Countable approximation of topological $G$-manifolds: compact Lie groups $G$
Qayum Khan



Let G be a compact Lie group.
(Compact) topological G-manifolds have the G-homotopy type of (finite-dimensional) countable G-CW complexes. 
This partly generalizes Elfving's theorem for locally linear G-manifolds <cit.>.

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    December 30, 2023
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§ TOPOLOGICAL MANIFOLDS


By topological manifold, we shall mean a separable metrizable space M such that each point x ∈ M has a neighborhood homeomorphic to some ^n.
Notice we allow n to depend on x, so that different components of M may have different dimensions.
Note separable and metrizable imply paracompact <cit.> and second-countable.


Let M be a topological manifold.
There exists an increasing sequence M_0 ⊆ M_1 ⊆…⊆ M_i ⊆… of open sets whose union is M such that each closure M_i in M both is compact and is a neighborhood deformation retract in M_i+1.


It is a lament of the author to not have discovered a proof from basic topology.


If M is compact, then simply use the constant sequence M_i=M.
Otherwise, assume M is noncompact.
Since M has only countably many components, say {M^c}_c=0^∞ and by forming the finite unions M_i := ⋃_c+j=i M^c_j when such sequences {M^c_j}_j=0^∞ exist, we may further assume that M is connected, say with n:= M.

First suppose n ⩽ 4.
If n=1, then M is homeomorphic to either S^1 or  <cit.> hence admits a (C^∞) smooth structure.
If n=2, then M admits a triangulation by Radó <cit.> hence a smooth structure by Richards <cit.>[One can avoid this later classification of Richards and instead use elementary (1920s) methods: the regular neighborhoods of simplices in the second barycentric subdivision are PL handles (see <cit.> for all n); each attaching map of a 1-handle or a 2-handle is isotopic to a smooth one.].
If n=3, then M admits a triangulation by Moïse <cit.> hence a smooth structure by Cairns <cit.>.
If n=4, then M admits a smooth structure by Freedman–Quinn <cit.>, since M is noncompact connected.
In any case, we may select a smooth structure on M.
There exists a Morse function f: M  [0,∞) with each i ∈ a regular value <cit.> and each M_i := f^-1[0,i) precompact <cit.>.
Furthermore, each M_i is a neighborhood deformation retract in M_i+1 <cit.>.

Now suppose n>4.
The topological manifold M admits a topological handlebody structure M_0⊂…⊂M_i⊂…, by Kirby–Siebenmann <cit.> if n>5 and by Quinn <cit.> if n=5.
That is, each M_i is open in M, the closure M_i is compact, the frontier M_i := M_i-M_i is a bicollared topological submanifold of M (that is, M_i is clean in M), and M_i+1 is the union of M_i and a handle.
So, since M_i is bicollared, each M_i is a neighborhood deformation retract in M_i+1.


An absolute neighborhood retract (with respect to the class of metric spaces, denoted ANR) is a metrizable space such that any closed embedding into a metric space admits a retraction of a neighborhood to the embedded image <cit.>.

A 1951 theorem of O Hanner has this nonseparable generalization <cit.>.


Topological manifolds are absolute neighborhood retracts.





§ EQUIVARIANT TOPOLOGICAL MANIFOLDS


A G-cofibration is a G-map with the G-homotopy extension property to any G-space.
Some G-homotopy commutative squares can be made strictly commutative.


Let G be a topological group.
Let A,B,C,D be (topological) G-spaces.
Let i: A  B be a G-cofibration.
Let f: A  C and g: B  D and h: C  D be (continuous) G-maps.
Suppose that g∘ i is G-homotopic to h ∘ f.
Then g is equivariantly homotopic to a G-map g': B  D such that g' ∘ i = h ∘ f.



Write H: A × [0,1]  D for the G-homotopy from g ∘ i to h ∘ f.
Since i: A  B is a G-cofibration, there exists a G-homotopy E: B × [0,1]  D from g to some g' such that E ∘ (i ×𝕀_[0,1]) = H.
Then g' ∘ i = H|(A ×{1}) = h ∘ f.


Recall G-cofibrant is equivalent to G-NDR for closed G-subsets <cit.>.


Let G be a topological group.
A G-space X is proper if, for the continuous map

    G × X  X × X  ;  (g,x) ⟼ (gx,x),

the image of closed is closed and the preimage of compact is compact <cit.>.



