(δ̅)(x̅) for 
another injective morphism δ̅:l̅_+→ m_+ 
and a non-degenerate element x̅∈ F(l̅_+)^ nd,
then l=l̅ and there exists a bijective morphism λ:l_+→ l_+
such that δ=δ̅λ and x̅=F(λ)(x).

  * Let K be a based set. Let (x,t)∈ F(l_+)× K^l be an element
of minimal dimension l within its equivalence class.
Let (y,s)∈ F(m_+)× K^m be equivalent to (x,t).
Then there exists a surjective morphism
σ:m_+→ k_+, an injective morphism δ:l_+→ k_+
and u∈ C_k(K) such that

    F(σ)(y) =  F(δ)(x) ,    
    s =  σ^*(u)      and     t =δ^*(u)  .


  * If (x,t) and (y,s) are equivalent elements, both of minimal dimension l
in their equivalence class, then
there is a unique isomorphism α:l_+→ l_+ such that
(F(α)(y),s)=(x,α^*(t)). 



Part (i) is an analog of the `Eilenberg-Zilber lemma' <cit.>
for simplicial sets, and a special case of <cit.>.
For the convenience of the reader, we give a direct proof. 
The existence part is proved by induction on m, starting with m=0,
where there is nothing to show. If m is positive and y is non-degenerate,
then x=y and δ= do the job. Otherwise 
y=F(α)(z) for some morphism α:(m-1)_+→ m_+ and z∈ F((m-1)_+).
We factor α=δ'∘β such that β:(m-1)_+→ k_+ is surjective
and δ':k_+→ m_+ is injective. We must have k<m, so the inductive
hypothesis provides an injective morphism
δ:l_+→ k_+ and a non-degenerate element x∈ F(l_+)^nd
such that F(β)(z)=F(δ)(x). But then δ'δ is also injective and

    y  =  F(α)(z) =  F(δ')(F(β)(z)) = F(δ')(F(δ)(x)) =  F(δ'δ)(x) .

This proves the first statement.

For the second statement we consider injective morphisms δ:l_+→ m_+
and δ̅:l̅_+→ m_+, and non-degenerate elements x∈ F(l_+)^nd
and x̅∈ F(l̅_+)^nd such that F(δ)(x)=F(δ̅)(x̅).
We let σ̅:m_+→l̅_+ be the unique morphism
that sends all elements not in the image of δ̅ to the basepoint 0
and satisfies σ̅δ̅=.
Then

    F(σ̅δ)(x) =  F(σ̅)(F(δ)(x)) =  
    F(σ̅)(F(δ̅)(x̅)) =  
    F(σ̅δ̅)(x̅) =  x̅ .

If σ̅δ:l_+→l̅_+ were not surjective,
then it could be factored through (l̅-1)_+, and x̅ would be degenerate,
contradicting the assumptions. So σ̅δ is surjective,
and hence l≥l̅. Reversing the roles of (x,δ) and (x̅,δ̅)
gives l≤l̅, and hence l=l̅. 
Since l=l̅ and σ̅δ is surjective, it must in fact
be bijective.

Now we claim that the morphisms δ,δ̅:l_+→ m_+ have the same image. 
If this were not the case,
then σ̅δ:l_+→ l_+ would not be surjective,
which again would contradict the assumption that x̅ is non-degenerate.
Since δ and δ̅ have the same image, there is 
a bijection λ:l_+→ l_+ such that δ=δ̅λ.
Hence 

    F(λ)(x) =  F(σ̅δ̅λ)(x) =  
    F(σ̅)(F(δ)(x)) =  
    F(σ̅)(F(δ̅)(x̅)) =  x̅ .

So λ is the bijection with the desired properties.

(ii)
We call a quadruple (σ,δ,u,x) consisting of
a surjective morphism σ:m_+→ k_+, 
an injective morphism δ:l_+→ k_+, 
a tuple u∈ C_k(K) and a non-degenerate element x∈ F(l_+)
a reduction datum for (y,s)∈ F(m_+)× K^m if 

    F(σ)(y)=F(δ)(x)     and      s=σ^*(u)
      .

Since δ is injective, the map δ^*:K^k→ K^l omits
some coordinates and reorders the remaining ones. In particular,
δ^* does not introduce duplicates or basepoints, 
so it sends the subset C_k(K) to the subset C_l(K), and hence
δ^*(u)∈ C_l(K).

Claim 1:
Every pair (y,s) has a reduction datum.
We let u=(u_1,…,u_k) be the non-basepoint elements in the set {s_1,…,s_m}, 
without repetition and in some chosen order.
We define a surjective morphism σ:m_+→ k_+ by

    σ(i)  = 
    j      if s_i=u_j, and
    
    0     if s_i is the basepoint of K.

Then u∈ C_k(K) and s=σ^*(u). 
Part (i) now provides an injective morphism δ:l_+→ k_+
and x∈ F(l_+)^nd such that F(σ)(y)=F(δ)(x).
So (σ,δ,u,x) is a reduction datum for (y,s).

Claim 2:
We show that the reduction datum is `essentially unique' in the following sense.
If

    (σ:m_+→ k_+,δ:l_+→ k_+,u,x)   and   
    (σ̅:m_+→k̅_+,δ̅:l̅_+→k̅_+,u̅,x̅)

 are reduction data for the same element (y,s),
then k=k̅, l=l̅ and there are bijective morphisms
λ:l_+→ l_+ and β:k_+→ k_+ such that

    σ̅=βσ ,   δ̅λ = βδ ,  
    u = β^*(u̅)     and    x̅ =F(λ)(x) .

Indeed, because s=σ^*(u)=σ̅^*(u̅)
and u and u̅ don't contain duplicates or basepoints, we must have k=k̅
and u and u̅ can only differ by the ordering.
So there is a bijective morphism β:k_+→ k_+ such that
u=β^*(u̅). For all i∈{1,…,m} we then have

    u̅_σ̅(i) =  s_i =  u_σ(i) =  u̅_β(σ(i)) .

Since the coordinates of u̅ are all distinct, this implies
that σ̅=βσ.
Using that both quadruples are reduction data, we know that 

    F(δ̅)(x̅) = F(σ̅)(y) =   F(β)(F(σ)(y))  =  
    F(β)(F(δ)(x)) =  F(βδ)(x) .

Since δ̅ and βδ are both injective and
x and x̅ are both non-degenerate, then essential uniqueness
statement in part (i)
shows that l=l̅ and provides a bijective morphism λ:l_+→ l_+
such that βδ=δ̅λ and F(λ)(x)=x̅.



We can now define a reduction map

    ρ :  ∐_n≥ 0  F(n_+)× K^n → ∐_l≥ 0  F(l_+)^nd×_Σ_l C_l(K)

by choosing a reduction datum (σ,δ,u,x) 
for a given element (y,s) and setting

    ρ(y,s) =  [ x,δ^*(u) ]  ,

where [-,-] denotes the Σ_l-orbit.
If (σ,δ,u,x) is another reduction datum for (y,s), then

    [x,δ^*(u)]  =  
    [x,δ^*(β^*(u̅))]  =  
    [x,λ^*(δ̅^*(u̅))]  =  
    [F(λ)(x),δ̅^*(u̅)]  =  [x̅,δ̅^*(u̅)] ,

by Claim 2. So ρ(y,s) is well-defined.



Claim 3:
If (y,s)∈ F(m_+)× K^m and (y̅,s̅)∈ F(m̅_+)× K^m̅
are equivalent, then ρ(y,s)=ρ(y̅,s̅).
It suffices to show the claim whenever (y,s) and (y̅,s̅)
are related by a generating relation (<ref>),
i.e., we can assume that y=F(α)(y̅) and s̅=α^*(s)
for some morphism α:m̅_+→ m_+.
We let (σ,δ,u,x) be a reduction datum for (y,s).
We choose a factorization

    σ∘α =  δ̅∘σ̅

as a surjective morphism σ̅:m̅_+→k̅_+
followed by an injective morphism δ̅:k̅_+→ k_+.
Using part (i) we write

    F(σ̅)(y̅)  =  F(δ')(x̅)

for an injective morphism δ':l'_+→k̅_+
and a non-degenerate element x̅∈ F(l'_+)^nd.
Then

    F(δ)(x)    =  F(σ)(y) =  F(σ)(F(α)(y̅)) 
       =  F(δ̅)(F(σ̅)(y̅))  
    =  F(δ̅)(F(δ')(x̅))  =  F(δ̅δ')(x̅)  .

Since x and x̅ are non-degenerate and δ and δ̅δ'
are injective, the essential uniqueness of part (i)
shows that l=l' and provides a bijection λ:l_+→ l_+
such that δ=δ̅δ'λ and x̅=F(λ)(x).
We conclude that

    F(σ̅)(y̅) =  F(δ')(x̅) =  
    F(δ')(F(λ)(x)) =  
    F(δ'λ)(x)

 and

    s̅ =  α^*(s) =  α^*(σ^*(u)) =  σ̅^*( δ̅^*(u))  .

Since u∈ C_k(K) and δ̅ is injective, δ̅^*(u)
belongs to C_k̅(K).
This shows that (σ̅, δ'λ, δ̅^*(u),x)
is a reduction datum for (y̅,s̅).
So

    ρ(y̅,s̅) =  [x, (δ'λ)^*(δ̅^*(u))]  = 
    [x, δ^*(u)]  =  ρ(y,s) .

This finishes the proof of Claim 3.

Now we can prove part (ii) of the proposition.
If (x,t)∈ F(l_+)× K^l is of minimal dimension in its equivalence class, then x
is non-degenerate and t∈ C_l(K), for otherwise (x,t) would be equivalent
to an element of smaller dimension. So (,,t,x) is a reduction datum
for (x,t), and hence ρ(x,t)=[x,t].
Now we let (y,s) be equivalent to (x,t), 
and we let (σ,δ,u,x') be a reduction datum for (y,s).
Then

    [x',δ^*(u)]  =  ρ(y,s) =  ρ(x,t) =  [x,t]  ,

where the second equality is Claim 3. So there is a bijective morphism
λ:l_+→ l_+ such that

    (x',λ^*(δ^*(u)))  =  (F(λ)(x),t) .

Hence

    F(σ)(y) =  F(δ)(x') =  
    F(δλ)(x) ,    
    s =  σ^*(u)    and   t =(δλ)^*(u)  .


(iii)
By the minimality assumptions, the data provided by part (ii)
must satisfy m=k=l, and the morphisms σ and δ 
must both be isomorphisms. So (F(α)(y),s)=(x,α^*(t)) 
with α=δ^-1σ:l_+→ l_+.
The coordinates of t are distinct, by minimality of dimension,
so there is only one permutation with s=α^*(t).



Now that we understand the equivalence relation generated by (<ref>), we can analyze the topology of the space F(K)
when F is a -space and K is a based topological space.
In particular, we show now that the prolongation of a -space
with values in the category  of compactly generated spaces
to a compactly generated space is automatically compactly generated,
with the weak Hausdorff property being the issue.
The statement and proof of the following proposition
are analogous to the ones in Proposition <ref>.


  Let F:→ be a -space with values in the
  category of compactly generated spaces, and K a compactly generated based space.
  
  
  * The space F(K) is compactly generated, and hence 
    a coend internal to the category ,
    of the functor (<ref>).
  
  * Let E be a -subspace of F such that
    E(n_+) is closed in F(n_+) for every n≥ 0.
    Let L be a closed based subset of K. 
    Then the inclusions E→ F and L→ K
    induce a closed embedding ι:E(L)→ F(K).
  


  (i) 
  The category  is closed under products inside the category ,
  so the functor (<ref>) takes values in ;
  the issue is that a priori, the quotient topology need not be weak Hausdorff.
  We use Proposition <ref>
  to show that the quotient space F(K) is compactly generated.
  We let E⊂ (∐_n≥ 0 F(n_+)× K^n)^2
  be the equivalence relation generated by (<ref>),
  which we had simply denoted `∼' above. We will show that E 
  is closed in the k-topology of (∐_n≥ 0 F(n_+)× K^n)^2.
  Since products distribute over disjoint unions, we may show that
  
    E_m,n =  E∩ (F(m_+)× K^m× F(n_+)× K^n)

  is closed in F(m_+)× K^m× F(n_+)× K^n for all m,n≥ 0.
  We consider (y,s,y̅,s̅)∈ E_m,n, i.e., the pairs (y,s)
  and (y̅,s̅) are equivalent.
  Proposition <ref> (ii) provides
  surjective morphisms
  σ:m_+→ k_+ and σ̅:n_+→k̅_+, 
  injective morphisms δ:l_+→ k_+ and δ̅:l_+→k̅_+
  and u∈ K^k, u̅∈ K^k̅  and (x,t)∈ F(l_+)× K^l
  such that
  
    F(σ)(y) =  F(δ)(x) ,    
      s =  σ^*(u)      and     t =δ^*(u)

  and
  
    F(σ̅)(y̅) =  F(δ̅)(x) ,   s̅ =  σ̅^*(u̅)      and     
      t =δ̅^*(u̅)  .

  Equivalently, E_m,n is the union,
  indexed over l,σ,σ̅,δ,δ̅ as above,
  of the finite number of sets
  
    (F(σ)×   K^m× F(σ̅)× K^n)^-1
       (
        (F(δ)×σ^*× F(δ̅)×σ̅^*)
        ((F(l_+)×δ^*× F(l_+)×δ̅^*)^-1
          ( Δ_F(l_+)× K^l)))  .

  The diagonal Δ_F(l_+)× K^l is closed in (F(l_+)× K^l)^2
  because F(l_+)× K^l is compactly generated.
  So its inverse image under the continuous map
  F(l_+)×δ^*× F(l_+)×δ̅^*
  is closed in F(l_+)× K^k× F(l_+)× K^k̅.
  The map F(δ)×σ^*× F(δ̅)×σ̅^*
  has a continuous retraction, and is thus a closed embedding by
  Proposition <ref>.
  So the set 
  
    (F(δ)×σ^*× F(δ̅)×σ̅^*)
      ((F(l_+)×δ^*× F(l_+)×δ̅^*)^-1
        ( Δ_F(l_+)× K^l))

  is closed in  F(k_+)× K^m× F(k̅_+)× K^n.
  Since E_m,n is the inverse image of this latter closed set
  under a continuous map, this show the claim that E_m,n
  is a closed subset of F(m_+)× K^m× F(n_+)× K^n.

  (ii)
  Our first claim is that the map ι:E(L)→ F(K) is injective.
  We let (x; l_1,…,l_m)∈ E(m_+)× L^m 
  be a minimal representative of an element of E(L).
  So the element x is non-degenerate, and the l_i's are pairwise distinct
  and different from the basepoint of L.
  We claim that x is also non-degenerate when viewed as an element of the
  ambient -space F. To see this, we argue by contradiction and assume
  that there was an injective based map δ:k_+→ n_+ with k<n 
  and an element y∈ F(k_+) such that F(δ)(y)=x. 
  We let σ:n_+→ k_+ be a retraction to δ.
  Then y=F(σ)(F(δ)(y))=F(σ)(x). Since E is a -subspace
  of F, we conclude that y∈ E(k_+). Then x=E(δ)(y),
  contradicting the non-degeneracy of x.
  Altogether this shows that a representative
  of minimal dimension for a class in E(L) remains a minimal representative
  for its class in F(K). Because minimal representatives are unique
  up to a permutation (Proposition <ref> (iii)), 
  this implies that the map ι:E(L)→ F(K) is injective.

  It remains to show that the continuous injection ι is a closed map.
  We consider the commutative square
  
    @C=12mm∐_n≥ 0 E(n_+)× L^n[r]^-incl[d]_p   ∐_n≥ 0 F(n_+)×  K^n[d]^q
    
        E(L)[r]_-ι    F(K)

  where the vertical maps are the quotient maps.
  We consider a point (y,s)∈ F(m_+)× K^m whose class in F(K)
  is in the image of ι. 
  Then (y,s) is equivalent to an element (x,t)∈ E(l_+)× L^l, 
  which we can take as a minimal representative in its (E,L)-equivalence class.
  As we argued in the injectivity statement, (x,t) is then also 
  a minimal representative in its (F,K)-equivalence class.

  Proposition <ref> (ii) 
  provides a surjective morphism σ:m_+→ k_+, 
  an injective morphism δ:l_+→ k_+ and u∈ K^k such that
  
    F(σ)(y) =  F(δ)(x) ,    
      s =  σ^*(u)      and      t  = δ^*(u)  .

  So for every subset A⊂ E(L), we have
  
    ( F(m_+   )× K^m) ∩ q^-1( ι(A) )  =   
    ⋃_σ,δ      (F(σ)× K^m)^-1( (F(δ)×σ^*) 
     (  (F(l_+)×δ^*)^-1( (E(l_+)× L^l)∩ p^-1(A) ) ) ) .

  The union is over the finite set of pairs (σ:m_+→ k_+,δ:l_+→ k_+)
  as above.

  Now we assume that A is closed inside E(L). Because p is continuous,
  E(l_+) is closed in F(l_+) and L is closed in K,
  the set (E(l_+)× L^l)∩ p^-1(A) is then closed inside F(l_+)× K^l.
  So
  (F(l_+)×δ^*)^-1((E(l_+)× L^l)∩ p^-1(A)) 
  is a closed subset of F(l_+)× K^k.
  Since F(δ)×σ^* has a continuous retraction,
  it is a closed embedding by Proposition <ref>.
  So the set (F(δ)×σ^*) 
  (  (F(l_+)×δ^*)^-1((E(l_+)× L^l)∩ p^-1(A) ))
  is closed in F(k_+)× K^m.
  As the inverse image of a closed set under a continuous map,
  each set in the finite union above is closed inside F(m_+)× K^m.
  We conclude that ( F(m_+)× K^m) ∩ q^-1( ι(A) )
  is closed in F(m_+)× K^m for every m≥ 0, 
  hence the set q^-1(ι(A)) is closed.
  Since q is a quotient map, this shows that ι(A) is closed in F(K).




We let F:→ be a -k-space and K and L two based k-spaces.
The prolongation comes with a continuous, 
based assembly mapsubjectassembly map!of a -space

    α :  K F(L)  → F(K L)  , α(k [x;  l_1,…,l_n]) =  [x; k l_1,…,k l_n] .

The assembly map is natural in all three variables and associative and unital.
We define a `shifted' 
-space F_K=F∘(K -) as the composite

    K- _*   F  _* .

Then for every based k-space L, we consider the maps

    ϕ_n  :  F_K(n_+)× L^n   = 
    F(K n_+)× L^n   → F(K L)

defined by

    ϕ_n( [x;k_1 i_1,…,k_m i_m]; l_1,…,l_n )  =  
    [x;  k_1 l_i_1,…, k_m l_i_m]  ,

where x∈ F(m_+), k_1,…,k_m∈ K and i_1,…,i_m∈{1,…,n}.
Another way to say this is that ϕ_n is the composite 

    F(K n_+)× L^n   proj  F(K n_+)  L^n   ass   F(K n_+ L^n)   F(Kϵ)  F(K L) ,

where

    ϵ :  n_+ L^n → L  ,   i (l_1,…,l_n) ⟼ l_i .

is the evaluation map.
These maps ϕ_n are compatible with the equivalence relation defining F_K(L),
so they combine into a continuous map

    ϕ :  F_K(L) → F(K L)  .

The map ϕ is natural in F, K and L.


Let F:→ be a -k-space and K and L based k-spaces. 
Then the natural map ϕ: F_K(L) →  F(K L) is a homeomorphism.


We let q:K× L→ K L denote the projection.
In the category , product with any space preserves proclusions 
(Proposition <ref>), so the map

    F(m_+)× q^m :  F(m_+)× (K× L)^m   → F(m_+)× (K L)^m

is a proclusion for every m≥ 0.
The continuous map

    ψ_m  :  F(m_+)× (K× L)^m   → F_K(L)

defined by

    ψ_m(x;(k_1,l_1),…,(k_m,l_m))  =  
    [[x;  k_1 1,…, k_m m]; l_1,…,l_m]

is constant on the fibers of F(m_+)× q^m,
so it factors over a continuous map

    ψ̅_m  :  F(m_+)× (K L)^m   → F_K(L) .

These maps are compatible with the equivalence relation defining F(K L),
so they combine into a continuous map

    ψ :  F(K L) → F_K(L)  .

Since ϕ and ψ are inverse to each other, this completes the proof.






Now we prove an interchange relation between geometric realization and 
prolongation of a -space. 
We let A:^→_* be a simplicial based k-space and
F:→ a -k-space. 
The composite

    ^  A  _*   F   _*

is a simplicial space, where the second functor is the prolongation of F,
denoted by the same symbol. The composite has a geometric realization 

    |F∘ A| =  | [n]⟼ F(A_n)|  .