By a topological G-manifold, we mean a topological G-space M so, for each closed subgroup H of G, the H-fixed subspace is a topological manifold:

    M^H  := {x ∈ M  | ∀ g ∈ H : gx=x }.



Indeed the restriction of M^H being locally euclidean is not automatic <cit.>.

[Bing]
A C_2-action on ^4 exists with fixed set not a C^0-manifold.



Let M be a topological manifold with the action of a Lie group G.
For any prime p, the fixed set M^H is an _p-cohomology manifold if H ⩽ G is a finite p-group (Smith <cit.>) or toral group (Connor–Floyd <cit.>).


T Matumoto introduced the notion of a G-CW complex for any G <cit.>.


Let G be a compact Lie group.
Any topological G-manifold is G-homotopy equivalent to a G-equivariant countable CW complex.
Furthermore, if the manifold is compact, then the CW complex can be selected to be finite-dimensional.



Let M be a topological G-manifold.
By Lemma <ref>, each M^H is an ANR.
Also, by the Bredon–Floyd theorem, each compact set in M has only finitely many[An example with infinitely many orbit types is M=× S^1 with U_1-action u(n,z) := (n,u^n z).] conjugacy classes of isotropy group <cit.>.
Consider the increasing sequence {M_i}_i=0^∞ of Lemma <ref>.
We may assume that each open set M_i is G-invariant by replacement with its G-saturation G M_i; indeed, the compactness of G implies the compactness of M_i <cit.>.
Then each M_i, hence M_i, has finitely many orbit types.
Also, the open set M_i^H = M_i ∩ M^H in M^H is an ANR by Hanner <cit.>.
So, since M_i is a separable metrizable finite-dimensional locally compact space, by a criterion of Jaworowski <cit.>[Beware the G-Wojdysławski theorem therein is incorrect and must be amended by <cit.>.], we obtain that M_i is a G-ENR.
That is, there exist a closed G-embedding of M_i in a euclidean G-space (equipped with a smooth orthogonal representation of G), an open G-neighborhood U of M_i, and a G-retraction r: U  M_i.
By a theorem of Illman <cit.>, the smooth G-manifold U admits a G-CW structure, finite-dimensional and countable.

Write e: M_i  U for the G-inclusion.
By the G-version of Mather's trick <cit.>, the mapping torus (r ∘ e) is G-homotopy equivalent to (e ∘ r).
Note (r ∘ e) is G-homeomorphic to M_i × S^1 with trivial G-action on S^1.
By G-cellular approximation <cit.>, the G-map e ∘ r: U  U is G-homotopic to a cellular G-map c: U  U.
Then (e ∘ r) is G-homotopy equivalent to the finite-dimensional countable G-CW complex (c).
Thus M_i × S^1, hence the infinite cyclic cover M_i ×≃ M_i, is G-homotopy equivalent to a finite-dimensional countable G-CW complex K_i, namely the bi-infinite mapping telescope of c.

Thus, for each i, we obtain a G-homotopy equivalence f_i: M_i  K_i.
Choose a G-homotopy inverse f_i: K_i  M_i.
Write c_i: M_i  M_i+1 for the G-cofibrant inclusion.
By G-cellular approximation <cit.>, the composite f_i+1∘ c_i ∘f_i is G-homotopic to cellular G-map g_i : K_i  K_i+1.
Recursively by Lemma <ref> for each i, we may reselect f_i+1 up to G-homotopy such that f_i+1∘ c_i = g_i ∘ f_i holds.
These assemble to a G-homotopy equivalence of the mapping telescopes

    f: _i ∈ (M_i,c_i)  K := _i ∈ (K_i, g_i).

Thus, as the c_i are cofibrations, the topological G-manifold M = _i (M_i, c_i) is G-homotopy equivalent to the countable G-CW complex K (<cit.>).





Notice that, since the dimension of the euclidean space equivariantly containing M_i may be unbounded over all i, the CW complex K may not be finite-dimensional.




§ EXAMPLES AND RELATED RESULTS


[Bing]
Consider the double D := E ∪_A E of the closed exterior E in S^3 of the Alexander horned sphere A ≈ S^2.
This double has an obvious involution r_B that interchanges the two pieces and leaves the horned sphere fixed pointwise.
R H Bing showed that D is homeomorphic to the 3-sphere <cit.>.

By <ref>, (S^3,r_B) has the C_2-homotopy type of a finite-dimensional countable C_2-CW complex.