We exhibit a natural homeomorphism from |F∘ A| to F(|A|),
the value of F at the realization of A.
We let κ_n:A_nΔ^n_+→ |A|
be the canonical based map sending a point (a,t) to its equivalence class in A.
Then the maps

    F(A_n)×Δ^n assembly F(A_nΔ^n_+) F(κ_n) F(|A|)

are compatible with the equivalence relation defining |F∘ A|,
as [n] varies over the objects of .
So the maps define a continuous map

    κ :  |F∘ A| → F(|A|) .

In terms of elements, κ is thus given by

    κ[[x; a_1,…,a_m],t]  =  [x; [a_1,t],…,[a_m,t]] ,

for x∈ F(m_+), a_1,…,a_m∈ A_n and t∈Δ^n.


The following proposition is <cit.>.


For every simplicial based k-space A:^→_* and every 
-space F:→,
the map κ:|F∘ A|→ F(|A|) is a homeomorphism.


This is a special case of the `Fubini theorem' for coends.
In more detail we consider the functor

    ×^×^× →  ,   
    (k_+,l_+,[m],[n]) ⟼ F(k_+)× A_m^l×Δ^n .

If we fix objects [m] and [n] of , 
and exploit that coends commute with product with Δ^n,
we obtain a homeomorphism

    ∫^k_+∈ F(k_+)× A_m^k×Δ^n     (  ∫^k_+∈ F(k_+)× A_m^k) ×Δ^n  
    F(A_m)×Δ^n .

These homeomorphisms are natural as ([m],[n]) vary in ^×.
So taking coends over  gives a homeomorphism

    ∫^[m]∈∫^k_+∈ F(k_+)× A_m^k×Δ^m     ∫^[m]∈ F(A_m)×Δ^m =  |F∘ A| .

On the other hand, if we fix k and l and exploit that geometric
realization commutes with product with the space F(k_+) and with products 
(see parts (i) and (ii) of Proposition <ref>),
we obtain a homeomorphism

    ∫^[m]∈ F(k_+)× A_m^l×Δ^m     
    F(k_+)×∫^[m]∈ A_m^l×Δ^m
       =  F(k_+)× |A^l|   F(k_+)× |A|^l .

These homeomorphisms are natural as (k_+,l_+) vary in ×^.
So taking coends over  gives a homeomorphism

    ∫^k_+∈∫^[m]∈ F(k_+)× A_m^k×Δ^m     ∫^k_+∈ F(k_+)× |A|^k =  F(|A|) .

The `Fubini theorem' for iterated coends 
(the dual to <cit.>)
says that these two ways of iteratively taking coends are canonically isomorphic;
the isomorphism is in fact the map κ of the proposition.




subjectGamma-space@-space!equivariant|(  
Now we add group actions to the discussion of -spaces.subjectGamma-G-space@-G-space|see-space, equivariant
By a -G-space, for a topological group G,
we simply mean a reduced functor from  to the category G of G-spaces.
The previous proposition reduces the study of prolonged -G-spaces
on the realizations of G-simplicial sets to the study of simplicial G-spaces
of the form F∘ A.
A sufficient condition for the realization of a simplicial G-space 
to be homotopically meaningful is Reedy cofibrancy. This explains why we look
for a practical condition to ensure that simplicial G-spaces of the form F∘ A
are Reedy cofibrant; the concept of `G-cofibrancy' introduced in
Definition <ref> below does the job.




We let (n) denote the power set of {1,…,n}, 
i.e., the set of subsets.
We also write (n) for the associated poset category, i.e., with object set (n)
and exactly one morphism U→ T whenever U⊆ T.
Given a -space F, we obtain a functor from the
poset category (n) to based spaces by sending a subset U
to F(U_+), with the maps F(U_+)→ F(T_+) induced by the inclusions.
We obtain a latching mapsubjectlatching map!of a -space

    l_n :  _U⊊{1,…,n}  F(U_+) →  F(n_+)  .

The symmetric group Σ_n acts on the target of the latching map 
by functoriality of F. The symmetric group also acts on the source of l_n,
by letting σ∈Σ_n send F(U_+) to F(σ(U)_+) via the map

    F( (σ· -)_+)  :  F(U_+) → F(σ(U)_+) .

The latching map is equivariant for these two Σ_n-actions.

We will often consider a -G-space for a topological group G.
Then G also acts on source and target of the latching map l_n, which
is thus (G×Σ_n)-equivariant.



Now we observe that the latching map
for a -space F is also a latching map 
for a certain simplicial space, whence the terminology and notation.
We recall that the `simplicial circle' 𝐒^1:^→symbol𝐒^1 - simplicial circlesubjectsimplicial circle  
is given on objects by 𝐒^1_n  =  n_+,
with face maps d_i:n_+→(n-1)_+ given by

    d_i(j)  = 
    j-1     for i <j, and
       j     for i ≥ j and j n,
       0     for i=j=n,

and degeneracy maps s_i:n_+→(n+1)_+ given by

    s_i(j)  = 
    j+1     for i  <j, and
       j     for i≥ j.

The simplicial set 𝐒^1 is isomorphic to Δ[1]/∂Δ[1],
and its realization is homeomorphic to a circle, whence the name.
  


 
Let F be a -space and n≥ 0.


  * There is a homeomorphism

    L_n(F∘𝐒^1)    _U⊊{1,…,n}   F(U_+)

whose composite with the latching map (<ref>) 
is the latching map l_n:L_n(F∘𝐒^1)→ F(𝐒^1_n)=F(n_+)
of the simplicial space F∘𝐒^1.

  * The latching map l_n (<ref>) is a closed embedding.  

  * Suppose that F is a -G-space for a topological group G.
Then the canonical map

    _U⊊{1,…,n}  ( F(U_+))^G → ( _U⊊{1,…,n}  F(U_+))^G

is a homeomorphism.



(i) We recall that (n) is the category with objects the weakly 
monotone surjections σ:[n]→ [k];
a morphism from σ to σ'
is a morphism α:[k]→ [k'] in  with α∘σ=σ'.
Moreover (n)_∘ is the full subcategory with all objects except 
the identity of [n].

As we recalled in Remark <ref>,
the category (n)^ is isomorphic to the
poset category (n).
Indeed, an isomorphism is given on objects by

    κ(σ:[n]→ [k])  =  { i∈{1,…, n} :  σ(i) > σ(i-1)} .

In the other direction, a subset U⊂{1,…,n} 
is taken to  the monotone surjection σ_U:[n]→[|U|]
defined by

    σ_U(i) =  |U∩{1,…,i}| .

The structure map σ^*:𝐒^1_k→𝐒^1_n=n_+
is injective with image

    σ^*(𝐒^1_k)  =  κ(σ)∪{0} .

This means that the maps σ^*:𝐒^1_k=k_+→σ^*(k_+)=κ(σ)_+ 
define a natural isomorphism
between the two composites in the square of functors:

    @C=2mm@R=2mm(n)^[ddd]_u[rrr]^-κ_-         (n)[ddd]^U↦ U_+
             
       @=>[ur]^σ^*      
    ^[rrr]_-𝐒^1

So as σ varies over the objects of (n)_∘,
the homeomorphisms

    (F∘𝐒^1∘ u)(σ)  =  F(k_+) []F(σ^*) 
    F(κ(σ)_+) →_U⊊{1,…,n}   F(U_+)

assemble into the desired isomorphism from L_n(F∘𝐒^1).

(ii) Part (i) shows that the latching map (<ref>) 
is an instance of the latching map of a simplicial compactly generated space.
Claim (ii) is then a special case of 
Proposition <ref> (iii).

(iii) We contemplate the commutative square:

    L_n( (F∘𝐒^1)^G)[d]_[r]
        (L_n(F∘𝐒^1))^G[d]^
    _U⊊{1,…,n}  ( F(U_+))^G[r]    ( _U⊊{1,…,n}  F(U_+))^G

The left vertical map is the homeomorphism of part (i)
for the non-equivariant -space (F∘𝐒^1)^G.
The right vertical map is the effect on G-fixed points of the homeomorphism of part (i).
The upper horizontal map is a homeomorphism by 
Proposition <ref> (iv),
for the simplicial G-space F∘𝐒^1.
So the lower horizontal map is a homeomorphism.



Let G be a compact Lie group. A -G-space F is G-cofibrant
if for every n≥ 1 the latching map

    l_n :  _U⊊{1,…,n}  F(U_+) →  F(n_+)

is a (G×Σ_n)-cofibration.subjectGamma-space@-space!cofibrant



We discuss a relevant example of a cofibrant -G-space 
in Example <ref>, namely the -G-space 
of pairwise orthogonal, finite-dimensional subspaces of 
a complete complex G-universe. 

[Equivariant -simplicial sets]
We let G be a finite group.
In the literature about equivariant -spaces with values in simplicial sets,
no cofibrancy condition is needed to ensure good homotopical behavior.
One way to explain this is to observe that the geometric realization
of a -G-simplicial set E:→ Gsset 
is automatically G-cofibrant.
Indeed, applying Proposition <ref> (ii) 
to the -set in any given simplicial dimension
shows that the latching morphism
 
    l_n :  _U⊊{1,…,n}  E(U_+) →  E(n_+)  ,

taken internal to simplicial sets, is dimensionwise injective.
Geometric realization commutes with colimits and takes injective morphisms
of (G×Σ_n)-simplicial sets to relative (G×Σ_n)-CW-inclusions, 
which are in particular
(G×Σ_n)-cofibrations. So the map |l_n| 
is a (G×Σ_n)-cofibration without any hypotheses on E.  

Our notion of `G-cofibrant' should not be confused with 
cofibrancy in the strict model structure that Bousfield and Friedlander
introduce for non-equi­variant -simplicial sets 
in <cit.>,
and generalized to -G-simplicial sets 
by Ostermayr <cit.>. 
Being `strictly cofibrant' includes the condition that the symmetric group
Σ_n acts freely on the complement of the image of the latching map l_n;
for our purposes, no such freeness is necessary.


As we shall now see, the notion of cofibrancy for equivariant -spaces
is stable under passage to closed subgroups and fixed points by normal subgroups.
If F is a -G-space and H a closed normal subgroup of G,
we obtain a -G/H-space F^H by taking H-fixed points objectwise.
In other words, F^H is the composite functor

    F   G (-)^H (G/H) .




Let H be a closed subgroup of a compact Lie group G 
and F a G-cofibrant -G-space.


  * The underlying -H-space of F is H-cofibrant.

  * If H is normal, then
the -G/H-space F^H is (G/H)-cofibrant.



Part (i) is clear because restriction from G×Σ_n to H×Σ_n
preserves colimits, and it preserves equivariant cofibrations 
by Proposition <ref> (i).

(ii) 
The n-th latching map for F^H factors as the composite

    _U⊊{1,…,n}  F^H(U_+) → ( _U⊊{1,…,n}  F(U_+))^H   (l_n)^H  F(n_+)^H

of the canonical map and the restriction of the latching map for F 
to H-fixed points.
The first map is an isomorphism of (G/H×Σ_n)-spaces 
by Proposition <ref> (iii). 
Since l_n is a (G×Σ_n)-cofibration, the
second map (l_n)^H is a (G/H×Σ_n)-cofibration by 
Proposition <ref>.  




 
Let G be a compact Lie group and F a G-cofibrant -G-space.
Let K be a finite group and T a finite K-set.
Let ⊂(T) be a K-invariant set
of subsets of T that is closed under passage to subsets.
Then the canonical map

    _A∈  F(A_+) →  F(T_+)

is a (G× K)-cofibration.


We prove the following more general statement.
We let ⊂⊂(T) be two K-invariant sets 
that are both closed under passage to subsets.
We show that then the canonical morphism

    _A∈  F(A_+) → _A∈  F(A_+)

is a (G× K)-cofibration. The claim is the special case
=(T), which has a terminal object T.

We start with the special case where 
=∪{ k· B :  k∈ K} 
for some subset B of T that does not belong to .
Since  is closed under taking subsets, 
every proper subset of B then belongs to , and hence to .
We let L≤ K be the stabilizer group of B, i.e., the subgroup
of those k∈ K that map B to itself.
Then the square

    @C=15mm 
    K×_L _U⊊ B  F(U_+) [d][r]^-K×_L l_B    
    K×_L F(B_+)[d]
    _A∈  F(A_+)[r]    _A∈  F(A_+)

is a  pushout of (G× K)-spaces. 
The latching map

    l_B  :  _U⊊ B  F(U_+) → F(B_+)

is a (G×Σ_B)-cofibration by the hypothesis that F is G-cofibrant. 
The L-action on B specifies a homomorphism L→Σ_B, 
so the latching map l_B is a (G× L)-cofibration by
Proposition <ref> (i). 
Hence the upper horizontal map in the square is a 
(G× K)-cofibration by Proposition <ref> (ii).
Since equivariant cofibrations are stable under cobase change,
the lower horizontal morphism is a (G× K)-cofibration.  

In the general case we choose a chain of intermediate K-invariant subsets

    =  _0  ⊂ _1  ⊂ … ⊂_n  =

such that each _i is closed under taking subsets and _i has exactly one
K-orbit more than _i-1. 
The claim then holds for each pair (_i,_i-1).
Since (G× K)-cofibrations are stable under composition, 
this proves the general case.





  Let G be a finite group, F a G-cofibrant -G-space,
  and A a simplicial finite based G-set.
  
  
  * For all m,n≥ 0 the `double latching map'
    
    L_m ( F∘(A n_+)) ∪__V⊊{1,…,n} L_m( F∘(A V_+))   (_V⊊{1,…,n} F(A_m V_+)) 
       →  F(A_m n_+)

    is a (G×Σ_n)-cofibration.
  
  * The shifted -G-space F_|A| is G-cofibrant.
  
  * The simplicial G-space F∘ A is Reedy G-cofibrant.
  
  * For every subgroup H of G the simplicial space (F∘ A)^H 
is Reedy cofibrant.
  


(i) 
The double latching space is the colimit of the functor

    ( (m)^×(n) )^∘ (σ:[m]→[k],V)↦ A_k V_+    F   G_*  ;

here ( (m)^×(n) )^∘ is the `punctured' poset category with 
objects those pairs (σ:[m]→[k],V) 
that are not both maximal, i.e., such that k<m or V is a proper subset
of {1,…,n}.
As we explained in Remark <ref>,
the category (m)^ is isomorphic to the poset category (m),
with U⊂{1,…,m} corresponding to σ_U:[m]→[|U|]
defined as 

    σ_U(i) =  |U∩{1,…,i}| .

Under this isomorphism of categories, the first functor above
becomes isomorphic to the functor

    ( (m)×(n) )^∘ (U,V)↦σ_U^*(A_k) V_+   .

We rewrite the functor (<ref>).
We set S=A_m{∗}, identify S_+ with A_m, and set

    I_U  =  Im(σ_U^*: A_|U|→ A_m){∗} ,

which is a G-invariant subset of S.
We let  denote the subposet of
(S×{1,…,n}) consisting of all subsets that are
contained in I_U×{1,…,n} 
for some (U,V)∈ ((m)×(n))^∘.
Then the functor (<ref>) factors as the composite

    ( (m)×(n) )^∘  φ   incl (S×{1,…,n})  (-)_+   ,

where φ(U,V)=I_U× V.
Altogether this exhibits the double latching space
as the colimit of the composite 

    ( (m)×(n) )^∘  φ    (-)_+    F   G_*  .


We claim that the poset map φ is final, 
i.e., for every subset B∈ the comma category B↓φ 
is non-empty and connected.
The comma category B↓φ is non-empty by the very definition of .
Now we let (U,V) and (U',V') be elements of the poset 
( (m)×(n) )^∘ such that

    B ⊆ I_U× V      and    
     B ⊆ I_U'× V'  .

We let σ=σ_U:[m]→[k] and σ'=σ_U':[m]→[k'] 
be the corresponding monotone surjections.
There is then a unique pushout in the category :

    [m][r]^-σ'[d]_σ    [k'][d]^α'
    
    [k] [r]_-α    [l]

The morphisms α and α' are again surjective.
We set τ=α∘σ=α'∘σ' and claim that

    τ^*(A_l) =  σ^*(A_k) ∩ (σ')^*(A_k')  .

This is essentially the content of <cit.>,
slightly reformulated; 
we reproduce the argument for the convenience of the reader.
Since τ^*(A_l)=σ^*(α^*(A_l)), the set τ^*(A_l) 
is contained in σ^*(A_k),
and similarly for (σ')^*(A_k').
Conversely, we let a∈ A_m be a simplex such that
a=σ^*(x)=(σ')^*(y) for some x∈ A_k and y∈ A_k'.
We write x=β^*(z) and y=(β')^*(z̅)
for surjective homomorphisms β:[k]→[p],β':[k']→[p'] 
and non-degenerate simplices z and z̅. Then

    (βσ)^*(z) =   a  =   (β'σ')^*(z̅) .

By the `Eilenberg-Zilber lemma' (<cit.>, 
see also <cit.>),
the representation of a simplex as a degeneracy of a non-degenerate
element is unique, so p=p', z=z̅ 
and βσ=β'σ'.
Since the square (<ref>) is a pushout, 
there is a unique morphism λ:[l]→[p] 
such that λα=β and λα'=β'.
So

    a  =  (βσ)^*(z) 
     =  (λασ)^*(z) 
     =  (λτ)^*(z) 
     =  τ^*(λ^*(z)) ∈ τ^*(A_l) .

This completes the proof of the relation (<ref>).

We define

    T =  { i∈{1,…, m} :  τ(i) > τ(i-1)} ,

so that τ^*(A_l{∗})=I_T. We now observe that

    B ⊆ (I_U× V) ∩ (I_U'× V')  =  τ^*(A_l{∗})× (V∩ V')  =  I_T× (V∩ V')  .

We have thus found an element (T,V∩ V') 
in the poset ( (m)×(n) )^∘ that is less than or equal
to both (U,V) and (U,V'), 
and such that B⊆φ(T,V∩ V').
So the comma category  B↓φ is connected.
Since the functor φ is final, we can conclude that
the canonical (G×Σ_n)-equivariant map

    _ (U,V)∈ ( (m)×(n) )^∘  
    F( (I_U× V)_+) → _B∈   F(B_+)

is a homeomorphism.
On the other hand, the set  is closed under passage to subsets
and (G×Σ_n)-invariant inside (S×{1,…,n}).
So we can apply Proposition <ref> 
with K=G×Σ_n and T=S×{1,…,n}.
We conclude that the canonical map

    _B∈  F(B_+) →  F( (S×{1,…,n})_+)  F(A_m n_+)

is a (G× G×Σ_n)-cofibration.
When we restrict to the diagonal G-action,
the same map is a (G×Σ_n)-cofibration by 
Proposition <ref> (i). 
Combining these two facts shows that the 
double latching morphism (<ref>) is a (G×Σ_n)-cofibration. 

(ii) Part (i) says that the morphism of simplicial (G×Σ_n)-spaces 

    _V⊊{1,…,n} F∘ (A V_+)   → 
    F∘ (A n_+)

is a Reedy (G×Σ_n)-cofibration.
Geometric realization is a left Quillen functor, 
so it takes Reedy (G×Σ_n)-cofibrations
of simplicial (G×Σ_n)-spaces to (G×Σ_n)-cofibrations.
So the map 

    | _V⊊{1,…,n} F∘ (A V_+) |  → 
    |F∘ (A n_+)|

is a (G×Σ_n)-cofibration.
Realization commutes with colimits, so the source is a colimit, 
over proper subsets of {1,…,n}, of the spaces |F∘(A V_+)|. 
The Fubini isomorphism of Proposition <ref>
identifies the G-space |F∘ (A V_+) | 
with F(|A V_+|) F(|A| V_+)=F_|A|(V_+).  
This proves the claim.

Part (iii) is the special case of part (i) for n=1.

(iv) The restriction functor from G-spaces to H-spaces
preserves colimits, and hence latching objects, 
and takes G-cofibrations to H-cofibrations
by Proposition <ref> (i); so the latching map
l_n:L_n(F∘ A)→ F(A_n) is an H-cofibration by part (iii).
The H-fixed point map

    (l_n)^H  : ( L_n(F∘ A))^H → (F(A_n))^H

is then a non-equivariant cofibration by 
Proposition <ref>.
Taking H-fixed points commutes with taking latching objects, 
by Proposition <ref> (iv).
So the n-th latching map for the simplicial space (F∘ A)^H
is a cofibration.



The following proposition provides a way to reduce certain questions
about -G-spaces for compact Lie groups to the special case
of finite groups.

 
  Let G be a connected compact Lie group and F a -G-space.
  Then for every based G-space K the map
    
    (F^G)(K^G) → (F(K))^G

    induced by the fixed point inclusions F^G→ F and K^G→ K is a homeomorphism.


  Since F(n_+)^G is closed inside F(n_+) and K^G is a closed subset of K, 
  Proposition <ref> (ii)
  shows that the inclusions induce a closed embedding (F^G)(K^G)→ F(K).
  The image of this map is contained in F(K)^G, 
  so it only remains to show that every G-fixed point of F(K)
  is the image of a point in (F^G)(K^G).
  We consider a point of F(K) represented by
  a tuple (x; k_1,…,k_n) in F(n_+)× K^n.
  We assume that the number n has been chosen minimally, so that
  x is non-degenerate and the entries k_i are pairwise distinct 
  and different from the basepoint of K.
  If the point  [x; k_1,…,k_n] of F(K) is G-fixed, 
  then for every group element g
  the tuple (g x; g k_1,…,g k_n) is equivalent to the original tuple.
  By Proposition <ref> (iii)
  there is a unique permutation σ(g)∈Σ_n such that
  
    (g x;  g k_1,…,g k_n) =  
      (F(σ(g)^-1)(x);  k_σ(g)(1),…,k_σ(g)(n) ) .

  The map σ:G→Σ_n is a homomorphism,
  and it is continuous since G acts continuously on K.
  Since G is connected, the homomorphism must be trivial, i.e.,
  σ(g)=1 for all g∈ G. Thus the points x 
  and k_1,…,k_n are all G-fixed.



For us, the main purpose of equivariant -spaces is to 
construct equivariant spectra by evaluation on spheres.
In more detail, we let G be a compact Lie group
and F a -G-space.
We define an orthogonal G-spectrum F() by
  
    F()(V)  =  F(S^V)   .

The structure map σ_V,W:S^V F()(W)→ F()(V⊕ W)
is the assembly map for K=S^V and L=S^W,
followed by the effect of F on the canonical homeomorphism 
S^V S^W S^V⊕ W.
The O(V)-action on F()(V) is via the action on S^V and the continuous
functoriality of F.
If V is a G-representation, then F(S^V) has the diagonal G-action,
from its actions on F and on V.subjectGamma-space@-space!evaluation on spheres



  Let G be a compact Lie group and F a G-cofibrant -G-space.
  
  
  * For every G-representation V,
  the fixed point space F(S^V)^G is ((V^G)-1)-connected.

  * The orthogonal G-spectrum F() is equivariantly connective.
  


(i) 
  We start with the special case where G is a finite group.
  We set d=(V^G).
  A choice of linear isometry V^G^d induces a G-equivariant homeomorphism
  between F(S^V) and F(S^d S^V^⊥), 
  where V^⊥ is the orthogonal complement of V^G inside V. 
  So F(S^V)^G is homeomorphic to F(S^d S^V^⊥)^G.

  The sphere S^d is homeomorphic to the geometric realization of the 
  based simplicial set Δ[d]/∂Δ[d], the represented simplicial set
  Δ[d]=(-,[d]) modulo its simplicial boundary. 
  The representation sphere S^V^⊥ is homeomorphic to the geometric realization 
  of a finite based G-simplicial set. Indeed, S^V^⊥ 
  admits the structure of a smooth G-manifold; 
  Illman's triangulation theorem <cit.> then provides
  a G-equivariant triangulation. Passing to the barycentric subdivision 
  provides a finite G-simplicial set B that realizes to S^V^⊥.
  Altogether, S^V is G-homeomorphic to the geometric realization
  of the G-simplicial set A=(Δ[d]/∂Δ[d]) B.

  Proposition <ref>
  provides a G-equivariant homeomorphism
  
    F(S^V)  F(|A|)   |F∘ A |  .

  Taking G-fixed points commutes with realization 
  by Proposition <ref> (iv),
  so F(S^V)^G is homeomorphic to the realization of the simplicial space
  (F∘ A)^G that takes [m]∈^
  to the space F(A_m)^G.
  In dimensions below d, the simplicial set Δ[d]/∂Δ[d],
  and hence also the simplicial set A=(Δ[d]/∂Δ[d]) B,
  consists only of the base point. So for m<d, the space F( A_m)^G is a single point.
  Moreover, the simplicial space (F∘ A)^G 
  is Reedy cofibrant by Proposition <ref> (iv).
  So its realization is (d-1)-connected by Proposition <ref> (ii).
  Altogether this establishes the claim that the space F(S^V)^G 
  is (d-1)-connected.

  Now we treat the case of a general compact Lie group.
  We let G^∘ denote the connected component of the identity and
  G̅= G/G^∘ the finite group of components of G.
  The -G̅-space F^G^∘ is G̅-cofibrant 
  by Proposition <ref> (ii).
  Proposition <ref>  provides a homeomorphism
  
    F(S^V)^G  =  ( F(S^V)^G^∘)^G̅  ( (F^G^∘)(S^V^G^∘))^G̅ .

  Because (V^G^∘)^G̅=V^G, the right hand side is ((V^G)-1)-connected 
  by the special case above, for the finite group G̅, 
  the -G̅-space F^G^∘
  and the G̅-representation V^G^∘.  

  (ii)
  We let H be any closed subgroup of G and show that the group π_-k^H(F())
  is trivial for all k≥ 1.
  We let V be any H-representation, K a closed subgroup of H, 
  and we set d_K=(V^K).
  The underlying -K-space of F is K-cofibrant
  by Proposition <ref> (i).
  So F(S^^k⊕ V)^K is (k+d_K-1)-connected by part (i).

  On the other hand, the cellular dimension of S^V at K,
  in the sense of <cit.>,
  is at most d_K. 
  Because k is positive, the cellular dimension of S^V at K does not exceed
  the connectivity of F(S^^k⊕ V)^K.
  So every based continuous H-map S^V→ F(S^^k⊕ V) is equivariantly
  null-homotopic by <cit.>,
  and the set [S^V,F(S^^k⊕ V)]^H has only one element.
  Passage to the colimit over V∈ s(_H) proves the claim.  



We can also show that prolongation of G-cofibrant -G-spaces
is homotopical in the -space variable, as long as
we evaluate on finite based G-CW-complexes.


Let G be a compact Lie group. 
A morphism ψ:E→ F of -G-spaces
is a strict equivalencesubjectstrict equivalence!of -G-spaces
if for every n≥ 1 the map of (G×Σ_n)-spaces 
ψ(n_+):E(n_+)→ F(n_+) is an
(G;Σ_n)-weak equivalence, where (G;Σ_n)
is the family of graph subgroups of G×Σ_n.subjectgraph subgroup



 
A strict equivalence ψ:E→ F of -G-spaces
also satisfies the following condition:
for every closed subgroup H of G and every finite H-set S, the map
ψ(S_+):E(S_+)→ F(S_+) is an H-weak equivalence with respect
to the diagonal H-actions. 
Indeed, if S has n elements, we may suppose that S={1,…,n}
with H-action specified by a continuous homomorphism
ρ:H→Σ_n.
We let K be a closed subgroup of H and Γ
the graph of ρ|_K:K→Σ_n.
The map
ψ(S_+)^Γ:E(S_+)^Γ:→ F(S_+)^Γ
is a weak equivalence by the hypothesis on ψ.
Moreover,

    F(S_+)^K  =   F(S_+)^Γ ,

where the fixed points on the left hand side are with respect to the diagonal K-action.
This proves the claim.




The following result will allow us to extend equivariant homotopical properties
from the class of finite G-simplicial sets to the class of finite G-CW-complexes.
The non-equivariant case is treated for example in <cit.>.
I am sure that also the equivariant result is well-known; 
however, I don't know a reference, so I provide a proof.



Let G be a finite group.


  * Let X be a finite G-CW-complex, A a finite G-simplicial set
and f:|A|→ X a continuous G-map. 
Then there is a finite G-simplicial set B, a monomorphism of
G-simplicial sets i:A→ B and a G-homotopy equivalence 
h:|B|→ X such that

    h∘ |i|  =  f  :  |A| → X .


  * Every finite based G-CW-complex is based G-homotopy equivalent
to the realization of a finite based G-simplicial set.



(i)
We let c Y= (Y×[0,1]) / (Y×{1}) denote 
the unreduced cone of a space Y.
We start with a very special case, namely when there is 
a pushout square of G-spaces

    G/H×|∂Δ[k]|[r]^-incl[d]_α   
    G/H× c|∂Δ[k]| [d]
    
    |A|[r]_-f     X

for some subgroup H of G and some k≥ 0;
in other words, we suppose that X is obtained from |A| 
by attaching one equivariant cell. 

The continuous map

    α(e H,-) :  |∂Δ[k]| → |A|

lands in the H-fixed points.
Since fixed points commute with realization
(Proposition <ref> (iv)), 
we may view it as a continuous map to |A^H|.
Now we use a `simplicial approximation', by which we mean the following data: 

  
  * a finite simplicial set D, 
  
  * a morphism of simplicial sets
    Φ : D→ A^H, and
  
  * a homotopy equivalence φ :|D|→ |∂Δ[k]|
    such that α(e H,-)∘φ is homotopic to the realization of Φ.

For example, we can use  D=Sd^m(∂Δ[k]) for a suitably
large m≥ 0, where Sd is Kan's subdivision functor <cit.>; 
the remaining data is then provided 
by Lemma 7.5 and Theorem 8.5 of <cit.>.
Alternatively, we can take D=SD^m(∂Δ[k]) for a suitably
large m≥ 0, where SD is the `double simplicial subdivision'
of <cit.>; the other data is provided by <cit.>.

 
We let Φ̃:G/H× D→ A be the G-equivariant extension of Φ,
i.e., Φ̃_n(g H,x)=g·Φ(x) for g∈ G and x∈ D_n.
The geometric realization of Φ̃ 
is then G-equivariantly homotopic to the composite

    G/H× |D|  G/H×φ 
    G/H× |∂Δ[k]|
      α  |A| .

We choose a G-equivariant homotopy

    K  :  G/H×|D|×[0,1]  → |A|

from |Φ̃| to the map α∘(G/H×φ).
We absorb the homotopy into the mapping cylinder of the map 
|Φ̃|:G/H×|D|→|A|, and obtain a commutative diagram of G-spaces:

    |A|∪_ |Φ̃| (G/H×|D|×[0,1])
     [d]__|A|∪ K     G/H× |D| [r]^-incl[l]_-(-,1)[d]^G/H×φ    
    G/H× c| D|[d]^G/H×φ̅
    
      |A|     G/H× |∂Δ[k]| [r]_-incl[l]^-α    G/H× c|∂Δ[k]|

All vertical maps in the diagram are G-homotopy equivalences,
and the right horizontal maps are G-cofibrations.
The gluing lemma (Proposition <ref>)
then shows that the induced map on pushouts

    |A|∪_|Φ̃| (G/H× |D|× [0,1])    ∪_G/H× |D| (G/H× c|D|)
       → 
    |A|∪_α (G/H× c|∂Δ[k]|)   X

is a G-weak equivalence;
moreover, its restriction to |A| is the original map f:|A|→ X.
The source of (<ref>)
 is homeomorphic to the unreduced mapping cone of the map
|Φ̃|.
Mapping cones can also be formed in the category of G-simplicial sets,
so the source of (<ref>) is equivariantly homeomorphic
to the realization of the unreduced mapping cone 
of Φ̃:G/H× D→ A.
This simplicial mapping cone is the desired G-simplicial set B.

The rest of the argument is now straightforward.
Induction over the number of relative equivariant cells
proves the case where (X,|A|) is a finite relative G-CW-pair 
and where f:|A|→ X is the inclusion.
In the general case, the filtration of A by simplicial skeleta induces a filtration
on the geometric realization that gives |A| the structure of a 
finite G-CW-complex.
If the map f is cellular for this structure,
then the mapping cylinder Z=|A|× [0,1]∪_f X
inherits a finite G-CW-structure in which |A| is an equivariant subcomplex. 
So the previous case provides a finite G-simplicial set B, a monomorphism of
G-simplicial sets i:A→ B and a G-homotopy equivalence 
h:|B|→ Z such that the composite

    h∘ |i|  :  |A| → Z

is the `front inclusion' (-,0):|A|→ Z.
The projection Z→ X is a G-equivariant homotopy equivalence
and f=p∘(-,0):|A|→ X.
So the triple (B,i,p∘ h) has the desired properties.
If the map f is arbitrary, we use the equivariant cellular approximation theorem 
(see for example <cit.>, <cit.>
or <cit.>)
and the equivariant homotopy extension property of the G-map |i|:|A|→|B|
to reduce to the cellular case.

(ii) We let X be a finite based G-CW-complex. 
We apply part (i) to the terminal simplicial G-set A=∗
and the inclusion of the basepoint |∗|→ X; 
part (i) provides a finite based G-simplicial set B 
and a based G-homotopy equivalence h:|B|→ X.


Now we have all tools ready to show that prolongation
preserves strict equivalences between cofibrant -G-spaces,
at least on finite G-CW-complexes.


Let G be a finite group and ψ:E→ F a strict equivalence between   
cofibrant -G-spaces.
Then for every finite based G-CW-complex X,
the map ψ(X)^G:E(X)^G→ F(X)^G is a weak equivalence.


We start with the special case where X=|B| is 
the geometric realization of a finite based G-simplicial set B.
In this situation, the morphism 
of simplicial spaces (ψ∘ B)^G:(E∘ B)^G→ (F∘ B)^G
is levelwise a weak equivalence by Remark <ref>.
Moreover, source and target are Reedy cofibrant 
by Proposition <ref> (iv).
Geometric realization takes levelwise weak equivalences
between Reedy cofibrant simplicial spaces to weak equivalences
(Proposition <ref>).
This proves the special case.

In the general case we choose a finite based G-simplicial set B 
and a based G-homotopy equivalence h:|B|→ X,
as provided by Proposition <ref> (ii).
Prolonged -spaces are continuous functors,
so they preserve equivariant based  homotopies.
In the commutative square

    @C=12mm 
    E(|B|)^G [r]^-ψ(|B|)^G[d]_E(h)^G   
    F(|B|)^G [d]^F(h)^G
    
    E(X)^G [r]_-ψ(X)^G    F(X)^G

both vertical maps are thus homotopy equivalences.
The upper map is a weak equivalence by the previous paragraph,
hence so is the lower map.




Now we move on to the analysis of special and very special -G-spaces.
We will eventually assume that the -G-space is G-cofibrant,
in order to have homotopical control over its prolongation.
The final aim is to show that for finite G and very special (respectively special) F,
the evaluation on spheres F() is a G-Ω-spectrum
(respectively `positive' G-Ω-spectrum),
see Theorem <ref> respectively
Theorem <ref> below.


If F is any -space and S a finite set, then we define the map

    P_S  :   F(S_+) → (S, F(1_+))

by P_S(x)(s)= F(p_s)(x), where p_s:S_+→ 1_+ sends s to 1 and all other elements
of S_+ to the basepoint. If a group G acts on F and S,
then the map P_S is G-equivariant for the diagonal G-action on the source
and the conjugation action on the target.


Let G be a compact Lie group. 
A -G-space F is specialsubjectGamma-space@-space!special
if for every closed subgroup H of G and every finite H-set S the map

    (P_S)^H  :  F(S_+)^H → ^H(S,F(1_+))

is a weak equivalence. 



We showed in Proposition <ref>
that the notion of cofibrancy for equivariant -spaces
is stable under passage to closed subgroups and fixed points by normal subgroups.
Now we show the analogous statement for specialness.



Let H be a closed subgroup of a compact Lie group G 
and F a special -G-space.


  * The underlying -H-space of F is special.

  * If H is normal, then the -G/H-space F^H is special.



Part (i) is clear by definition.
For part (ii) we consider a closed subgroup of G/H, which must be of
the form Δ/H for a closed subgroup Δ of G with H≤Δ.
We let S be a finite Δ/H-set,
which we can also view as a finite Δ-set by restriction along
the projection Δ→Δ/H.
The claim then follows from the hypothesis that F is special and
the relations

    ( F^H(S) )^Δ/H =  F(S)^Δ   and  ^Δ/H(S, F^H(1_+)) =  ^Δ(S, F(1_+))  .





The functor obtained by prolonging a -G-space 
comes with `Wirthmüller type' maps, defined as follows.
We let H be a closed subgroup of finite index in G and Z a based H-space.
We recall that l_Z:G⋉_H Z→ Z denotes the H-equivariant
projection to the wedge summand indexed by the preferred coset e H, i.e.,

    l_Z[g,z] = 
      g· z     if g∈ H, and
     ∗   else.

The H-equivariant map F(l_Z): F(G⋉_H Z) → F(Z)
is then adjoint to a G-equivariant Wirthmüller mapsubjectWirthmüller map!of an equivariant -space

    ω_Z :  F(G⋉_H Z)  → ^H(G,F(Z))  .


We recall some equivalent characterizations of special -G-spaces.
The equivalence of the first two conditions 
was first noted in <cit.>. 


  Let G be a compact Lie group and F a -G-space. 
  Then the following conditions are equivalent.
  
  
  * The -G-space F is special. 
  
  * For every n≥ 1 the map
    
    P_n  :  F(n_+) → ({1,…,n},F(1_+))  =  F(1_+)^n

    is an (G;Σ_n)-equivalence, where
    (G;Σ_n) is the family of graph subgroups.subjectgraph subgroup
  
  * For all pairs of finite G-sets T and U, the map
    
    (F(p_T),F(p_U)) :  F((T⨿ U)_+) → F(T_+)× F(U_+)

    is a G-weak equivalence, and for every finite index subgroup H of G
    and every finite H-set S, the Wirthmüller map
    ω_S_+: F( (G×_H S)_+)→^H(G,F(S_+))
    is a G-weak equivalence. 
  


  (i)⟺ (ii)
  We let H be a closed subgroup of G.
  Every finite H-set is isomorphic to α^*{1,…,n}
  for some continuous homomorphism α:H→Σ_n.
  If Γ≤ G×Σ_n denotes the graph of α,
  then F(n_+)^Γ=F(α^*{1,…,n})^H and
  (F(1_+)^n)^Γ= ^H(α^*{1,…,n},F(1_+)).
  So the map (P_n)^Γ is a weak equivalence if and only if
  the map (P_α^*{1,…,n})^H  is a weak equivalence.

  (i)⟹ (iii) 
  In the commutative square
  
    F( (T⨿ U)_+)[r]^-(F(p_T),F(p_U))[d]_P_T⨿ U^≃    
        F(T_+)× F(U_+) [d]^p_T× p_U_≃
    ( T⨿ U,F(1_+))[r]_-   (T,F(1_+))×(U,F(1_+))

  both vertical maps are G-weak equivalences because F is special,
  and the lower horizontal map is a homeomorphism.
  So the upper horizontal map is a G-weak equivalence.

  Now we suppose that H has finite index in G and we let S be a finite H-set.
  Then the Wirthmüller map participates in the commutative diagram:

    F( (G×_H S)_+)[r]^-ω_S_+[d]_P_G×_H S   ^H(G,F(S_+)) [d]^^H(G,P_S)
    ( G×_H S, F(1_+)) [r]_-   ^H(G,(S,F(1_+)))

The left and right vertical maps are G-weak equivalences by specialness, 
applied to the G-set G×_H S respectively the H-set S.
The lower horizontal homeomorphism sends f:G×_H S→ F(1_+)
to the adjoint H-map f̂:G→(S,F(1_+)) 
with f̂(g)(s)=f[g,s].
Altogether this shows that the Wirthmüller map ω_S_+
is a G-weak equivalence.

(iii)⟹ (i) 
We start by showing that for every finite G-set S the map
P_S : F(S_+)→(S,F(1_+)) is a G-weak equivalence. 
If S=T⨿ U is the disjoint union of two finite G-sets T and U,
we contemplate the commutative square (<ref>).
The upper horizontal map is a G-weak equivalence by hypothesis.
The lower horizontal map is an equivariant homeomorphism.
So if the claim holds for the G-sets T and U, then it holds for
their disjoint union. This reduces the claim to the case of transitive G-sets.
For every finite index subgroup H of G, there is a commutative square
of G-maps

    F( (G/H)_+)[r]^-P_G/H[d]_   (G/H,F(1_+)) [d]^
    
    F(G⋉_H 1_+)[r]_-ω_1_+   ^H(G,F(1_+))

where the right vertical map is adjoint to the H-map

    (G/H,F(1_+))  → F(1_+) ,   f  ⟼ f(e H)  .