[Montgomery–Zippin]
Adaptation of Bing's ideas produces an involution r_MZ of S^3 with fixed set a wildly embedded circle <cit.>.
So (S^3,r_MZ) also has the C_2-homotopy type of a finite-dimensional countable C_2-CW complex.


[Lininger]
For each k ⩾ 3, there exist uncountably many inequivalent free U_1-actions on S^2k-1 with quotient not a C^0-manifold <cit.>.
Each has the U_1-homotopy type of a finite-dimensional countable U_1-CW complex.
Indeed, using <cit.> for mutation[His first step is C^0-existence of a product tube around a principal orbit <cit.>.] of S^2k-1 P_k keeps isotropy groups trivial.




The case of G finite in Theorem <ref> was proven by S Kwasik <cit.>.
However, in his first step, he implicitly assumed the continuity of the G-action, defined by g · f := (x ↦ f(g^-1 x)), on the Banach space B(X) of bounded continuous functions X equipped with the sup-norm.
This popular assumption was implicit in the infamous assertion of an equivariant version of Wojdysławski's theorem.
Non-equivariantly it states that the Kuratowski embedding, defined by x ⟼ (y ↦ d(x,y)), of a metrizable space X into B(X) <cit.>, has image closed in its convex hull, where d is any bounded metric on X <cit.>.
Nonetheless, Kwasik's assumption is indeed true and proven later by S Antonyan; it follows from <cit.> that it holds when G is a discrete group.
Continuity of the G-action on B(X) fails if G = U_1 <cit.>, a flaw in <cit.>.



Let  be a virtually torsionfree[Virtually torsionfree holds necessarily for all finitely generated < GL_n() by Selberg <cit.>.  Deligne gives an example of this property failing for a lattice  in a nonlinear Lie group <cit.>. I conjecture that this corollary is true if “virtually torsionfree” is replaced with “residually finite.”], discrete group.
Any topological -manifold with proper action has the -homotopy type of a countable -CW complex.
Moreover if the action is cocompact, then the CW complex can be finite-dimensional.



Using intersection with finitely many conjugates, there is a normal, torsionfree subgroup ^+ of .
Write G := /^+ for the finite quotient group.
Since the -action on the given topological -manifold M^+ is properly discontinuous and ^+ is torsion-free, M^+ is a regular ^+-covering space of the orbit space M := M^+/^+.
Thus, by the evenly covered property, M is a topological G-manifold.
Therefore, by Theorem <ref> (or just <cit.> amended by Remark <ref>), there is a G-homotopy equivalence f: X  M from a G-equivariant countable CW complex X.

Write X^+ := f^* M^+ and consider the pullback diagram of -spaces and -maps:

    X^+ @>f^+>> M^+
    
    @Vf^*qVV @VqVV
    
    X @>f>> M.

Since q is a regular ^+-covering map, the induced map f^*q given by inclusion-projection is also a regular ^+-covering map.
Then the covering space X^+ is a CW complex <cit.>.
This -invariant CW structure on X^+ is a countable -CW complex <cit.>.
The induced map f^+ is a -homotopy equivalence.


An application of Corollary <ref>: any such topological -manifold, whose fixed set of any finite subgroup is contractible, is a model for the classifying space E_; the -CW condition is assumed for the existence of classifying maps.
This corollary thus simplifies the fundamentally complicated proofs of <cit.> <cit.>.


Let G be a topological group.
A locally linear G-manifold <cit.> (`locally smooth' therein) is a proper G-space M such that, for each x ∈ M, there are a G_x-invariant subset x ∈ S ⊂ M and a homeomorphism S ^k equivariant with respect to an O(k)-representation of the isotropy group

    G_x  := { g ∈ G  |  gx=x }

such that the multiplication map G ×_G_x S  M is an open embedding.
Any smooth G-manifold is a locally linear G-manifold <cit.>.
M Freedman constructed a nonsmoothable free involution r_F on the 4-sphere (more in <cit.>).
Hence (S^4,r_F) is a locally linear C_2-manifold that is not a smooth C_2-manifold.
F Quinn constructed a locally linear G-disc Δ where G=C_15⋊_2 C_4 such that Δ does not contain a finite G-CW complex in its G-homotopy type <cit.>.
So for non-smooth actions there is no G-analog of the Borsuk Conjecture <cit.>.



Let G be any Lie group, and further assume that all actions are proper in the more specialized sense of Palais <cit.>.
E Elfving concluded that any locally linear G-manifold has the G-homotopy type of a G-CW complex <cit.>.
To contrast with Theorem <ref>, the input type of group is broader, but the input type of manifold is restricted, and the output type of CW complex less sharp.
Philosophically, for a homotopy-type result, one does not need the homeomorphism structure of locally linear tubes, but simply equivariant neighborhood retractions.






§ COUNTABILITY FROM SEPARABILITY


By a G-ANR we mean that a G-spacePalais-proper? such that every closed G-embedding into a G-space with G-invariant metric has its image a G-retract of a G-neighborhood.
A G-simplicial complex is countable means countably many G-simplices G/H ×Δ^k.

Antonyan–Elfving showed, for any locally compact group G, any metric G-ANR has the G-homotopy type of a G-CW complex <cit.>.
We rework their proof to provide a parallel result, one which has a restricted input to get a restricted output.


Let G be a locally compact Hausdorff group.
If a metric G-ANR is separable, then it is G-homotopy equivalent to a countable G-simplicial complex.


A special case is when G is a compact Lie group and was proven incorrectly by Murayama <cit.>.
We now sharpen Theorem <ref> to include countability.


Let G be a compact Lie group.
Any topological G-manifold with only finitely many G-orbit types has the G-homotopy type of a countable G-complex.


S Kwasik proved the case of G a finite group <cit.>, modulo Remark <ref>.


Let M be a topological G-manifold.Palais-proper automatic.
Recall that any topological manifold is an absolute neighborhood retract (ANR).citation of ANR fact?
Then each fixed subspace M^H is an ANR.
So, since M is paracompact metrizable, by a Jaworowski's criterion, M is a G-ANR <cit.>[The main proof therein implicitly uses a G-version of the ANR criterion <cit.>.].
Therefore, since M is separable, we are done by Theorem <ref>.


To begin, <cit.> is an equivariant generalization of Wojdysławski's properties of Kuratowski's embedding i: X  B(X), which was stated in Remark <ref>.
Therein the action of G on the set B(X) makes the function i equivariant.


Let G be a countably compact Hausdorff group.
Let X be a metric G-space.
There is a G-invariant closed linear subspace L ⊃ i(X) of B(X) on which the G-action is continuous, such that i(X) is closed in the L-convex hull.


Recall a G-Banach space is a G-linear space with a complete G-invariant norm.


Let G be a countably compact Hausdorff group.
Any separable metric G-space equivariantly embeds into a separable G-Banach space so that the embedded image is closed in its convex hull.



Let X be a separable metric G-space.
By Lemma <ref>, there is a G-Banach space L and a G-embedding i: X  L so that i(X) is closed in its convex hull.
There exists a countable dense subset S of X.
Write B for the closure of the linear span of i(S) in L.
Observe B is separable, contains i(X), and is G-invariant.



INCOMPLETE






§ APPLICATION OF THE HILBERT–SMITH CONJECTURE


An early attempt on Hilbert's fifth problem (1900) is this approximation <cit.>.


Any compact Hausdorff group is isomorphic to the filtered limit of an infinite tower of compact Lie groups and smooth epimorphisms.



We outline a later proof, as it is a beautiful interaction of algebra, analysis, and topology <cit.>. 
Haar (1933) showed that any locally compact Hausdorff group G admits a translation-invariant measure μ on its Borel σ-algebra.
Since G is compact, we take μ to be a probability measure.
Urysohn's lemma (1925) implies G is completely Hausdorff.
So on the Hilbert space := L^2(G,μ;) the G-action via pre-multiplication is faithful.
Peter–Weyl (1927) showed the sum of all finite-dimensional G-invariant -linear subspaces is dense in .
Thus each nonidentity element of G acts nontrivially on such a subspace of .

So, since G is compact, it follows that each neighborhood B of the identity e in G contains the kernel N of a finite product of (finite-dimensional) unitary representations.
That is, B ⊃ N ◃ G and G/N is monomorphic to a finite product of unitary groups.
So G/N is a Lie group (von Neumann, 1929).
If G is first-countable, then e has a countable decreasing basis {B_i}_i=0^∞, so we can also assume decreasing for the corresponding sequence {N_i}_i=0^∞ of closed normal subgroups of G with G/N_i a (compact) Lie group.
So, since ⋂_i=0^∞ N_i is trivial, the homomorphism G lim_i (G/N_i) is injective.
It is surjective, since for any {a_i N_i }_i=0^∞ in the limit, the intersection of the descending chain of the compact sets a_i N_i ⊂ G is nonempty.