The two vertical maps are homeomorphisms.
The lower horizontal map is the Wirthmüller map 
for the H-set {1};
this map is a G-weak equivalence by hypothesis.
So the map P_S is a G-weak equivalence for transitive G-sets;
this completes the proof of the first claim.

Now we let H be a finite index subgroup of G and S a finite H-set.
The H-maps [1,-]:S_+→ (G×_H S)_+ and l_S:(G×_H S)_+→ S_+
express S_+ as an H-equivariant retract of the based G-set (G×_H S)_+.
The middle vertical map in the commutative diagram

    @C=12mm 
    F( S_+)^H[d]_ (P_S)^H[r]^-F([1,-])^H    
    F( (G×_H S)_+)^H[r]^-F(l_S)^H[d]_(P_G×_H S)^H   
    F(S_+)^H[d]^ (P_S)^H
    ^H(S,F(1_+)) [r]_-(l_S)^*   ^H(G×_H S,F(1_+)) [r]_-[1,-]^*   ^H(S,F(1_+))

is a weak equivalence by the previous paragraph,
and both horizontal composites are the identity maps.
Since weak equivalences are closed under retracts, the map (P_S)^H
is a weak equivalence. 
This proves that the -G-space F is special.








  Let G be a compact Lie group and F a G-cofibrant special -G-space. 
  
  
  * For all finite based G-CW-complexes X and Y the map
    
    (F(p_X),F(p_Y)) :  F(X∨ Y) → F(X)× F(Y)

    is a G-weak equivalence.
  
  * For every finite index subgroup H of G and every
    finite based H-CW-complex Z, the Wirthmüller mapsubjectWirthmüller map!of an equivariant -space
    
    ω_Z :  F(G⋉_H Z)  → ^H(G,F(Z))

    is a G-weak equivalence. 
  
  * For every finite based G-CW-complex X,
    the shifted -G-space F_X is special.
  


(i)
We wish to show that for every closed subgroup H of G the map
    
    (F(p_X)^H,F(p_Y)^H) :  F(X∨ Y)^H → F(X)^H× F(Y)^H

is a weak equivalence.
The underlying -H-space of F is H-cofibrant 
by Proposition <ref> (i)
and special by Proposition <ref> (i).
Moreover, the underlying H-spaces of X and Y are H-homotopy
equivalent to finite H-CW-complexes, 
by <cit.>.
So we may assume without loss of generality that H=G.

We start with the special case where the group G is finite and
X=|A| and Y=|B| are realizations of two finite based G-simplicial sets A and B.
For every n≥ 0, the map

    (F(p_A_n)^G,F(p_B_n)^G) :  F(A_n∨ B_n)^G → F(A_n)^G× F(B_n)^G

is a weak equivalence by Proposition <ref> (iii).
The map of simplicial spaces

    ( (F∘ p_A)^G, (F∘ p_B)^G) :  
     ( F∘(A∨ B))^G → (F∘ A)^G× (F∘ B)^G

is thus levelwise a weak equivalence.
The simplicial spaces 
( F∘(A∨ B))^G, (F∘ A)^G and (F∘ B)^G are Reedy cofibrant by 
Proposition <ref> (iv).
The product simplicial space (F∘ A)^G× (F∘ B)^G is then 
Reedy cofibrant by Proposition <ref>.
So the map induced on geometric realizations 

    | (F∘(A∨ B))^G| → |(F∘ A)^G× (F∘ B)^G|

is a weak equivalence by Proposition <ref>.
Realization commutes with fixed points 
(Proposition <ref> (iv))
and products (Proposition <ref> (ii)), so the map 

    ( |F ∘ p_A|, |F∘ p_B|)^G :  
     | F∘(A∨ B)|^G  → ( |F∘ A|× |F∘ B| )^G

is a weak equivalence.
The Fubini isomorphism |F∘ A|^G F(|A|)^G 
of Proposition <ref> then translates this into the claim 
for the geometric realizations of A and B.

Now we continue to assume that G is finite, but X and Y
are arbitrary finite based G-CW-complexes.
We choose based G-homotopy equivalences 
|A|≃ X and |B|≃ Y for suitable finite based G-simplicial sets
A and B as provided by Proposition <ref> (ii).
Prolonged -G-spaces preserve equivariant homotopy equivalences,
so the general case follows from the special case.

It remains to treat the general case of a compact Lie group.
We let G^∘ be the identity component of G
and we write G̅=G/G^∘ for the finite group of path components.
The -G̅-space F^G^∘ is G̅-cofibrant 
by Proposition <ref> (ii)
and special by Proposition <ref> (ii).
Since X and Y are finite G-CW-complexes, X^G^∘ 
and Y^G^∘ are finite G̅-CW-complexes.
So the map

    ( F^G^∘(p_X^G^∘),F^G^∘(p_Y^G^∘)) :  
     F^G^∘(X^G^∘∨ Y^G^∘) → 
    F^G^∘(X^G^∘)× F^G∘(Y^G^∘)

induces a weak equivalence on G̅-fixed points by the previous paragraph.
Moreover,

    (F^G^∘(X^G^∘))^G̅  (F(X)^G^∘)^G̅ =    F(X)^G

by Proposition <ref>, and similarly for the
G-fixed points of F(Y) and F(X∨ Y).
This completes the proof.

(ii) 
We start by showing that the G-fixed point map

    (ω_Z)^G :  F(G⋉_H Z)^G  → ( ^H(G,F(Z)) )^G

is a weak equivalence. 
Evaluation at 1∈ G identifies the target of (ω_Z)^G
with F(Z)^H, so we may show that the composite map

    F(G⋉_H Z)^G  incl F(G⋉_H Z)^H  F(l_Z)^H F(Z)^H

is a weak equivalence.

We start with the special where G is finite and 
Z=|B| is the geometric realization 
of a finite based H-simplicial set B;
then G⋉_H Z is G-homeomorphic to the geometric realization 
of the finite based G-simplicial set G⋉_H B.
By Proposition <ref>
and <ref> (ii), 
the space F(|G⋉_H B|)^G is homeomorphic to | (F∘ (G⋉_H B))^G|. 
Similarly,  F(|B|)^H is homeomorphic to | (F∘ B)^H|.
Under these homeomorphisms, the map (<ref>) 
becomes the geometric realization
of the morphism of simplicial spaces

    (F∘ l_B)^H∘incl :  (F∘ (G⋉_H B))^G  → (F∘ B)^H

whose value at the object [n] of ^ is the map
F(l_B_n)^H∘incl.
The morphism of simplicial spaces (<ref>)
is a weak equivalence in every simplicial dimension
by Proposition <ref> (iii);
moreover, source and target are Reedy cofibrant 
by Proposition <ref> (iv).
As a levelwise weak equivalence between Reedy cofibrant simplicial spaces,
the morphism (<ref>) induces
a weak equivalence on geometric realizations,
by Proposition <ref>. 
This completes the proof that the map (<ref>)
is a weak equivalence in the special case Z=|B|.

Now we treat the case where G is finite, but Z is any finite based H-CW-complex.
Proposition <ref> (ii)
provides a finite based H-simplicial set B 
and a based H-homotopy equivalence h:|B|→ Z.
The map G⋉_H h:G⋉_H |B|→ G⋉_H Z
is then a based G-homotopy equivalence.
In the commutative square

    @C=18mm 
    F(G⋉_H |B|)^G[r]^-F(l_|B|)^H∘incl[d]_F(G⋉_H h)^G   
    F(|B|)^H[d]^F(h)^H
    
    F(G⋉_H Z)^G [r]_-F(l_Z)^H∘incl    F(Z)^H

both vertical maps are then homotopy equivalences.
The upper map is a weak equivalence by the previous paragraph,
hence so is the lower map.
This completes the proof that the map (<ref>)
is a weak equivalence in the special case when G is finite.

It remains to treat the case of a general compact Lie group.
We let G^∘ be the identity component of G
and we write G̅=G/G^∘ for the finite group of path components.
The -G̅-space F^G^∘ is G̅-cofibrant 
by Proposition <ref> (ii)
and special by Proposition <ref> (ii).
Since H has finite index in G, the identity components 
of H and G are the same, i.e., H^∘=G^∘.
Moreover, H̅=H/H^∘ is a subgroup of G̅.


Since Z is a finite H-CW-complex, X^H^∘ is a finite H̅-CW-complex.
So the map

    ( F^G^∘(l_Z^H^∘) )^H̅∘incl :  
    ( F^G^∘( G̅⋉_H̅ Z^H^∘) )^G̅ → 
    ( F^G^∘(Z^H^∘))^H̅

is a weak equivalence by the previous paragraph.
Proposition <ref> lets us rewrite 
source and target of this map as

    ( F^G^∘( G̅⋉_H̅ Z^H^∘))^G̅     ( F^G^∘( (G⋉_H Z)^G^∘))^G̅
         ( F( G⋉_H Z)^G^∘)^G̅ =   F(G⋉_H Z)^G

respectively 

    ( F^G^∘(Z^H^∘))^H̅    =  
    ( F^H^∘(Z^H^∘))^H̅   (F(Z)^H^∘)^H̅  F(Z)^H .

Under these homeomorphisms, the map 
( F^G^∘(l_Z^H^∘) )^H̅∘incl becomes 
the map (<ref>).
This completes the proof that the map (<ref>)
is a weak equivalence for all compact Lie groups G and all finite based
H-CW-complexes Z.
Hence the map (ω_Z)^G is a weak equivalence. 


Now we let K be any closed subgroup of G. Since H has finite index in G,
the set G/H is finite, and there is a finite set {γ_i}_i=1,…,n
of K-H-double coset representatives.
These determine a K-equivariant wedge decomposition

    D :  ⋁_i=1,…,n K⋉_K∩^γ_i H c_γ_i^*(Z)
      G⋉_H Z ,   
    [k,z]_i  ⟼ [kγ_i, z] .

We obtain a commutative square:

    @C=12mm 
     F(G⋉_H Z)^K[r]^-(ω_Z)^K[d]_≃     ( ^H(G,F(Z)) )^K [d]^
    ∏_i=1^n F(K⋉_K∩^γ_i H c_γ_i^*(Z))^K
    [r]_-∏ω_c_γ_i^*(Z)   ∏_i=1^n F(Z)^K^γ_i∩ H

The right vertical map is evaluation at the chosen coset representatives,
and it is a homeomorphism.
The left map is F( D^-1)^K 
followed by F(-)^K applied to the projections to the wedge summands.
The latter is a weak equivalence by the additivity property of part (i),
applied to the underlying -K-space of F,
and the spaces K⋉_K∩^γ_i H c_γ_i^*(Z);
there is a slight caveat, namely that the underlying (K^γ_i∩ H)-space
of Z need not admit the structure of an equivariant CW-complex. However, it is always
(K^γ_i∩ H)-homotopy equivalent to a finite 
(K^γ_i∩ H)-CW-complex, by <cit.>;
this is enough to run the argument because prolonged
-K-spaces preserve equivariant homotopy equivalences.
The lower horizontal map in the above square is a weak equivalence
by the previous paragraph,
applied to the underlying -K-space of F,
and the subgroups K∩^γ_i H.
So we can conclude that the map (ω_Z)^K is a weak equivalence. 
Since K was an arbitrary closed subgroup of G,
this completes the proof that the Wirthmüller
map ω_Z is a G-weak equivalence.

(iii)
We verify the criterion of Proposition <ref> (iii)
for the shifted -G-space F_X.
Given two finite G-sets T and U, part (i) shows that 
the map

    (F_X(p_T),F_X(p_U)) :  F_X((T⨿ U)_+)    =  F(X (T⨿ U)_+) 
    → F(X T_+)   × F(X U_+)  =  F_X(T_+)× F_X(U_+)

is a G-weak equivalence.
For every finite index subgroup H of G and every finite H-set S,
the effect of the shearing isomorphism

    X (G×_H S)_+   G⋉_H(X S_+)

identifies the Wirthmüller map

    ω_S_+ :  F_X((G×_H S)_+)    =  F(X (G×_H S)_+)
        → ^H(G,F(X S_+)) =  ^H(G,F_X(S_+))

for the shifted -G-space and the H-space S_+
with the Wirthmüller map

    ω_X S_+ :  F(G⋉_H (X S_+))  → ^H(G,F(X S_+))

of the original -G-space and the H-space X S_+.
The latter map is a G-weak equivalence by part (ii),
hence so is the former.
Proposition <ref> now applies and shows that
the shifted -G-space F_X is special.




We still have to recall the notion of a `very special' -G-space.
We let G be a compact Lie group and F a special -G-space.
We let p_1,p_2:2_+→ 1_+ denote the two projections.
The map

    (F(p_1),F(p_2)) :  F(2_+) → F(1_+)× F(1_+)

is a G-weak equivalence by specialness.
We let ∇:2_+→ 1_+ denote the fold map.
We obtain a diagram of set maps

    π_0( F(1_+)^G)×π_0(F(1_+)^G) [](π_0(F(p_1)^G),π_0(F(p_2)^G)) π_0( F( 2_+)^G) π_0(F(∇)^G)π_0( F(1_+)^G)

the left of which is bijective.
So the map

    π_0(F(∇)^G)∘ (π_0(F(p_1)^G),    π_0(F(p_2)^G))^-1 : 
       π_0( F(1_+)^G)×π_0(F(1_+)^G) → π_0( F(1_+)^G)

is a binary operation on the set π_0(F(1_+)^G).

If τ:2_+→ 2_+ is the involution that interchanges 1 and 2,
then composition with τ 
fixes ∇ and interchanges p_1 and p_2;
this implies that the operation + is commutative.
Contemplating the different ways to fold and project from the based set 3_+
leads to the proof that the operation is also associative,
and hence an abelian monoid structure on the set π_0(F(1_+)^G).

For every closed subgroup H of G, 
the underlying -H-space is again special. 
So the same argument provides an abelian monoid structure on π_0(F(1_+)^H).



Let G be a compact Lie group. 
A -G-space F is very specialsubjectGamma-space@-space!very specialsubjectGamma-space@-space!equivariant  
if it is special and for every closed subgroup H of G 
the abelian monoid π_0(F(1_+)^H) is an abelian group.




We let G be a compact Lie group and F a special -G-space. 
Then the construction of the abelian monoid structure
on π_0(F(1_+)^H) generalizes as follows. 
We let H be a closed subgroup of G and S a finite G-set.
We let p_1,p_2:(S⨿ S)_+→ S_+ denote the two projections.
The map

    (π_0(F(p_1)^H),π_0(F(p_2)^H)) :  π_0(F((S⨿ S)_+)^H)
     → π_0(F(S_+)^H)×π_0(F(S_+)^H)

is bijective by Proposition <ref> (iii);
inverting this map and composing with the effect of 
the fold map ∇:(S⨿ S)_+→ S_+ on π_0(F(-)^H) 
yields a binary operation on the set π_0( F(S_+)^H).
The same arguments as in the special case S={1} show that this operation
defines an abelian monoid structure on π_0(F(S_+)^H), which is moreover natural
for G-maps in S. 
If K is a closed subgroup of H, then the inclusion F(S_+)^H⊂ F(S_+)^K 
induces  a monoid homomorphism π_0(F(S_+)^H)→π_0(F(S_+)^K). 

We claim that for very special F, all the abelian monoids π_0(F(S_+)^H)
are in fact groups.
To see this, we choose a set of representatives s_1,…,s_n
for the H-orbits of S, and we let K_i be the stabilizer group
of s_i inside H. We consider the composite

    F(S_+)^H (P_S)^H ^H(S,F(1_+))  ∏_i=1^n F(1_+)^K_i ,

where the first map is a weak equivalence by specialness, 
and the homeomorphism is evaluation at the elements s_1,…,s_n.
The induced bijection

    π_0(F(S_+)^H)  ∏_i=1^n  π_0( F(1_+)^K_i)

is a homomorphism of abelian monoids by the previous paragraph. 
The target is an abelian group since F is very special.
So the abelian monoid π_0(F(S_+)^H) is also an abelian group.


 
We recall how very special -G-spaces
give rise to simplicial spaces that satisfy the π_*-Kan condition
(see <cit.> or Definition <ref>).
We let G be a finite group and F:→ G_* 
a very special -G-space. 
We let A be a simplicial finite based G-set. 
Then for every subgroup H of G we obtain a simplicial space
(F∘ A)^H whose space of n-simplices is F(A_n)^H
(where are usual, F(A_n) has the diagonal H-action by restriction of
the G-actions on F and A_n). We claim that this simplicial space
(F∘ A)^H satisfies the π_*-Kan condition.
To see this, we borrow the argument from Bousfield 
and Friedlander <cit.>:
since F is special, the space F(A_n)^H comes with
a homotopy-commutative H-space structure, so each of its path components is
a simple space. Moreover, the map of simplicial sets
β:π_t((F∘ A)^H)_free→π_0( (F∘ A)^H)
discussed in B.3 of <cit.>
is underlying a surjective morphism of simplicial groups,
and is thus a Kan fibration by <cit.>.
So <cit.> proves that
(F∘ A)^H satisfies the π_*-Kan condition.




Let H be a closed subgroup of a compact Lie group G 
and F a very special -G-space.


  * The underlying -H-space of F is very special.

  * If H is normal, then the -G/H-space F^H is very special.



Part (i) is clear by definition.
For part (ii) we recall from Proposition <ref> (ii)
that the -G/H-space F^H is special.
Any closed subgroup of G/H is of the form Δ/H 
for a closed subgroup Δ of G with H≤Δ.
Then (F^H(1_+))^Δ/H=F(1_+)^Δ.
Since the monoid π_0(F(1_+)^Δ) is a group by hypothesis,
we conclude that the monoid π_0(F^H(1_+)^Δ/H) is a group.


The following Theorem <ref>
is essentially a reformulation of the equivariant delooping results 
of Segal <cit.> respectively Shimakawa <cit.>;
however, I am not aware that the result has been established in this form. 
The difference is that we formulate the result for the
prolongation (i.e., categorical Kan extension), whereas Segal and Shimakawa
work with a bar construction (also known as a homotopy coend
or homotopy Kan extension) instead.
We also give a partial extension of the machinery to compact Lie groups. 

A minor technical difference is that in Shimakawa's 
paper <cit.>, an equivariant -space is assumed to take
values in non-degenerately based G-spaces with the homotopy type
of a based G-CW-complex; similarly, Segal <cit.> 
is implicitly assuming that the values of equivariant -spaces are G-ANRs.
Our proof includes the observation that the key arguments work
in the larger category of compactly generated G-spaces and G-weak equivalences,
a long as we restrict to G-cofibrant -G-spaces.

Our proof verifies that the prolonged cofibrant -G-space 
satisfies the delooping criterion of Blumberg <cit.>.
An advantage of this approach is that Blumberg does not quote
Segal or Shimakawa; 
rather, he works in the category of compactly generated G-spaces throughout,
and adapts the relevant arguments to this context.

The paper <cit.> by May, Merling and Osorno also contains a modern
perspective on the work of Segal and Shimakawa, 
and there is some overlap with our present discussion.
In fact, our exposition of equivariant -spaces
partially arose from discussions with Mona Merling and Peter May,
to whom I am indepted for valuable feedback. 
In <cit.> the method of producing an equivariant spectrum 
by evaluating a -space
on spheres is referred to as the conceptual Segal machine.
In various respects, <cit.> goes further than we do here;
for example, it contains detailed comparisons of prolongation 
(the `conceptual Segal machine'),
bar construction (the `homotopical Segal machine')
and the operadic approach to equivariant delooping (via a `generalized Segal machine').
Moreover, for special -G-spaces,
the value at S^1 is identified as an equivariant group completion;
this is particularly relevant since many -G-spaces
that arise in practice are special, but not very special.



  Let G be a compact Lie group and F a G-cofibrant 
  very special -G-space.
  
  
  * The prolonged functor F takes G-homotopy pushout squares 
    of finite based G-CW-complexes to G-homotopy pullback squares.
  
  * For every finite based G-CW-complex X and every G-representation V
    on which the identity component G^∘ acts trivially, the adjoint assembly map
    
    α̃ :  F(X) → _*(S^V,F(X S^V))

    is a G-weak equivalence.
    
  * If G is finite, then the orthogonal G-spectrum F()
      is a G-Ω-spectrum.subjectG-Omega-spectrum@G-Ω-spectrum
  


(i) We start with the special case where G is finite, and we consider a monomorphism
of finite based G-simplicial sets i:A→ B. We show that then the sequence

    F(|A|)  F(|i|) F(|B|)  F(|q|) F(|B/A|)

is a G-homotopy fiber sequence, where q:B→ B/A is the projection.
We adapt an argument that Bousfield and Friedlander
use for non-equivariant -spaces of simplicial sets 
in <cit.>. 
The space F(|A|) is homeomorphic to the geometric realization
of the simplicial space F∘ A, by Proposition <ref>.
Moreover, for H≤ G, taking H-fixed points commutes with geometric realization by 
Proposition <ref> (iv),
so F(|A|)^H is homeomorphic to the geometric realization of the
simplicial space (F∘ A)^H, sending [n] to F(A_n)^H.
The same applies to B and B/A,
so we may show that the sequence

    |(F∘ A)^H| |(F∘ i)^H||(F∘ B)^H| |(F∘ q)^H| |(F∘ (B/A))^H|

is a homotopy fiber sequence.
We let r:B_n→ A_n be the retraction to i_n that sends the complement of i_n(A_n)
to the basepoint.
For fixed n≥ 0 the lower row in the commutative diagram

    @C=15mm 
    F(A_n)^H[r]^-F(i_n)^H@=[d]    
    F(B_n)^H[r]^-F(q_n)^H[d]^(F(r)^H,F(q_n)^H)_≃    
    F(B_n/A_n)^H  @=[d]
     
    F(A_n)^H [r]     F(A_n)^H× F( B_n/A_n)^H[r]_-proj     F( B_n/A_n )^H

is a homotopy fiber sequence.
The middle vertical map is a weak equivalence by 
Proposition <ref> (iii).
So the upper row in the diagram is a homotopy fiber sequence.

Because F is G-cofibrant, 
the three simplicial spaces (F∘ A)^H,  (F∘ B)^H
and (F∘ (B/A))^H are Reedy cofibrant by
Proposition <ref> (iv).
So we are considering a sequence of simplicial spaces that is a 
homotopy fiber sequence in every simplicial dimension.
Moreover, the simplicial spaces (F∘ B)^H
and (F∘ (B/A))^H satisfy the π_*-Kan condition 
by Example <ref>.
The morphism of simplicial sets 
π_0((F∘ q)^H):π_0( (F∘ B)^H)→π_0( (F∘ (B/A))^H)
underlies a homomorphism of simplicial abelian groups
by Remark <ref>;
this morphism is split surjective in every simplicial dimension.
Every surjective homomorphism of simplicial groups is a Kan fibration,
see for example <cit.> or <cit.>.
So the sequence (F∘ A)^H→ (F∘ B)^H→ (F∘ (B/A))^H
satisfies the hypotheses of Bousfield and Friedlander's 
Theorem <ref>; 
we conclude that the sequence (<ref>)
is a homotopy fiber sequence.
This completes the proof that the sequence (<ref>)
is a G-homotopy fiber sequence.

Now we consider a G-cofibration j:X→ Y 
between finite based G-CW-complexes, where G is still a finite group.
We claim that then the sequence

    F(X)  F(j) F(Y)  F(p) F(Y/X)

is a G-homotopy fiber sequence.
Proposition <ref>
provides a finite G-simplicial set A 
and a G-homotopy equivalence k:|A|→ X.
Another application of Proposition <ref>
provides a finite G-simplicial set B, 
a monomorphism of G-simplicial sets i:A→ B and a 
G-homotopy equivalence h:|B|→ Y such that h∘|i|=j∘ k:|A|→ Y.
The induced map on cokernels

    h/k  :  |B / A|   |B| /  |A|  →  Y/X

is then a G-weak equivalence by the gluing lemma 
(Proposition <ref>).
Source and target of h/k are cofibrant as G-spaces,
so this G-weak equivalence is even a G-homotopy equivalence.
The upper row in the commutative diagram of G-spaces

    @C=18mm 
    F(|A|)[r]^-F(|i|)[d]_F(k)^≃   
    F(|B|)[r]^-F(|q|)[d]_F(h)^≃    
    F(|B/A|)  [d]^F(h/k)_≃
     
    F(X) [r]_-F(j)    F(Y)[r]_-F(p)     F(Y/X)

is a G-homotopy fiber sequence by the previous paragraph.
The prolongation of every -G-space preserves
based G-homotopy equivalences; so all vertical maps are G-homotopy equivalences.
Hence the lower row in the diagram is a G-homotopy fiber sequence.

Now we let G be an arbitrary compact Lie group, and j:X→ Y 
a G-cofibration between finite based G-CW-complexes.
We claim that then the sequence (<ref>)
is a G-homotopy fiber sequence.
We let H be a closed subgroup of G with identity component H^∘ 
and group of path components H̅=H/H^∘.
The -H̅-space F^H^∘ is H̅-cofibrant 
by Proposition <ref>
and very special by Proposition <ref>.
The map j^H^∘:X^H^∘→ Y^H^∘ is an H̅-cofibration
by Propositions <ref> (i) 
and <ref>.
Since j is in particular a closed embedding, 
the canonical map X^H^∘/Y^H^∘→ (X/Y)^H^∘ is a homeomorphism
by Proposition <ref> (i).
So by the previous paragraph, the sequence

    F^H^∘(X^H^∘)  F^H^∘(j^H^∘) F^H^∘(Y^H^∘)  F^H^∘(p^H^∘) F^H^∘( (Y/X)^H^∘)

is an H̅-homotopy fiber sequence.
We may thus take H̅-fixed points and obtain a non-equivariant
homotopy fiber sequence.
Proposition <ref> provides a homeomorphism
(F^H^∘(X^H^∘))^H̅ (F(X)^H^∘)^H̅=F(X)^H,
and similarly for the other two terms.
So we conclude that the sequence

    F(X)^H  F(j)^H F(Y)^H  F(p)^H F(Y/X)^H

is a non-equivariant homotopy fiber sequence.
This completes the proof that the sequence (<ref>)
is a G-homotopy fiber sequence.

Now we treat a general homotopy cocartesian square 
of finite based G-CW-complexes:

    X[r]^-j[d]_k     Y[d]^h 
     Z [r]_i     P

By replacing Y and P by the reduced mapping cylinders
of j respectively i, 
we may assume that the horizontal maps are G-cofibrations.
Because the square is homotopy cocartesian, the induced map
h/k:Y/X→ P/Z is a G-weak equivalence, and hence a G-homotopy equivalence.
We obtain a commutative diagram of based G-spaces:

    @C=12mm 
    F(X)[r]^-F(j)[d]_F(k)    F(Y) [d]_F(h)[r]    
    F(Y/X)  [d]^F(h/k)_≃
     
    F(Z) [r]_-F(i)    F(P)[r]      F(P/Z)

Because F preserves equivariant homotopy equivalences, 
the right vertical map  F(h/k) is a G-homotopy equivalence.
Both rows are G-homotopy fiber sequences by the previous paragraph;
so the left square is G-homotopy cartesian.
This completes the proof that the prolonged functor 
takes G-homotopy cocartesian squares to G-homotopy cartesian squares.

(ii)
As usual we start with the special case where G is finite.
Restricting the prolonged functor F to finite G-CW-complexes
gives a 𝒲_G-space in the sense of <cit.>.
This 𝒲_G-space satisfies conditions (1) and (2') 
of <cit.>, by part (i) of this theorem
respectively Proposition <ref> (ii).
So <cit.> applies to the prolonged functor,
and gives the desired conclusion.

Now we let G be a general compact Lie group.
We want to show that for every closed subgroup H of G the map
    
    α̃^H :  F(X)^H → ^H_*(S^V,F(X S^V))

    is a weak equivalence.
The underlying -H-space of F is H-cofibrant 
by Proposition <ref> (i)
and very special by Proposition <ref> (i).
Moreover, the underlying H-space of X is H-homotopy
equivalent to a finite H-CW-complex, 
by <cit.>.
Finally, the identity component of H is contained in G^∘,
so it acts trivially on V.
So we may assume without loss of generality that H=G.

As usual we write G̅=G/G^∘ for the finite group of path components.
The -G̅-space F^G^∘ is G̅-cofibrant 
by Proposition <ref> (ii)
and very special by Proposition <ref> (ii).
Since X is a finite G-CW-complex, X^G^∘ is a finite G̅-CW-complex.
Since G^∘ acts trivially on V, we can view V as a representation
of the factor group G̅. 
The previous paragraph shows that the map

    α̃ :  F^G^∘(X^G^∘) → _*(S^V,F^G^∘(X^G^∘ S^V))

induces a weak equivalence on G̅-fixed points.
Proposition <ref> provides a homeomorphism

    ( F^G^∘( X^G^∘) )^G̅  (F(X)^G^∘)^G̅ =   F(X)^G ;

similarly,

    ^G̅_*(S^V,F^G^∘(X^G^∘ S^V))      ^G̅_*(S^V, F(X S^V)^G^∘) 
       =  ^G_*(S^V, F(X S^V) )  .

Under these identifications, the map α̃^G̅
corresponds to the G-fixed points of the adjoint assembly map
for the -G-space F. This completes the proof.

Part (iii) is simply the special case of (ii) with X=S^W
for some G-re­presen­tation W. Finite groups have a trivial identity
component, which must necessarily act trivially on V.




Now we let F be a cofibrant -G-space that is special
(but not necessarily very special). Then Theorem <ref>
need not hold for F, as the adjoint assembly map F(1_+)→Ω F(S^1)
need not be a weak equivalence.
However, there is a standard argument to deduce 
from Theorem <ref> that the adjoint assembly map
is still a G-weak equivalence on suspensions;
this argument uses the shift of the given -G-space by S^1.
Since we work with prolongations throughout, there is a new ingredient, 
not already present in the classical references, 
namely that the S^1-shift of a G-cofibrant -G-space 
is again G-cofibrant.



  Let G be a compact Lie group and F a G-cofibrant, special  -G-space. 
  Then for every pair of G-representations V and W 
  such that W^G 0  and G^∘ acts trivially on V,
  the adjoint assembly map
  
    α̃ :  F(S^W) → _*(S^V,F(S^V⊕ W))

  is a G-weak equivalence.


One more time we start with the special case when G is finite.
We consider the shifted -G-space F_S^1=F∘(S^1 -),
which is G-cofibrant by Proposition <ref> (ii)
and special by Proposition <ref> (iii).
Moreover, for every subgroup H of G the space

    ( F_S^1(1_+))^H  F(S^1)^H

is path connected by Proposition <ref> (i).
So the abelian monoid π_0( F_S^1(1_+)^H) has only one element,
and is thus an abelian group. Hence F_S^1 is very special.

So Theorem <ref> applies to the 
G-cofibrant and very special -G-space F_S^1.
Since W has a non-zero G-fixed point, there is 
a G-representation U and a G-equivariant linear isometry
φ:⊕ U W.
We let ϕ:F_S^1(L)→ F(S^1 L) denote the natural homeomorphism established in
Proposition <ref>. 
In the commutative diagram

    @C=15mm@R=7mm 
    F_S^1(S^U)[d]^_ϕ[r]^-α̃   _*(S^V,F_S^1(S^V⊕ U))[d]^_*(S^V,ϕ)_
    
    F(S^1 S^U) [r]^-α̃[d]_F(S^φ)^   _*(S^V,F(S^1 S^V⊕ U)) [d]_^_*(S^V,F(S^V⊕φ))
    
    F(S^W)  [r]_-α̃   _*(S^V,F(S^V⊕ W))

all vertical maps are then homeomorphisms,
and the upper horizontal map is a G-weak equivalence
by Theorem <ref>. 
So the lower horizontal map is a G-weak equivalence.
This completes the proof in the case of finite groups.

The case of general compact Lie groups can be reduced to the case of finite groups
by taking fixed points with respect to the identity component G^∘,
and using the residual action of the finite group G/G^∘;
the argument is almost literally the same as in the last paragraph of the proof of 
Theorem <ref> (ii), and we omit it.




The previous Theorems <ref> 
and <ref>
indicate why equivariant -spaces are usually considered
for finite groups only – 
this is where the equivariant delooping machine reveals its full power.
We want to briefly explain why the spectrum F() cannot be 
a G-Ω-spectrum when G has positive dimension,
except in degenerate cases.
The reason is that when H is a closed subgroup of strictly smaller
dimension than G, the `dimension shifting Wirthmüller map'
cannot be a G-weak equivalence, except in degenerate cases.
Given a based H-space Z, the H-equivariant collapse map

    l_Z  :  G⋉_H Z → Z S^L

was introduced in (<ref>); here L=T_e H(G/H)
is the tangent H-representation.
The dimension shifting Wirthmüller map 

    ω_Z   :  F(G⋉_H Z) → ^H(G,F(Z S^L) )

is the adjoint of the H-map F(l_Z):F(G⋉_H Z)→ F(Z S^L).

Blumberg's recognition theorem <cit.> 
says that his conditions (1) and (2') are not only sufficient, 
but also necessary for F() to be a G-Ω-spectrum.
If F is G-cofibrant and very special, then the prolonged functor
always satisfies conditions (1), by Theorem <ref> (i).
But condition (2') requires that the dimension shifting Wirthmüller map ω_Z
is a G-weak equivalence; 
Proposition <ref> (ii) shows 
that this is true for compact Lie groups, 
provided H has finite index in G (in which case L=0).
However, the map ω_Z is typically not a G-weak equivalence
when H has strictly smaller dimension than G. To illustrate this
we take the extreme case H=e and Z=1_+,
in which case G⋉_H Z is simply G_+.
If the identity component G^∘ of G is non-trivial, then
F(G_+)^G^∘ is a single point by Proposition <ref>.
On the other hand, evaluation at a set of representatives
of the path components of G is a homeomorphism

    ( (G,F(S^L)))^G^0  (G/G^∘,F(S^L))   F(S^L)^m  ,

where m=[G:G^∘].
So if the Wirthmüller map ω_1_+ is a G-weak equivalence,
then the space F(S^L) is non-equivariantly weakly contractible.
Since F() is a connective non-equivariant Ω-spectrum,
it must be altogether stably trivial.

subjectGamma-space@-space!equivariant|)  





CHAPTER: ENRICHED FUNCTOR CATEGORIES




In this final appendix we review definitions, properties and constructions
involving categories of enriched functors.
The general setup consists of:


  * a complete and cocomplete closed symmetric monoidal category  
(the `base category').
We denote the monoidal product in  by ;

  * a skeletally small -category  (the `index category').

We denote by ^∗ the category of covariant -functors
X:→ from the index category to the base category. 
We are mostly interested in the following special cases of such functor categories:


  * orthogonal spaces 
(where = and =, see Definition <ref>);

  * orthogonal spectra 
(where =_* and =, see Definition <ref>);

  * global functors 
(where = and =, see Definition <ref>).

We discuss the Day convolution product (Definition <ref>) 
that turns the functor category into a symmetric monoidal category, 
compare Theorem <ref>.
Construction <ref>
introduces the skeleton filtration,
which enters into the level model structure on
certain functor categories, see Proposition <ref>. 
For background on symmetric monoidal categories
and enriched category theory we refer the reader to <cit.>. 



[-enrichment of functor categories]
The objects of the functor category ^∗  
are the -functors X:→, and the morphisms
in ^∗ are the -natural transformations between such functors.
The functor category ^∗ is itself enriched in the base category , as follows. 
Given two objects X,Y in ^∗, a -object of -natural transformations   
^∗(X,Y) is defined 
as an enriched end <cit.>, 
i.e., an equalizer in  of two morphisms

    ∏_c  (X(c),Y(c))   @<.4ex>[r]^-α@<-.4ex>[r]_-β     ∏_d,e ((d,e) X(d),Y(e)) .

Here the products run over a set of representatives for the
isomorphism classes of -objects and (-,-) denotes 
the internal function objects in the closed monoidal structure.
The (d,e)-component of the  morphism α is the composite

    ∏_c  (X(c),Y(c))  proj_d(X(d),Y(d))     ((d,e) X(d),Y(e))

where the second morphism is adjoint to 

    (X(d),Y(d))(d,e) X(d)  eval(d,e) Y(d)   Y   Y(e)  .

The (d,e)-component of the morphism β is the composite 

    ∏_c  (X(c),Y(c))  proj_e(X(e),Y(e))     ((d,e) X(d),Y(e))

where the second morphism is adjoint to 

    (X(e),Y(e))(d,e) X(d)   X(X(e),Y(e)) X(e)  eval Y(e)  .

The underlying set of a -object V is 

    u V  =  (I,V) ,

the set of morphisms from the monoidal unit. 
The functor (I,-):→(sets) takes equalizers in  to 
equalizers of sets. When applied to the defining equalizer
for ^∗(X,Y) this shows that 
the set of morphisms ^∗(X,Y) in the functor category ^∗
(i.e., the set of -natural transformations)
can be recovered as the underlying set of the -object ^∗(X,Y),
via a natural bijection

    ^∗(X,Y)  (I,^∗(X,Y)) 
    =  u(^∗(X,Y))  .

The -objects ^∗(X,Y) even assemble into
an entire -category ^∗ of -functors
whose underlying category is ^∗.
We refer to <cit.> for details.

The category ^∗ is also tensored over the base category .
Given a functor Y∈^∗ and an object A of ,
we define a new functor Y A∈^∗
by objectwise product in , i.e., Y A is the composite -functor

    Y    -  A  .

This construction is an action of  on ^∗, which means
preferred natural associativity and unit isomorphisms 

    Y (A B)   (Y A) B      and    
    Y I   Y .

Indeed, the value of the above isomorphisms
at an object d∈ is simply the associativity (respectively unit)
isomorphism in the monoidal structure of , for the triple
of objects (Y(d), A,B). 



[Enriched Yoneda lemma]subjectYoneda lemma!enriched
The Yoneda lemma has enriched versions that we use frequently
to identify morphisms out of representable -functors.  
For every object d∈ the covariant hom functor

    d^* =  (d,-)  :   →

is a -functor, hence an object in the functor category ^∗.
The enriched Yoneda lemma comes in two flavors,  a -valued version
and a set-valued version.
For a -functor Y in ^∗,
the evaluation morphism is the composite

    ^∗(d^∗,Y) proj_d (d^∗(d),Y(d))     =  ((d,d),Y(d)) 
       (_d,Y(d)) (I,Y(d))   Y(d)

where _d:I→(d,d) is the identity of d.
This composite is an isomorphism in , see for example
<cit.>.
Passing to underlying sets produces a bijection

    ^∗(d^∗,Y)    u(Y(d))

between the set of -natural transformations and the underlying set of Y(d).
This is the weak form of the enriched Yoneda lemma, 
compare <cit.>.






Convolution product.subjectconvolution product!of enriched functors
We review a general method, due to B. Day <cit.>, 
for constructing symmetric monoidal structures on certain functor categories.
We now make the additional assumption that the index category 
is a symmetric monoidal -category.
We denote the monoidal product on  by ⊕.

We get the main cases of interest by specializing as follows:


  * For = the category of spaces under cartesian product
and = the topological category of inner product spaces
under orthogonal direct sum, 
^∗= is the category of orthogonal spaces.
This yields the box product of orthogonal spaces.

  * For =_* the category of based spaces under smash product
and = the Thom space category of the orthogonal complement bundles,
under orthogonal direct sum,
^∗= is the category of orthogonal spectra.
This yields the smash product of orthogonal spectra.

  * For = the category of abelian groups under tensor product
and = the pre-additive Burnside category
with the monoidal structure of Theorem <ref>,
^∗= is the category of global functors.
This yields the box product of global functors.




We denote by  the -category whose objects are
pairs of -objects, and with morphism -objects

    ()((d,d'),(e,e')) =  (d,e)(d',e')  .

Given two -functors X,Y:→, the composite

    X Y  :   X Y

is a -functor.


A bimorphism b:(X,Y)→ Zsubjectbimorphism
from a pair of objects (X,Y) of ^∗
to another object Z of ^∗ is a -natural 
transformation b:X Y → Z∘⊕ 
of -functors →.


So a bimorphism  b:(X,Y)→ Z consists of  -morphisms

    b(d,e)  :  X(d)  Y(e)  → Z(d⊕ e)

for all objects d,e of , that form a -natural transformation.
We can then define the box productsubjectbox product 
as a universal recipient of a bimorphism from X and Y.


A box product for objects X and Y of ^∗ 
is a pair (X Y,i) consisting of a ^∗-object X Y
and a universal bimorphism i:(X,Y)→ X Y,
i.e., a bimorphism 
such that for every ^∗-object Z the map

    ^∗(X Y,Z)  → Bimor((X,Y),Z)  ,   
    f⟼ f∘ i

is bijective.subjectuniversal property!of a convolution product 




 
Every pair of objects of ^∗ has a box product.