Weil removed first-countability by limiting over a larger directed set <cit.>.
Consider the set  of finite subsets of G-{e} partially ordered by inclusion; note it is directed as I,J ∈ implies I ∪ J ∈.
For each e ≠ x ∈ G, our unitary representation M_x: G  U_d(x) satisfies M_x(x)≠𝕀; call the kernel N_x ◃ G.
For each I ∈, define M_I := ∏_x ∈ I M_x with kernel N_I = ⋂_x ∈ I N_x, so that again G/N_I  M_I(G) ⩽∏_x ∈ I U_d(x) is a compact Lie group.
Thus I ⟼ G/N_I is a functor from ^op to the category of compact Lie groups and smooth epimorphisms.
Since each e ≠ x ∉ N_x, the homomorphism q: G lim_I ∈ (G/N_I) is injective.
Since {a_I N_I}_I ∈ has finite-intersection property, q is surjective <cit.>.


I was unable to apply this tower to generalize Theorem <ref> to compact Hausdorff G.
However, under certain circumstances, G becomes Lie, so that Theorem <ref> applies.
If G is compact, by Lemma <ref>, G has no small subgroups if and only if it is Lie.
Notice, for any prime p, the additive group _p of p-adic integers is compact Hausdorff with arbitrarily small subgroups.
So G is Lie implies it contains no _p.


Let G be a locally compact group.
If G admits a faithful action on a connected topological manifold M, then G must be a Lie group.
In short, G is Lie if G ⩽(M).
Hilbert's fifth problem is the case M=G.


Partial affirmations exist if one assumes some sorts of regularity of the action.


Conjecture <ref> is true for G, if there exists a C^1-manifold structure on M and the action is by C^1-diffeomorphisms.


If G is compact, it is unworthy to use Theorem <ref> as the C^1-action becomes C^ω <cit.>.
Any C^1-map (continuous first partial derivatives) is locally Lipschitz.


Conjecture <ref> is true for G, if the action is by Lipschitz homeomorphisms with respect to a riemannian metric on smooth M.


Their argument shows the impossibility of G containing _p for any prime p, by otherwise establishing inequalities with Hausdorff dimension for various metrics.
The riemannian hypothesis is only used to obtain equality with covering dimension.

Later it was noticed that the Baire-category part of the argument could be done locally in a euclidean chart, averaging a metric on it over a small subgroup of _p.
In this situation, the new lemma is: M and M/_p have equal covering dimensions.

Both short proofs rely on C T Yang's 1960 result: _(M/_p)=_(M)+2.


Conjecture <ref> is true for G, if the action is by locally Lipschitz homeomorphisms on the connected topological manifold M.


Notice any self-homeomorphism is Lipschitz if the metric space M is compact.
The following consequence is pleasant because of no smoothness on G or the action.


Let G be a compact group.
Let M be a topological G-manifold.
Suppose the action is faithful and that M is compact and connected.
Then M has the G-homotopy type of a G-equivariant finite-dimensional countable CW complex.


To such (M,G), including Examples <ref>–<ref> or others with C^0 dynamics, one can now apply equivariant homology, with variable coefficients in abelian groups (Bredon–Matumoto <cit.>) or more generally in spectra (Davis–Lück <cit.>).
Therefore, inductively calculable invariants of these wild objects are now available.





 §.§ Acknowledgements


Frank Connolly (U Notre Dame) provided early motivation (<ref>).
Craig Guilbault (U Wisconsin–Milwaukee) reminded me topological manifolds have handles (<ref>).
The following brief but fruitful discussions happened in June 2017, at Wolfgang Lück's 60th birthday conference (U Münster).
Wolfgang Steimle (U Augsburg) assured me that strictification of homotopy commutative diagrams is easy for squares (<ref>).
Elmar Vogt (Freie U Berlin) excited me further into classical topology (ANRs and Lemma <ref>).
Finally, individually Jim Davis (Indiana U), Nigel Higson (Penn State U), Wolfgang Lück (U Bonn), and Shmuel Weinberger (U Chicago) encouraged me toward the Hilbert–Smith Conjecture (<ref>).

The referee corrected some exposition and recommended Examples <ref> & <ref>.
Chris Connell (Indiana U) was helpful in discussing Hausdorff dimension (<ref>, <ref>).


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