The universal property of the box product precisely means that  
(X Y,i) is an enriched left Kan extension of the -functor
X Y:→ along the -functor
⊕:→. Such a Kan extension exists because  is cocomplete
and  is skeletally small, see <cit.>.



As we saw in the proof of Proposition <ref>,
the box product is an enriched Kan extension
along the functor ⊕:→. We can make this a little
more explicit. Given X,Y∈^∗, a box product X Y
is a coequalizer of the two morphisms of -functors

    ∐_e,e',d,d'(e⊕ e',-)(d,e)   (d',e') X(d) Y(d') 
       @<.4ex>[r]@<-.4ex>[r]    ∐_d,d'(d⊕ d',-) X(d) Y(d')
     .

The left coproduct is indexed over all quadruples (e,e',d,d') in a set
of representatives of the isomorphism classes of -objects.
The right coproduct is indexed over pairs (d,d') of such representatives.
One of the two morphisms to be coequalized is the coproduct
of the monoidal products of 

    (e⊕ e',-)(d,e)(d',e') → (d⊕ d',-) 
    ,     (φ,τ,τ')  ⟼ φ∘(τ⊕τ')

and the identity on X(d) Y(d').
The other morphism is the coproduct of the monoidal products of the identity on 
(e⊕ e',-)
and the action maps (d,e) X(d)→ X(e)
respectively  (d',e') Y(d')→ Y(e').



[Uniqueness of box products]
The universal property makes box products of two functors unique up to preferred
isomorphism. Indeed, if (X Y,i) and (X' Y,i') 
are two box products, then the universal properties provide unique morphisms
f:X Y→ X' Y and g:X' Y→ X Y that satisfy

    f∘ i  =  i'      respectively     g∘ i' =  i .

Then (g∘ f)∘ i=g∘ i' = i = _X Y∘ i,
so g∘ f=_X Y by the uniqueness part of the universal property. 
Reversing the roles gives f∘ g=_X' Y.



In the following we choose a box product (X Y,i)
for every pair of objects X and Y of ^∗.
The universal bimorphism i is often omitted from the notation, but one should
remember that the pair (X Y,i), and not the object X Y alone,
has a universal property.
While the box products are choices, the rest of
the structure is canonically determined by these.


[Functoriality]
The box product X Y automatically becomes a functor in both variables:
if f:X→ X' and g:Y→ Y' are morphisms in ^∗, 
then the -morphisms

    { X(d)  Y(e)  X'(d)   Y'(e)   
    (X' Y')(d⊕ e) }_d,e

form a bimorphism (X,Y)→ X' Y'. 
So there is a unique morphism f g:X Y→ X' Y'
such that (f g)(d⊕ e)∘ i_d,e=i'_d,e∘ (f(d) g(e))
for all objects d,e of .
The uniqueness part of the universal property implies that 
this is compatible with identities and composition in both variables.


Now that we have constructed a box product functor
: ^∗×^∗→^∗,
we recall that it `is' automatically symmetric monoidal.
Since `symmetric monoidal' is extra data, and not a property,
we are obliged to construct unit, associativity and symmetry isomorphisms.

[Coherence isomorphisms]
It will be convenient to make the -functor 0^∗=(0,-)
represented by the unit object 0 of 
into a strict unit for the box product (as opposed to a unit
up to coherent isomorphisms). So we make the following conventions:


  * (Right unit) We choose X 0^∗=X with universal bimorphism
i:(X,0^∗)→ X given by the maps

    X(d) 0^∗(e) =  X(d)(0,e) X(d) (d⊕-)    X(d)(d⊕ 0,d⊕ e)
      X(d)   (d,d⊕ e)  ∘   X(d⊕ e)  .


  * (Left unit) We choose 0^∗ Y=Y with universal bimorphism
i:(0^∗,Y)→ Y given by the composite

    0^∗(d) Y(e)  =  (0,d) Y(e)  (-⊕ e) Y(e)   (0⊕ e,d⊕ e) Y(e) 
        (e,d⊕ e) Y(e)   ∘  Y(d⊕ e)  .


The unnamed isomorphisms are induced by the unit isomorphisms in ,
and `∘' is the functor structure of X respectively Y.
For X=Y=0^∗ these two bimorphisms (0^∗,0^∗)→ 0^∗
are equal, so there is no ambiguity.

We also obtain the associativity and symmetry isomorphisms
from the universal property of the box products.
For the associativity isomorphism we notice that the family

    { i_d,e⊕ f∘ (X(d) i_e,f)  :  
    X(d)  Y(e)  Z(f)  → (X (Y Z))(d⊕ e⊕ f) }_d,e,f

and the family

    { i_d⊕ e,f∘(i_d,e Z(f))  :   
     X(d)  Y(e)  Z(f)  → ((X Y) Z)(d⊕ e⊕ f)}_d,e,f

both have the universal property of a tri morphism
(whose definition is hopefully clear) out of X, Y and Z.
The uniqueness of representing objects gives a unique isomorphism
of -functors

    α_X,Y,Z :  X (Y Z)    (X Y) Z

such that (α_X,Y,Z)_d⊕ e⊕ f∘ i_d,e⊕ f∘ (X(d) i_e,f)
=i_d⊕ e,f∘ (i_d,e Z(f)). 

The symmetry isomorphism τ_X,Y:X Y→ Y X corresponds to the
bimorphism from (X,Y) to Y X with components

    X(d)   Y(e) Y(e)   X(d)  (Y X)(e⊕ d)
     (Y X)(d⊕ e)  .






The following theorem is due to B. Day. 
In fact, Theorems 3.3 and 4.1 of <cit.> 
together imply the `monoidal'  part, and
Theorem 3.6 of <cit.> deals with the symmetries.



The associativity and symmetry isomorphisms make the box product
into a symmetric monoidal product on the category ^∗ with 
the functor 0^∗ as a strict unit object. 



The box product also commutes with -tensors up to a natural isomorphism
that we now construct.
The family of morphisms

    X(d)  ( Y(e)   A )
      α_X(d),Y(e),A  (X(d)  Y(e))   A  (X Y)(d⊕ e)  A

forms a bimorphism from (X,Y A) to (X Y) A.
So the universal property provides a distinguished morphism

    α_X,Y,A :  X(Y A) → ( X Y) A   .

Since the monoidal structure on  is closed, the functor - A
is a left adjoint, so it preserves coends. So the morphism
α_X,Y,A is an isomorphism.
The two types of associativity isomorphisms 
X (Y A) (X Y) A and 
Y (A B) (Y A) B
are compatible in the sense of a commuting pentagon:

    @C=-5mm       X (Y (A B)) [dll]_Xα_Y,A,B[drr]^α_X,Y,A B
    
    X ((Y A) B) [dr]_α_X,Y A,B             (X Y) (A B)[dl]^α_X Y,A,B
        (X(Y A)) B [rr]_α_X,Y,A B      
    ((X Y) A) B

A minimally more complicated construction that also involves
the symmetry constraint in  to move the object A past the functor Y
provides an isomorphism

    (X A) Y    (X Y) A  .


[Convolution product of representable functors]
The box product of the functors represented by two objects a,b of  
is related by a preferred isomorphism to the functor represented by a⊕ b.
Indeed, the -morphisms

    a^∗(d) b^∗(e)  =  (a,d)(b,e)   ⊕ (a⊕ b,d⊕ e) =  (a⊕ b)^∗(d⊕ e)

form a bimorphism as d and e vary. So there is a unique morphism

    j_a,b :   a^∗ b^∗ → (a⊕ b)^∗

such that j_a,b∘ i=⊕, where i is 
the universal bimorphism from (a^∗,b^∗) to a^* b^*. 
On the other hand, the morphism

    I   I I  _a_b(a,a)(b,b)  =  a^*(a) b^*(b)  i_a,b(a^* b^*)(a⊕ b)

is an element of the underlying set of (a^* b^*)(a⊕ b).
The enriched Yoneda lemma thus provides a unique morphism
of -functors (a⊕ b)^*→ a^* b^*
that evaluates to the element at a⊕ b. 
The universal properties of the convolution product and
of the representable functors show that these two morphisms
are inverse to each other. 



[Internal function objects]
The box product is a closed monoi­dal product
in the sense that for all objects Y and Z of ^∗ the functor

    ^∗(- Y, Z)  :  ^∗ → (sets) ,   
    X ⟼ ^∗(X Y,Z)

is representable.
In this book we don't make any substantial use of the internal function objects,
so we will not elaborate on the construction. The most relevant consequence
for us is that the box product preserves colimits
in both variables because the functors X - and - Y
are left adjoints. This consequence can also be shown 
without reference to function objects by observing that
the original monoidal product in  preserves
colimits in both variables (because the monoidal structure of  is closed),
because X Y can be constructed as an enriched coend,
and because enriched coends commute with colimits.



[Skeleton filtration]
For the rest of this appendix we assume that the index category  
is equipped with a dimension function :ob()→ 
to the natural numbers, satisfying the following conditions for all
objects d,e of :


  * If (e)<(d), then (d,e) 
is an initial object of the base category .

  * If (e)=(d), then d and e are isomorphic.

The two examples that we care about in this book are
the category of orthogonal spaces and the category of orthogonal spectra.
In both cases the dimension function is the dimension as an -vector space.


We denote by _≤ m the full -subcategory of 
spanned by all objects of dimension at most m.
We denote by ^∗_≤ m=(_≤ m)^∗ 
the category of enriched functors from _≤ m
to . The restriction functor 

    ^∗ → ^∗_≤ m ,    
    Y  ⟼ Y^≤ m =  Y|__≤ m

has a left adjoint 

    l_m  :   ^∗_≤ m → ^∗

given by an enriched Kan extension as follows. 
For every k≥ 0 we choose an object 𝐤 of  of dimension k.
The extension l_m(Z) of an enriched functor Z:_≤ m→ 
is a coequalizer of the two morphisms in ^∗:

    @C=8mm∐_0≤ j≤ k≤ m(𝐤,-)(𝐣,𝐤) Z(𝐣) 
     @<-.4ex>[r] @<.4ex>[r]     ∐_0≤ i≤ m(𝐢,-) Z(𝐢)

One morphism arises from the composition morphisms

    (𝐤,-)(𝐣,𝐤)  → (𝐣,-)

and the identity on Z(𝐣);
the other morphism arises from the action morphisms

    (𝐣,𝐤) Z(𝐣)  → Z(𝐤)

and the identity on the represented functor (𝐤,-).
Colimits in the functor category ^∗ are created objectwise, 
so the value l_m(Z)(d) at an object d can be calculated 
by plugging d into the variable slot 
in the coequalizer diagram (<ref>).
Kan extensions along a fully faithful functor do not change
the values on the given subcategory, 
see for example <cit.>. 
For the inclusion _≤ m→ this means that the adjunction unit

    Z  → (l_m Z)^≤ m

is an isomorphism for every functor Z:_≤ m→.




The m-skeleton,subjectskeletonsymbol^m X - m-skeleton of X
for m≥ 0, of an enriched functor Y:→ is the functor

    ^m Y =  l_m(Y^≤ m)  ,

the extension of the restriction of Y to _≤ m.
It comes with a natural morphism i_m:^m Y→ Y, the counit of
the adjunction (l_m,(-)^≤ m).
We set (m)=(𝐦,𝐦), a -monoid under composition.
The m-th latching objectsubjectlatching object!of an enriched functorsymbolL_m Y - m-th latching object of Y
of Y is the (m)-object

    L_m Y  =  (^m-1 Y)(𝐦)  ;

it comes with a natural (m)-equivariant morphism

    ν_m=i_m-1(𝐦) : L_m Y → Y(𝐦)  ,

the m-th latching morphism.



We agree to set ^-1 Y=∅, an initial object of ^∗;
then L_0 Y=∅, an initial -object.
The two morphisms i_m-1:^m-1 Y→ Y and
i_m:^m Y→ Y both restrict to isomorphisms on ^≤ m-1, so there is a 
unique morphism j_m:^m-1 Y→^m Y such that i_m∘ j_m=i_m-1.
The sequence of skeleta stabilizes to Y in a 
very strong sense. For every object d, the morphisms
j_m(d) and i_m(d) are isomorphisms as soon as m > (d).
In particular, Y(d) is a colimit, with respect to the morphisms i_m(d), 
of the sequence of morphisms j_m(d).
Since colimits in the functor category ^∗ are created objectwise,
we deduce that Y is a colimit, 
with respect to the morphisms i_m, of the sequence of morphisms j_m. 






[Latching objects of represented functors]
  Let d be an object of  of dimension m=(d), 
  and A an object of .
  Then the functor (d,-) A
  is `purely m-dimensional' in the following sense.
  The evaluation functor
  
    _d :  ^∗ →

  factors through the category _≤ m^* as the composite
  
    ^∗ → _≤ m^*  _d  .

  So the left adjoint free functor A↦(d,-) A 
  can be chosen as the composite
  of the two individual left adjoints
  
    (d,-) -  =  l_m ∘ (_≤ m(d,-) -)  .

  The object (d, e) A is initial for (e) < m, by hypothesis
  on the dimension function, and hence the 
  latching object L_n ((d,-) A) is initial for n≤ m.
  For n>m the latching morphism 
  ν_n:L_n((d,-)  A) →(d,𝐧) A 
  is an isomorphism. So the skeleton ^n ((d,-) A) is initial for n<m
  and ^n ((d,-) A)=(d,-) A is the entire functor for n ≥ m.




We denote by

    G_m  :  (m) → ^∗ ,    
    A  ⟼ (𝐦,-)_(m) A

the left adjoint to the evaluation functor Y↦ Y(𝐦).
As a consequence of the previous example, the enriched functor G_m A
is purely m-dimensional for every (m)-object A.




For every enriched functor Y:→ and every m≥ 0 the commutative square

    @C=15mm G_m L_m Y [r]^G_mν_m[d]     G_m Y(𝐦) [d]
    ^m-1 Y [r]_-j_m   ^m Y

is a pushout in the category ^∗.
The two vertical morphisms are adjoint to the identity of L_m Y 
respectively Y(𝐦).


All four functors are `m-dimensional', i.e., isomorphic
to the extensions of their restrictions to _≤ m;
for G_m L_m Y and G_m Y(𝐦) 
this follows from Example <ref>.
So it suffices to check that the square is a pushout in 
when evaluated at objects d of dimension at most m.
When the dimension of d is strictly less than m, 
then (G_m L_m Y)(d) and (G_m Y_m)(d) are initial objects and the morphism
j_m(d):(^m-1 Y)(d)→(^m Y)(d) is an isomorphism.
So for (d) < m both horizontal morphisms are isomorphisms at d.
When (d)=m, then d is isomorphic to 𝐦, 
so we may suppose that d=𝐦.
Then the square (<ref>) evaluates to

    @C=25mm(𝐦,𝐦)_(m) L_m Y
     [r]^(𝐦,𝐦)_(m)ν_m[d]    (𝐦,𝐦)_(m) Y(𝐦) [d]
    
    L_m Y [r]_-j_m(𝐦)     Y(𝐦)

So for (d)= m, both vertical maps are isomorphisms at d.







Given any morphism f:A→ B of enriched functors in ^∗, we
define a relative skeleton using the commutative square of enriched functors:

    @C=12mm^m A[r]^^m f[d]_i_m   ^m B [d]^i_m
    
    A[r]_f     B

The relative m-skeleton of f is the pushout

    ^m [f]  =  A∪_^m A^m B   ;

it comes with a unique morphism i_m:^m [f]→ B
which restricts to f:A→ B respectively to i_m:^m B→ B.
Since (^m-1 A)(𝐦)=L_m A we have

    (^m-1 [f])(𝐦)  =  A(𝐦)∪_L_m AL_m B ,

the m-th relative latching object.
A morphism j_m[f]:^m-1 [f]→^m [f] is obtained from the commutative
diagram

    @C=15mm
    A @=[d]    ^m-1 A [r]^-^m-1 f[l]_(.6)i_m-1[d]_j_m^A   ^m-1 B[d]^j_m^B
    
    A     ^m A [r]_-^m f[l]^i_m   ^m B

by taking pushouts.
The original morphism f:A→ B 
factors as the composite of the countable sequence

    A=^-1[f] j_0[f] ^0[f]  j_1[f] ^1[f]  →⋯ j_m[f] ^m [f]  → ⋯ .

If d has dimension n, then the sequence stabilizes to 
the identity map of B(d) from (^n[f])(d) onward; in particular,
the compatible morphisms j_m exhibit B as a colimit of the sequence.
The pushout square (<ref>)
also has a relative version.
We contemplate the following commutative diagram:

    @C=15mm 
    G_m(A(𝐦)∪_L_m AL_m B)  [r]^-G_m(ν_m f)[d]     
    G_m B(𝐦)[d] 
    ^m A∪_^m-1 A^m-1B  [r] [d]    ^m B[d] 
    ^m-1 [f] [r]_-j_m[f]   ^m [f]

All three horizontal morphisms are isomorphisms at objects of dimension less
than m. At the object 𝐦, the two upper vertical morphisms
are isomorphisms.
All four objects in the upper square are `m-dimensional', i.e., isomorphic
to the extensions of their restrictions to _≤ m;
so the upper square is a pushout of enriched functors.
The lower square is also a pushout, by definition of the relative skeleta.
We conclude that the outer composite square 
in the diagram (<ref>)    
is a pushout in the category ^∗.



The following proposition is an immediate application of the
relative skeleton filtration. It is the key ingredient to
the lifting properties of the various level model structures 
in this book.
We recall that a pair (i:A→ B,f:X→ Y) of morphisms in some category
has the lifting propertysubjectlifting property!of a pair of morphisms
if for all morphisms α:A→ X and β:B→ Y such that
fα=β i there exists a lifting, i.e.,
a morphism λ:B→ X such that λ i=α and fλ=β.
Instead of saying that the pair (i,f) has the lifting property
we also say `i has the left lifting property
with respect to f' or `f has the right lifting property with respect to i'.
subjectlifting property!left
subjectlifting property!right


 
Let i:A→ B and f:X→ Y be morphisms of enriched functors in ^∗.
If the pair 

    (ν_m i=i(𝐦)∪ν_m^B:A(𝐦)∪_L_m AL_m B→ B(𝐦),   
    f(𝐦):X(𝐦)→ Y(𝐦))

has the lifting property in the category of (m)-objects for
every m≥ 0, then the pair (i,f) has the lifting property 
in the functor category ^∗.


We consider the class f-cof of all morphisms in ^∗
that have the left lifting property with respect to f; this class is closed
under cobase change and countable composition.
Since the pair (ν_m i,f_m)
has the lifting property in the category of (m)-objects,
the morphism G_m(ν_m i) belongs to the class f-cof by adjointness.
The relative skeleton filtration (<ref>) 
shows that i is a countable composite of cobase changes 
of the morphisms G_m(ν_m i), so i belongs to the class f-cof.


Now we discuss  a general recipe for constructing 
level model structures on the functor category ^∗.
As input we need, for every m≥ 0, a model structure (m)
on the category of (m)-objects.
We call a morphism f:X→ Y in ^∗


  * a level equivalence if  f(𝐦):X(𝐦)→ Y(𝐦) 
is a weak equivalence in the model structure (m) for all m≥ 0;

  * a level fibration if the  morphism
f(𝐦):X(𝐦)→ Y(𝐦) is a fibration in the model structure 
(m) for all m≥ 0;

  * a cofibration if the latching morphism
ν_m f:X(𝐦)∪_L_m XL_m Y→ Y(𝐦) 
is a cofibration in the model structure 
(m) for all m≥ 0.

Proposition <ref> below shows 
that if the various model structures (m)
satisfy the following `consistency condition', then the 
level equivalences,  level fibrations and cofibrations define a model structure
on the functor category ^∗.


subjectconsistency condition
For all m,n≥ 0 and every acyclic cofibration i:A→ B in 
the model structure (m) on (m)-objects,
every cobase change, in the category of (m+n)-objects, of the morphism

    (𝐦,𝐦+𝐧)_(m) i : (𝐦,𝐦+𝐧)_(m) A → (𝐦,𝐦+𝐧)_(m) B

is a weak equivalence in the model structure (m+n).


For example, the consistency conditions holds if the functor

    (𝐦,𝐦+𝐧)_(m) - : (m) → (m+n)

is a left Quillen functor in the given model structures, for all m,n≥ 0.


 
  Let (m) be a model structure 
  on the category of (m)-objects, for m≥ 0,
  such that the consistency condition holds.
   
  
  * The classes of level equivalences, 
    level fibrations and cofibrations define a model structure
    on the functor category ^∗.
  
  * A morphism i:A→ B in ^∗ is simultaneously a 
    cofibration and a level equivalence if and only if 
    for all m≥ 0 the latching morphism 
    ν_m i:A(𝐦)∪_L_m AL_m B→ B(𝐦) is an acyclic cofibration 
    in the model structure (m).
  
  * Suppose that the fibrations in the model structure (m) 
    are detected by a set of morphisms J(m); 
    then the level fibrations are detected by the set of morphisms 
    
    {G_m j  |  m≥ 0,   j∈ J(m)} .

    Similarly, if the acyclic fibrations in the model structure (m) 
    are detected by a set of morphisms I(m), 
    then the level acyclic fibrations
    are detected by the set of morphisms 
    
    {G_m i  |  m≥ 0,   i∈ I(m)} .



 
We start by showing one of the directions of part (ii):
we let i:A→ B be a morphism 
such that the latching morphism ν_m i:A(𝐦)∪_L_m AL_m B→ B(𝐦) 
is an acyclic cofibration in the model structure 
(m) for all m≥ 0; we show that then i is a level equivalence.

The map i(𝐧):A(𝐧)→ B(𝐧) is the finite composite 

    A(𝐧)=(^-1[i])(𝐧)  (j_0[i])(𝐧) 
    (^0[i])(𝐧) (j_1[i])(𝐧)…(j_n[i])(𝐧) (^n[i])(𝐧)=B(𝐧)  ,

so it suffices to show that j_k[i]
is a level equivalence for all k≥ 0.
The pushout square (<ref>)
in level m+n is a pushout of (m+n)-objects

    @C=20mm(𝐦,𝐦+𝐧)_(m)(A(𝐦)∪_L_m AL_m B)
    [r]^-(𝐦,𝐦+𝐧)_(m) (ν_m i)[d]
        (𝐦,𝐦+𝐧)_(m) B(𝐦) [d] 
    
    (^m-1 [i])(𝐦+𝐧) [r]_-(j_m[i])(𝐦+𝐧)    (^m [i])(𝐦+𝐧)

The consistency condition guarantees that
the lower horizontal morphism is a (m+n)-weak equivalence.


(i)
  Several of the axioms are straightforward:
  the functor category ^∗ inherits all small limits and colimits
  from the base category ; 
  the level equivalences satisfy the 2-out-of-3 property;
  the classes of level equivalences, cofibrations and 
  fibrations are closed under retracts. 
  
  Now we prove the factorization axiom, i.e., we show that every morphism 
  f:A→ X in ^* can be factored as f=q i 
  where q is a level acyclic fibration and i a cofibration;
  and it can be factored as f=p j 
  where p is a level fibration and j a cofibration and level equivalence.
  We start with the first factorization and construct 
  an enriched functor B and morphisms
  i:A→ B and q:B→ X by induction over the dimension.
  Since the objects 𝐦 for m≥ 0 form a skeleton
  of the category , it suffices to construct all the relevant data on these 
  objects. 
  In level 0 we choose a factorization
  
    A(0)   i(0)  B(0)   q(0)  X(0)

  of f(0) in the category of (0)-objects such that i(0) 
  is a cofibration
  and q(0) is an acyclic fibration in the model structure (0).
  Now we suppose that the enriched functor B and the morphisms
  i and q have already been constructed for all objects of dimension less than m.
  Then we have all the data necessary to define the
  m-th latching object L_m B; moreover, the `partial morphism'
  q:B→ X provides a (m)-morphism L_m B→ X(𝐦)
  such that the square
  
    L_m A [r]^-L_m i[d]_ν_m^A    L_m B [d] 
    
        A(𝐦) [r]_-f(𝐦)    X(𝐦)

  commutes. We factor the resulting morphism 
  A(𝐦)∪_L_m A L_m B→ X(𝐦) 
  in the category of (m)-objects
  
    A(𝐦)∪_L_m A L_m B  ν_m i  B(𝐦) 
          q(𝐦)  X(𝐦)

  such that ν_m i is a cofibration
  and q(𝐦) is an acyclic fibration in the model structure (m).
  The intermediate (m)-object B(𝐦) defines the value of B
  at the object 𝐦, 
  and q(𝐦) defines the m-th level of the morphism q. 
  For 0≤ i<m, the structure morphism
  is the composite
  
    (𝐢,𝐦)  B(𝐢) → L_m B  →  
      A(𝐦)∪_L_m A L_m B  ν_m i B(𝐦)

  and the composite of ν_m i with the canonical morphism
  A(𝐦)→ A(𝐦)∪_L_m A L_m B is the m-th level of the morphism i.

  At the end of the day we have indeed factored f=q i
  in the category ^∗ such that q is a level
  equivalence and level fibration. Moreover, the m-th
  latching morphism comes out to be the morphism
  ν_m i:A(𝐦)∪_L_m A L_m B→ B(𝐦)
  in the factorization (<ref>), which is a cofibration
  in the model structure (m). So the morphism i is indeed a
  cofibration.

  The second factorization f=p j as a cofibration and level equivalence j
  followed by a level fibration p is similar, but instead of the 
  factorization (<ref>) we use a factorization,
  in the model category (m), as an acyclic cofibration followed by 
  a fibration. Then the resulting morphism p is a level fibration and
  the morphism j has the property that all its latching morphisms ν_m j 
  are acyclic cofibrations. So j is a cofibration
  (by definition) and a level equivalence 
  (by the part of (ii) established above).

  It remains to show the lifting axioms. 
  In each of the model structures (m) 
  the cofibrations have the left lifting property with respect 
  to the acyclic fibrations; so by 
  Proposition <ref> 
  the cofibrations in ^∗ have the left lifting property 
  with respect to level equivalences which are also level fibrations. 

  We postpone the proof of the other lifting property and 
  prove the remaining direction of (ii) next.
  We let i:A→ B be a cofibration and a level equivalence. 
  The second factorization axiom proved above provides a factorization
  i=p j where j:A→ D is a level equivalence such that 
  each latching morphism ν_m j is an acyclic cofibration
  in the model structure (m),
  and p:D→ B is a level fibration.
  Since i and j are level equivalences, so is p. 
  So the cofibration i has the left lifting property 
  with respect to the level equivalence and level fibration p 
  by the previous paragraph.
  In particular, a lift λ:B→ D in the square
 
    A[r]^-j[d]_i      D[d]^p
    
        B@=[r] @..>[ur]^(.4)λ    B

  shows that the morphism i is a retract of the morphism j.
  So the latching morphism ν_m i is a retract 
  of the latching morphism ν_m j, hence also an acyclic cofibration
  in the model structure (m). This proves (ii).

  Now we prove the remaining lifting property.
  We let i:A→ B be a cofibration that is also a level equivalence. 
  By (ii), which has just been shown,
  each latching morphism ν_mi is an acyclic cofibration
  in the model structure (m).
  So i has the left lifting property with respect to all level fibrations
  by Proposition <ref>.   

  Property (iii) is a straightforward consequence of
  the fact that the functor G_m is left adjoint to evaluation at 𝐦.








MMMN


adams-haeberly-jackowski-may:atiyah-segal
J. F. Adams, J.-P. Haeberly, S. Jackowski, J. P. May,
A generalization of the Atiyah-Segal completion theorem.
Topology 27 (1988), no. 1, 1–6. 

alexander
J. C. Alexander, The bordism ring of manifolds with involution.
Proc. Amer. Math. Soc. 31 (1972), 536–542. 

atiyah-Bott and elliptic
M. F. Atiyah, Bott periodicity and the index of elliptic operators.
Quart. J. Math. Oxford Ser. (2) 19 (1968), 113–140. 

atiyah-bott-shapiro
M. F. Atiyah, R. Bott, A. Shapiro, Clifford modules.
Topology 3 (1964) suppl. 1, 3–38. 

atiyah_segal-completion
M. F. Atiyah, G. B. Segal, Equivariant K-theory and completion.
J. Differential Geom. 3 (1969), 1–18.

becker-gottlieb
J. C. Becker, D. H. Gottlieb, The transfer map and fiber bundles.
Topology 14 (1975), 1–12. 

beem-2mod4
R. P. Beem, Generators for G bordism.
Proc. Amer. Math. Soc. 67 (1977), no. 2, 335–343. 

beem-C_4
R. P. Beem, The structure of Z/4 bordism.
Indiana Univ. Math. J. 27 (1978), no. 6, 1039–1047. 

beem-rowlett
R. B. Beem, R. J. Rowlett, 
The fixed point homomorphism for maps of period 2^k.
Indiana Univ. Math. J. 30 (1981), no. 4, 489–500. 

BBD
A. A. Beilinson, J. Bernstein, P. Deligne, 
Faisceaux pervers.
Analysis and topology on singular spaces, I (Luminy, 1981), 5–171, 
Astérisque, 100, Soc. Math. France, Paris, 1982. 

beligiannis-reiten
A. Beligiannis, I. Reiten, 
Homological and homotopical aspects of torsion theories.
Mem. Amer. Math. Soc. 188 (2007), no. 883, viii+207 pp. 

berger-moerdijk-Reedy
C. Berger, I. Moerdijk, 
On an extension of the notion of Reedy category. 
Math. Z. 269 (2011), no. 3-4, 977–1004. 

besse
A. L. Besse, Einstein manifolds.
Ergebnisse der Mathematik und ihrer Grenz­gebiete (3) 10. 
Springer-Verlag, Berlin, 1987. xii+510 pp.

blumberg
A. J. Blumberg, 
Continuous functors as a model for the equivariant stable homotopy category.
Algebr. Geom. Topol. 6 (2006), 2257–2295.

boardman-thesis
J. M. Boardman, On Stable Homotopy Theory and Some Applications.
PhD thesis, University of Cambridge (1964).

boardman-vogt-homotopy everything
J. M. Boardman, R. M. Vogt, Homotopy-everything H-spaces. 
Bull. Amer. Math. Soc. 74 (1968), 1117–1122. 

boardman-vogt-homotopy invariant
J. M. Boardman, R. M. Vogt,
Homotopy invariant algebraic structures on topological spaces. 
Lecture Notes in Mathematics, Vol. 347. Springer-Verlag, Berlin-New York, 
1973. x+257 pp. 

bohmann-thesis
A. M. Bohmann, Topics in equivariant stable homotopy theory.
PhD thesis, University of Chicago, 2011.

bohmann-orthogonal
A. M. Bohmann, Global orthogonal spectra.
Homology Homotopy Appl. 16 (2014), no. 1, 313–332.

boltje-canonical brauer
R. Boltje,
A canonical Brauer induction formula.
Astérisque No. 181–182 (1990) 5, 31–59. 

boltje-snaith-symonds
R. Boltje, V. Snaith, P. Symonds,
Algebraicisation of explicit Brauer induction.
J. Algebra 148 (1992), no. 2, 504–527.

bousfield-friedlander
A. K. Bousfield, E. M. Friedlander,
Homotopy theory of Γ-spaces, spectra, and bisimplicial sets.
 Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II, 
pp. 80–130,
Lecture Notes in Mathematics, Vol. 658, Springer-Verlag, Berlin, 1978. 

bott-periodicity
R. Bott, The stable homotopy of the classical groups.
Ann. of Math. (2) 70 (1959), 313–337. 

brauer
R. Brauer, On Artin's L-series with general group characters.
Ann. of Math. (2) 48 (1947), 502–514.

bredon-cohomology
G. E. Bredon, Equivariant cohomology theories. 
Lecture Notes in Mathematics, Vol. 34, Springer-Verlag, Berlin-New York, 1967. vi+64 pp.

bredon-intro
G. E. Bredon, Introduction to compact transformation groups.
Pure and Applied Mathematics, Vol. 46. 
Academic Press, New York-London, 1972. xiii+459 pp. 

broecker-hook
T. Bröcker, E. C. Hook, Stable equivariant bordism.
Math. Z. 129 (1972), 269–277. 

broecker-tomDieck
T. Bröcker, T. tom Dieck, Representations of compact Lie groups.
Graduate Texts in Math., Vol. 98, Springer-Verlag, New York, 1985. x+313 pp.

brown-abstract
K. S. Brown, Abstract homotopy theory and generalized sheaf cohomology.  
Trans. Amer. Math. Soc. 186  (1974), 419–458. 

brown-function spaces
R. Brown, Function spaces and product topologies.
Quart. J. Math. Oxford Ser. (2) 15 (1964), 238–250.

brun-Witt cobordism 
M. Brun, Witt vectors and equivariant ring spectra applied to cobordism.
Proc. London Math. Soc. (3) 94 (2007), 351–385.

brun-dundas-stolz 
M. Brun, B. I. Dundas, M. Stolz,
Equivariant structure on smash powers.


bruner-greenlees-connective_K_finite
R. Bruner, J. Greenlees, The connective K-theory of finite groups.
Mem. Amer. Math. Soc. 165 (2003), no. 785, viii+127 pp.

bruner-may-mcclure-steinberger
R. Bruner, J. P. May, J. McClure, M. Steinberger,
H∞ ring spectra and their applications.
Lecture Notes in Mathematics, Vol. 1176.
Springer-Verlag, Berlin, 1986. viii+388 pp.

carlsson-segal conjecture
G. Carlsson, Equivariant stable homotopy and Segal's Burnside ring conjecture.
Ann. of Math. (2) 120 (1984), no. 2, 189–224. 

Christensen-Hovey-relative
J. D. Christensen, M. Hovey, 
Quillen model structures for relative homological algebra.
Math. Proc. Cambridge Philos. Soc. 133 (2002), no. 2, 261–293. 

cohen-weak topology
D. E. Cohen, Spaces with weak topology.
Quart. J. Math. Oxford Ser. (2) 5, (1954), 77–80. 

cole-mixed
M. Cole, Mixing model structures. 
Topology Appl. 153 (2006), no. 7, 1016–1032. 

conner-floyd
P. E. Conner, E. E. Floyd, 
Differentiable periodic maps. 
Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Band 33,
Academic Press Inc., Publishers, New York; Springer-Verlag, 1964. vii+148 pp.

costenoble-thesis
S. R. Costenoble, Equivariant cobordism and K-theory.
PhD thesis, University of Chicago, 1985.

curtis
E. B. Curtis, Simplicial homotopy theory. 
Advances in Math. 6 (1971), 107–209.

day-closed
B. Day, On closed categories of functors. 
Reports of the Midwest Category Seminar, IV pp. 1–38.
Lecture Notes in Mathematics, Vol. 137.
Springer-Verlag, Berlin, 1970.

dowker-metric complexes
C. H. Dowker, Topology of metric complexes.
Amer. J. Math. 74 (1952), 555–577. 

dress-notes
A. Dress, 
Notes on the theory of representations of finite groups.
Duplicated notes, Bielefeld, 1971.

dugger-real K
D. Dugger, An Atiyah-Hirzebruch spectral sequence for KR-theory.
K-theory 35 (2005), 213–256.

dugger-isaksen
D. Dugger, D. C. Isaksen, 
Topological hypercovers and 𝔸^1-realizations. 
Math. Z. 246 (2004), no. 4, 667–689. 

dundas-goodwillie-mccarthy
B. Dundas, T. G. Goodwillie, R. McCarthy, 
The local structure of algebraic K-theory.
Algebra and Applications, 18. Springer-Verlag London, Ltd., 
London, 2013. xvi+435 pp.

dwyer-spalinski
W. G. Dwyer, J. Spalinski, 
Homotopy theories and model categories. 
In: Handbook of algebraic topology, ed. I. M. James, Elsevier (1995), 73–126. 

eilenberg-zilber-ssc
S. Eilenberg, J. A. Zilber,
Semi-Simplicial Complexes and Singular Homology.
Ann. of Math. (2) 51 (1950), 499–513. 

EKMM
A. D. Elmendorf, I. Kriz, M. A. Mandell, J. P. May,
Rings, modules, and algebras in stable homotopy theory.
With an appendix by M. Cole.
Mathematical Surveys and Monographs, 47, Amer. Math. Soc.,
Providence, RI, 1997, xii+249 pp.

elmendorf-mandell
A. D. Elmendorf, M. A. Mandell,
Rings, modules, and algebras in infinite loop space theory.
Adv. Math. 205 (2006), no. 1, 163–228.

fausk-pro
H. Fausk, Equivariant homotopy theory for pro-spectra.
Geom. Topol. 12 (2008), 103–176.

feshbach
M. Feshbach, The transfer and compact Lie groups. 
Trans. Amer. Math. Soc. 251 (1979), 139–169.

firsching
M. Firsching, Real equivariant bordism for elementary abelian 2-groups.
Homology Homotopy Appl. 15 (2013), no. 1, 235–251. 

fresse-module functors
B. Fresse, Modules over operads and functors.
Lecture Notes in Mathematics, Vol. 1967. Springer-Verlag, Berlin, 2009. x+308 pp.

friedlander-mazur
E. M. Friedlander, B. Mazur, 
Filtrations on the homology of algebraic varieties.
With an appendix by Daniel Quillen.
Mem. Amer. Math. Soc. 110 (1994), no. 529, x+110 pp. 

fritsch-piccinini
R. Fritsch, R. Piccinini, Cellular structures in topology.
Cambridge Studies in Advanced Mathematics, 19. 
Cambridge University Press, Cambridge, 1990. xii+326 pp. 

gabriel-zisman
P. Gabriel, M. Zisman, Calculus of fractions and homotopy theory. 
Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35. 
Springer-Verlag New York, Inc., New York, 1967. x+168 pp. 

galatius-madsen-tillmann-weiss
S. Galatius, U. Tillmann, I. Madsen, M. Weiss, 
The homotopy type of the cobordism category.
Acta Math. 202 (2009), no. 2, 195–239.

ganter-power
N. Ganter, Global Mackey functors with operations and n-special lambda rings.
 

gepner-henriques
D. Gepner, A. Henriques, Homotopy theory of orbispaces.



Goerss-Hopkins-moduli
P. Goerss, M. Hopkins, Moduli spaces of commutative ring spectra.
Structured ring spectra, 151–200,
London Math. Soc. Lecture Note Ser., 315, 
Cambridge University Press, Cambridge, 2004. 

goerss-jardine
P. G. Goerss, J. F. Jardine, Simplicial homotopy theory. 
Progress in Mathematics, 174. Birkhäuser Verlag, Basel, 1999. xvi+510 pp.

gor-gul-symmetric
S. Gorchinskiy, V. Guletskiĭ, 
Symmetric powers in abstract homotopy categories.
Adv. Math. 292 (2016), 707–754. 

green-axiomatic
J. A. Green, Axiomatic representation theory for finite groups.
J. Pure Appl. Algebra 1 (1971), 41–77.

greenlees-connective
J. P. C. Greenlees, Equivariant connective K-theory for
compact Lie groups. J. Pure Appl. Algebra 187 (2004), 129–152.

greenlees-may-Tate
J. P. C. Greenlees, J. P. May,
Generalized Tate cohomology.
Mem. Amer. Math. Soc. 113 (1995), no. 543, viii+178 pp.

greenlees-may-completion
J. P. C. Greenlees, J. P. May,
Localization and completion theorems for M U-module spectra.
Ann. of Math. (2) 146 (1997), 509–544. 

haag
U. Haag, Some algebraic features of _2-graded KK-theory.
K-Theory 13 (1998), no. 1, 81–108. 

harris-Bott
B. Harris, Bott periodicity via simplicial spaces. 
J. Algebra 62 (1980), no. 2, 450–454. 

hatcher
A. Hatcher, Algebraic topology. 
Cambridge University Press, Cambridge, 2002. xii+544 pp.

hausmann-master thesis
M. Hausmann, Global equivariant homotopy theory of symmetric spectra.
Master thesis, Universität Bonn, 2013.

hausmann-global
M. Hausmann, Symmetric spectra model global homotopy theory of finite groups.


hausmann-ostermayr
M. Hausmann, D. Ostermayr,
Filtrations of global equivariant K-theory.


higson-kasparov-trout
N. Higson, G. Kasparov, J. Trout,
A Bott periodicity theorem for infinite-dimensional Euclidean space.
Adv. Math. 135 (1998), no. 1, 1–40. 

higson-guentner
N. Higson, E. Guentner,
Group C^∗-algebras and K-theory. 
Noncommutative geometry, 137–251,
Lecture Notes in Mathematics, Vol. 1831, Springer-Verlag, Berlin, 2004. 

HHR-Kervaire
M. Hill, M. Hopkins, D. Ravenel,
On the nonexistence of elements of Kervaire invariant one.
Ann. of Math. (2) 184 (2016), 1–262. 

hirschhorn
P. S. Hirschhorn, Model categories and their localizations. 
Mathematical Surveys and Monographs, 99. Amer. Math. Soc., 
Providence, RI, 2003. xvi+457 pp. 

hohnhold-stolz-teichner
H. Hohnhold, S. Stolz, P. Teichner, 
From minimal geodesics to supersymmetric field theories.
A celebration of the mathematical legacy of Raoul Bott, 207–274, 
CRM Proc. Lecture Notes 50, Amer. Math. Soc., Providence, RI, 2010. 

hovey-book
M. Hovey, Model categories. Mathematical Surveys and Monographs, 63.
Amer. Math. Soc., Providence, RI, 1999, xii+209 pp.

HSS
M. Hovey, B. Shipley, J. Smith, Symmetric spectra.
J. Amer. Math. Soc. 13 (2000), 149–208.

hurewicz-beitraege IV
W. Hurewicz, 
Beiträge zur Topologie der Deformationen. IV. Asphärische Räume.
Proc. Kon. Akad. Wet. Amsterdam 39 (1936), 215–224. 

illman-finite G
S. Illman, 
Smooth equivariant triangulations of G-manifolds for G a finite group. 
Math. Ann. 233 (1978), no. 3, 199–220. 

illman
S. Illman, 
The equivariant triangulation theorem for actions of compact Lie groups.
Math. Ann. 262 (1983), 487–501.

illman-restricting equivariance
S. Illman,
Restricting the transformation group in equivariant CW complexes.
Osaka J. Math. 27 (1990), no. 1, 191–206. 

joachim-coherences
M. Joachim, Higher coherences for equivariant K-theory.
Structured ring spectra,  87–114, London Math. Soc. Lecture Note Ser., 315, 
Cambridge University Press, Cambridge, 2004. 

kan-on css
D. M. Kan, On c. s. s. complexes. 
Amer. J. Math. 79 (1957), 449–476. 

karoubi-sur la K-theorie
M. Karoubi, Sur la K-théorie équivariante.
Séminaire Heidelberg-Saar­brücken-Strasbourg sur la K-théorie (1967/68), 
187–253. 
Lecture Notes in Mathematics, Vol. 136. Springer-Verlag, Berlin, 1970. 

kelley-general topology
J. L. Kelley, General topology.
D. Van Nostrand Company, Inc., Toronto-New York-London, 1955. xiv+298 pp.

kelly-enriched category 
G. M. Kelly, Basic concepts of enriched category theory. 
Reprint of the 1982 original.
Repr. Theory Appl. Categ. No. 10 (2005), vi+137 pp.

kirillov
A. A. Kirillov, 
Elements of the theory of representations. 
Translated from the Russian by E. Hewitt. 
Grundlehren der Mathematischen Wissen­schaften, Band 220. Springer-Verlag, 
Berlin-New York, 1976. xi+315 pp.

krause-Brown coherent
H. Krause, A Brown representability theorem via coherent functors.
Topology 41 (2002), no. 4, 853–861. 

kriz-Z/p cobordism
I. Kriz, The Z/p-equivariant complex cobordism ring.
 Homotopy invariant algebraic structures (Baltimore, MD, 1998), 217–223,
Contemp. Math., 239, Amer. Math. Soc., Providence, RI, 1999. 

lashof-equivariant bundles
R. K. Lashof, Equivariant bundles. 
Illinois J. Math. 26 (1982), no. 2, 257–271.

lashof-may-segal
R. K. Lashof, J. P. May, G. B. Segal, 
Equivariant bundles with abelian structural group. 
Proceedings of the Northwestern Homotopy Theory Conference (Evanston, Ill., 1982), 
167–176, Contemp. Math., 19, Amer. Math. Soc., Providence, RI, 1983. 

lewis-thesis
L. G. Lewis, Jr., The stable category and generalized Thom spectra.
Ph.D. thesis, University of Chicago, 1978.

lewis-cofibration 
L. G. Lewis, Jr., When is the natural map X→ΩΣ X a cofibration? 
Trans. Amer. Math. Soc. 273 (1982), no. 1, 147–155. 

lewis-projective not flat
L. G. Lewis, Jr., When projective does not imply flat, 
and other homological anomalies. Theory Appl. Categ. 5 (1999), no. 9, 202–250. 

lewis-may-mcclure-classifying
L. G. Lewis, Jr., J. P. May, J. E. McClure,
Classifying G-spaces and the Segal conjecture. 
Current trends in algebraic topology, Part 2 (London, Ont., 1981), pp. 165–179,
CMS Conf. Proc., 2, Amer. Math. Soc., Providence, RI, 1982. 

lms
L. G. Lewis, Jr., J. P. May, M. Steinberger,
Equivariant stable homotopy theory. Lecture Notes in Mathematics,
Vol. 1213, Springer-Verlag, 1986. x+538 pp.

lillig
J. Lillig, A union theorem for cofibrations.
Arch. Math. (Basel) 24 (1973), 410–415.

lind-diagram
J. A. Lind, Diagram spaces, diagram spectra and spectra of units.
Algeb. Geom. Topol. 13 (2013), 1857–1935.

lindner-Mackey
H. Lindner, A remark on Mackey-functors.
Manuscripta Math. 18 (1976), no. 3, 273–278. 

loeffler-budva
P. Löffler, 
Equivariant unitary cobordism and classifying spaces. 
Proceedings of the International Symposium on Topology and its Applications 
(Budva, 1972), pp. 158–160. Savez Društava Mat. Fiz. i Astronom., Belgrade, 1973. 

maclane-working
S. Mac Lane, Categories for the working mathematician. Second edition. 
Graduate Texts in Mathematics, 5. Springer-Verlag, New York, 1998. xii+314 pp. 

mandell-algebraization  
M. A. Mandell,  Algebraization of E_∞ ring spectra. 
Preprint, 1998.

mmss
M. A. Mandell, J. P. May, S. Schwede, B. Shipley, 
Model categories of diagram spectra.
Proc. London Math. Soc. (3) 82  (2001),  no. 2, 441–512.

mandell-may
M. A. Mandell, J. P. May, Equivariant orthogonal spectra and S-modules. 
Mem. Amer. Math. Soc. 159 (2002), no. 755, x+108 pp.

matumoto
T. Matumoto, On G-CW complexes and a theorem of J. H. C. Whitehead.
J. Fac. Sci. Univ. Tokyo Sect. IA Math. 18 (1971), 363–374.

may-geometry iterated loop
J. P. May, The geometry of iterated loop spaces.
Lectures Notes in Mathematics, Vol. 271. 
Springer-Verlag, Berlin-New York, 1972. viii+175 pp. 

may-group completions
J. P. May, E_∞ spaces, group completions, and permutative categories. 
New developments in topology (Proc. Sympos. Algebraic Topology, Oxford, 1972), 
pp. 61–93. 
London Math. Soc. Lecture Note Ser., No. 11, 
Cambridge University Press, London, 1974. 

may-quinn-ray
J. P. May, E∞ ring spaces and E∞ ring spectra.
With contributions by F. Quinn, N. Ray, and J. Tornehave. 
Lecture Notes in Mathematics, Vol. 77.
Springer-Verlag, Berlin-New York, 1977. 268 pp.

may-alaska
J. P. May, Equivariant homotopy and cohomology theory.
With contributions by M. Cole, G. Comezaña, S. Costenoble, 
A. D. Elmendorf, J. P. C. Greenlees, L. G. Lewis, Jr., 
R. J. Piacenza, G. Triantafillou, and S. Waner. 
CBMS Regional Conference Series in Mathematics, 91. 
Amer. Math. Soc., Providence, RI, 1996. xiv+366 pp.

may-concise
J. P. May, A concise course in algebraic topology.
Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1999. 
x+243 pp.

mmo
J. P. May,  M. Merling, A. M. Osorno,
Equivariant infinite loop space theory, I. The space level story.


may-sigurdsson
J. P. May, J. Sigurdsson, Parametrized homotopy theory.
Mathematical Surveys and Monographs, 132. 
Amer. Math. Soc., Providence, RI, 2006. x+441 pp. 

mcclure-schwanzl-vogt
J. McClure, R. Schwänzl, R. Vogt,
THH(R) R S^1 for E_∞ ring spectra.
J. Pure Appl. Algebra 121 (1997), no. 2, 137–159. 

mccord
M. C. McCord, Classifying spaces and infinite symmetric products.
Trans. Amer. Math. Soc. 146 (1969), 273–298. 

meyer-equivariant generalized
R. Meyer, Equivariant Kasparov theory and generalized homomorphisms.
K-Theory 21 (2000), no. 3, 201–228.

milnor-universalII
J. Milnor, Construction of universal bundles. II.
Ann. of Math. (2) 63 (1956), 430–436. 

milnor-realization
J. Milnor, The geometric realization of a semi-simplicial complex.
Ann. of Math. (2) 65 (1957), 357–362. 

montgomery-zippin
D. Montgomery, L. Zippin, A theorem on Lie groups.
Bull. Amer. Math. Soc. 48 (1942), 448–452. 

morel-voevodsky
F. Morel, V. Voevodsky, 𝔸^1-homotopy theory of schemes.
Inst. Hautes Études Sci. Publ. Math. 90 (1999), 45–143. 

mostow-embeddings
G. D. Mostow, 
Equivariant embeddings in Euclidean space.
Ann. of Math. (2) 65 (1957), 432–446. 

murayama-GANRs 
M. Murayama, On G-ANRs and their G-homotopy types.
Osaka J. Math. 20 (1983), no. 3, 479–512. 

murphy-Cstar
G. J. Murphy, C^∗-algebras and operator theory.
Academic Press, Inc., Boston, MA, 1990. x+286 pp. 

neeman-Grothendieck
A. Neeman,
The Grothendieck duality theorem
via Bousfield's techniques and Brown representability.
J. Amer. Math. Soc. 9 (1996), 205–236.

neeman-triangulated categories
A. Neeman, Triangulated categories. 
Annals of Mathematics Studies, 148. 
Princeton University Press, Princeton, NJ, 2001. viii+449 pp.

ostermayr
D. Ostermayr, Equivariant Γ-spaces.
Homology Homotopy Appl. 18 (2016), no. 1, 295–324. 

palais-imbedding
R. S. Palais,
Imbedding of compact, differentiable transformation groups 
in orthogonal representations.
J. Math. Mech. 6 (1957), 673–678. 

palais-classification
R. S. Palais,
The classification of G-spaces.
Mem. Amer. Math. Soc. 36 (1960), iv+72 pp.

puppe-homotopiemengen
D. Puppe, Homotopiemengen und ihre induzierten Abbildungen. I.
Math. Z. 69 (1958), 299–344. 

puppe-bemerkungen
D. Puppe, 
Bemerkungen über die Erweiterung von Homotopien.
Arch. Math. (Basel) 18 (1967), 81–88. 

Q
D. Quillen, Homotopical algebra. Lecture Notes in Mathematics,
Vol. 43, Springer-Verlag, 1967. iv+156 pp. 

quillen-higher algebraic
D. Quillen, Higher algebraic K-theory. I. 
Algebraic K-theory, I: Higher K-theories 
(Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 85–147. 
Lecture Notes in Mathematics, Vol. 341, Springer-Verlag, Berlin 1973. 

reedy
C. L. Reedy Homotopy theory of model categories.
Unpublished manu­script, 1974.

rezk-thesis
C. Rezk, Spaces of algebra structures and cohomology of operads.
PhD thesis, Massachusetts Institute of Technology, 1996.

rezk-1truncated
C. Rezk, Classifying spaces for 1-truncated compact Lie groups.
Algebr. Geom. Topol. 18 (2018), 525–546.

rotman-intro at
J. J. Rotman, An introduction to algebraic topology. 
Graduate Texts in Mathematics, 119. Springer-Verlag, New York, 1988. xiv+433 pp. 

rotman-homological algebra
J. J. Rotman, An introduction to homological algebra. 
Second edition. Universitext. Springer-Verlag, New York, 2009. xiv+709 pp.

sagave-schlichtkrull-diagram
S. Sagave, C. Schlichtkrull, 
Diagram spaces and symmetric spectra.
Adv. Math. 231 (2012), no. 3-4, 2116–2193. 

dosSantos
P. dos Santos, 
A note on the equivariant Dold-Thom theorem.
J. Pure Appl. Algebra 183 (2003), 299–312. 

schwede-topological
S. Schwede, The p-order of topological triangulated categories.
J. Topol. 6 (2013), no. 4, 868–914.

schwede-infinite symmetric
S. Schwede, Equivariant properties of symmetric products.
J. Amer. Math. Soc. 30 (2017), 673–711.

schwede-orbispaces 
S. Schwede, Orbispaces, orthogonal spaces, and the universal compact Lie group.


schwede-shipley-monoidal
S. Schwede, B. Shipley, 
Algebras and modules in monoidal model categories.
Proc. London Math. Soc. 80 (2000), 491–511.

schwede-shipley-uniqueness
S. Schwede, B. Shipley, 
A uniqueness theorem for stable homotopy theory.
Math. Z. 239 (2002), 803–828.

schwede-shipley-modules
S. Schwede, B. Shipley, 
Stable model categories are categories of modules.  
Topology 42  (2003),  no. 1, 103–153. 

segal-classifying spaces
G. Segal, Classifying spaces and spectral sequences. 
Inst. Hautes Études Sci. Publ. Math. 34 (1968), 105–112.

segal-representation
G. Segal, The representation ring of a compact Lie group.
Inst. Hautes Études Sci. Publ. Math. 34 (1968), 113–128. 

segal-equivariant K-theory
G. Segal, Equivariant K-theory. 
Inst. Hautes Études Sci. Publ. Math. 34 (1968), 129–151. 

segal-ICM
G. Segal, Equivariant stable homotopy theory. 
Actes du Congrès International des Mathématiciens (Nice, 1970), 
Tome 2, pp. 59–63, 1971.

segal-cat coho
G. Segal, Categories and cohomology theories. Topology 13 (1974),
293–312.

segal-K-homology
G. Segal, K-homology theory and algebraic K-theory.  
K-theory and operator algebras (Proc. Conf., Univ. Georgia, Athens, 
Ga., 1975),  pp. 113–127. Lecture Notes in Mathematics, Vol. 575, 
Springer-Verlag, Berlin, 1977. 

segal-some_equivariant
G. Segal, Some results in equivariant homotopy theory. 
Preprint, 1978.

seguinspazzis-realize
C. de Seguins Pazzis, 
The geometric realization of a simplicial Hausdorff space is Hausdorff.
Topology Appl. 160 (2013), no. 13, 1621–1632. 

shimakawa
K. Shimakawa, 
Infinite loop G-spaces associated to monoidal G-graded categories.
Publ. Res. Inst. Math. Sci. 25 (1989),  239–262. 

shimakawa-note
K. Shimakawa, A note on Γ_G-spaces. 
Osaka J. Math. 28 (1991), no. 2, 223–228.

singer
J. Singer, Äquivariante λ-Ringe und kommutative
Multiplikationen auf Moore-Spektren. Dissertation, Universität Bonn, 2007.
 


Snaith-Brauer
V. P. Snaith, Explicit Brauer induction.
Invent. Math. 94 (1988), no. 3, 455–478. 

steenrod-convenient
N. E. Steenrod, 
A convenient category of topological spaces.
Michigan Math. J. 14 (1967), 133–152. 

stenstrom
B. Stenström, Rings of quotients. An introduction to methods of ring theory.
Die Grundlehren der Mathematischen Wissenschaften, Band 217. 
Springer-Verlag, New York-Heidelberg, 1975. viii+309 pp.

stolz-thesis
M. Stolz, Equivariant structure on smash powers of commutative ring spectra.
PhD thesis, University of Bergen, 2011.

stong-unoriented finite
R. E. Stong, Unoriented bordism and actions of finite groups.
Mem. Amer. Math. Soc. 103 (1970), 80 pp. 

strickland-MU involution 
N. P. Strickland, Complex cobordism of involutions.
Geom. Topol. 5 (2001), 335–345.

strickland-realizing formal groups
N. P. Strickland, Realising formal groups.
Algebr. Geom. Topol. 3 (2003), 187–205. 

strickland-CGWH 
N. P. Strickland, The category of CGWH spaces.
Preprint, available from the author's homepage.

sullivan-localization reprint
D. P. Sullivan, 
Geometric topology: localization, periodicity and Galois symmetry.
The 1970 MIT notes. Edited and with a preface by A. Ranicki. 
K-Monographs in Mathematics, 8. Springer-Verlag, Dordrecht, 2005. xiv+283 pp. 

suslin-beilinson 
A. A. Suslin, 
The Beilinson spectral sequence for the K-theory of the field of real numbers.
Mat. Metody i Fiz.-Mekh. Polya No. 28 (1988), 51–52, 105; 
translation in J. Soviet Math. 63 (1993), no. 1, 57–58. 

symonds-splitting
P. Symonds, A splitting principle for group representations.
Comment. Math. Helv. 66 (1991), no. 2, 169–184. 

tambara
D. Tambara, On multiplicative transfer.
Comm. Algebra 21 (1993), no. 4, 1393–1420. 

thevenaz-some results
J. Thévenaz, 
Some remarks on G-functors and the Brauer morphism.
J. Reine Angew. Math. 384 (1988), 24–56. 

thom-quelques
 R. Thom, Quelques propriétés globales des variétés 
différentiables. Comment. Math. Helv. 28 (1954), 17–86. 

tomDieck-faserbundel
T. tom Dieck, Faserbündel mit Gruppenoperation.
Arch. Math. (Basel) 20 (1969), 136–143. 

tomDieck-bordism
T. tom Dieck, Bordism of G-manifolds and integrality theorems.
Topology 9 (1970), 345–358.

tomDieck-OrbittypenI
T. tom Dieck, Orbittypen und äquivariante Homologie. I.
Arch. Math. (Basel) 23 (1972), no. 1, 307–317. 

tomDieck-OrbittypenII
T. tom Dieck, Orbittypen und äquivariante Homologie. II.
Arch. Math. (Basel) 26 (1975), no. 6, 650–662. 

tomDieck-and representation theory
T. tom Dieck, Transformation groups and representation theory.
Lecture Notes in Mathematics, Vol. 766. Springer-Verlag, Berlin, 1979. viii+309 pp.

tomDieck-transformation
T. tom Dieck, Transformation Groups.
De Gruyter Studies in Mathematics, 8. 
Walter de Gruyter & Co., Berlin, 1987. x+312 pp.

tomDieck-algebraic topology
T. tom Dieck, Algebraic topology. 
EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zürich, 2008. 
xii+567 pp. 

ullman-tambara 
J. Ullman, Tambara functors and commutative ring spectra.



verona-triangulation
A. Verona, Triangulation of stratified fibre bundles.
Manuscripta Math. 30 (1979/80), no. 4, 425–445. 

waner-equivariant classifying
S. Waner, Equivariant classifying spaces and fibrations.
Trans. Amer. Math. Soc. 258 (1980), 385–405. 

wasserman
A. G. Wasserman, Equivariant differential topology. 
Topology 8 (1969), 127–150.

webb
P. Webb, Two classifications of simple Mackey functors with applications
to group cohomology and the decomposition of classifying spaces. 
J. Pure Appl. Algebra 88 (1993), no. 1–3, 265–304. 

wegge-olsen
N. E. Wegge-Olsen, K-theory and C^∗-algebras. A friendly approach.
 Oxford Science Publications. 
The Clarendon Press, Oxford University Press, New York, 1993. xii+370 pp.

weibel-homological algebra
C. A. Weibel, 
An introduction to homological algebra. 
Cambridge Studies in Advanced Mathematics, 38. Cambridge University Press, 
Cambridge, 1994. xiv+450 pp

white-commutative monoids
D. White, Model structures on commutative monoids in general model categories. 
J. Pure Appl. Algebra 221 (2017), no. 12, 3124–3168. 

whitehead-note borsuk
J. H. C. Whitehead, Note on a theorem due to Borsuk.
Bull. Amer. Math. Soc. 54 (1948), 1125–1132. 

whitehead-combinatorialI
J. H. C. Whitehead, Combinatorial homotopy. I.
Bull. Amer. Math. Soc. 55, (1949). 213–245.

wirthmuller
K. Wirthmüller, Equivariant homology and duality.
Manuscripta Math. 11 (1974), 373–390. 

woolfson
R. Woolfson, Hyper-Γ-spaces and hyperspectra.
Quart. J. Math. Oxford Ser. (2) 30 (1979), no. 118, 229–255. 

yang-montgomery
Ch.-T. Yang, On a problem of Montgomery.
Proc. Amer. Math. Soc. 8 (1957), 255–257. 



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