The homotopy groups of the η-periodic motivic sphere spectrum
Kyle Ormsby Oliver Röndigs




Reed College
ormsbyk@reed.edu



Universität Osnabrück
oroendig@uni-osnabrueck.de

K.O. was partially supported by NSF grant DMS-1709302; O.R. was partially supported by the DFG SPP 1786.


We compute the homotopy groups of the η-periodic motivic sphere spectrum over a field  of finite cohomological dimension with characteristic not 2 and in which -1 a sum of four squares.  We also study the general characteristic 0 case and show that the α_1-periodic slice spectral sequence over  determines the α_1-periodic slice spectral sequence over all extensions /.  This leads to a speculation on the role of a “connective Witt-theoretic J-spectrum” in η-periodic motivic homotopy theory.


The homotopy groups of the η-periodic motivic sphere spectrum
    Oliver Röndigs
    Received: date / Revised version: date
=============================================================





§ INTRODUCTION


The motivic sphere spectrum  is the unit object in the tensor triangulated stable homotopy category of motivic spectra ((),∧) over a field .  In this category, both the simplicial circle S^1 and the punctured affine line Å^1∖ 0 are ∧-invertible, so it is crucial that we understand the bigraded homotopy groups π_⋆ := ⊕_m,n∈π_m+nα where π_m+nα := [(S^1)^∧ m∧ (Å^1∖ 0)^∧ n,]_().  See the introduction to <cit.> for a more complete discussion of the importance of this ring.

The motivic Hopf map η∈π_α represented by the canonical _m-torsor Å^2∖ 0→^1 plays an especially important role in π_⋆.  This class is non-nilpotent over all fields <cit.> and thus represents a first example of exotic behavior in π_⋆, differentiating it from the classical stable stems.  (Recall that η^4=0 classically, and that the Nishida Nilpotence Theorem <cit.> tells us that all classes of nonzero degree in the classical stable stems are nilpotent.)  Let

    η^-1 := hocolim(Σ^-αΣ^-2αΣ^-3α⋯)

denote the η-periodic sphere spectrum.[Other authors have referred to this object as the η-local or η-inverted sphere.  We have chosen our terminology to match the language of classical v_n-periodic homotopy theory, which seems appropriate given the emerging role of η in motivic nilpotence and periodicity <cit.>.]  We have π_⋆η^-1≅η^-1π_⋆ (where the latter term represents the localization of the ring π_⋆ at the multiplicative set {η,η^2,…}), so inverting η annihilates Γ_η := {x∈π_⋆| xη^N=0 for some N} and induces an injection π_⋆/Γ_η→π_⋆η^-1.  

A number of authors have studied π_⋆η^-1 over particular fields, including M. Andrews and H. Miller <cit.> over , B. Guillou and D. Isaksen <cit.> over  and , and G. Wilson <cit.> over finite fields, local fields, and .  Over , π_m+nα≅ W()≅/2 for nonnegative m congruent to 0 or 3 mod 4, whereas more complicated “image of J”-style patterns occur in π_⋆η^-12 (the bigraded homotopy groups of the η-periodic 2-complete sphere) over  and .  These authors work with either the motivic Adams-Novikov or motivic Adams spectral sequence in order to produce their results.  In addition to these results, Röndigs <cit.> has shown that π_1η^-1=π_2η^-1=0 over all fields of characteristic different from 2.

In this note, we use the α_1-periodic slice spectral sequence to completely determine π_⋆η^-1 over finite-cohomological dimension fields with characteristic different from 2 in which -1 as a sum of four squares.[The smallest r such that -1 is a sum of r squares in  is called the level of  and is often denoted s(). (The s is for Stufe).  By a theorem of Pfister <cit.>, s() is always a power of 2.  See <cit.> for examples of fields of various levels.]  Let W() denote the Witt ring of quadratic forms over  modulo the hyperbolic plane.

[see thm:main]
Let  be a field of finite cohomological dimension with characteristic not 2.  If -1 is a sum of four squares in , then

    π_⋆η^-1≅ W()[η^± 1,σ,μ]/(σ^2)

where |σ|=3+4α and |μ| = 4+5α.  In particular, the bigraded homotopy groups of η^-1 are

    π_m+nαη^-1≅
        W()   if m≥ 0 and m≡ 0 or 34,
    
        0   otherwise.



In cor:nonperiodic (see also <cit.>), we see that for fields satisfying the same hypotheses, π_m+nα≅π_m+nαη^-1 for 2n≥max{3m+5,4m}, so we have also computed a bi-infinite range of homotopy groups of the motivic sphere spectrum.

The picture is less clear for fields of characteristic 0 with cohomological dimension greater than 2, but we are able to produce some partial results in sec:char0.  Let α_1^-1() denote the α_1-periodic slice spectral sequence over .  In thm:profile, we show that α_1^-1() determines α_1^-1() for any field extension /.  This leads to a conjecture on the differentials in α_1^-1() and some speculations regarding the structure of π_⋆η^-1 in general.



 §.§ Acknowledgments

The first named author thanks Paul Arne Østvær, Glen Wilson, and participants at the conference Motivic Homotopy Groups of Spheres III who gave particularly helpful feedback during an early stage of this project.  Both authors thank Tom Bachmann for a helpful discussion regarding conj:jw.

The spectral sequence diagrams in this paper were produced with Hood Chatham's  package https://www.ctan.org/pkg/spectralsequences.




§ THE Α_1-PERIODIC SLICE SPECTRAL SEQUENCE


In this section, we set up the α_1-periodic slice spectral sequence and discuss its convergence properties and first two pages over a general field of characteristic different from 2.

We refer to <cit.> for the setup of the slice spectral sequence, and only briefly recall some pertinent facts here.  The sphere spectrum  is effective and thus has a slice tower of the form

    ⋯   f_3   f_2   f_1   f_0=
       s_3   s_2   s_1   s_0

where f_q is the q-th effective cover of  and s_q is the q-th slice of .  Associated with this tower is the slice spectral sequence  with E_1-page _1^q,m+nα = π_m+nαs_q.  (If we need to refer to the base field, then we will denote this spectral sequence ().)  The differentials take the form d_r:_r^q,m+nα→_r^q+r,m-1+nα.

The following theorem states the basic convergence properties of ; it is a concatenation of <cit.> and <cit.>. Recall that the η-complete sphere spectrum is η := holim/η^n where /η^n is the cofiber of η^n:Σ^nα→.


The slice spectral sequence for  conditionally converges to π_⋆η.  Moreover, if k<∞, then η≃ and the slice spectral sequence for  converges conditionally to π_⋆.


We also have control over _1 via the following slice computation.  We let _MU_*MU^s,t denote the cohomology of the MU-Hopf algebroid in cohomological degree s and internal degree t.  (Recall that this is the E_2-page of the Novikov, i.e., MU-Adams, spectral sequence from topology.)


The q-th slice of the motivic sphere spectrum is

    s_q≃⋁_s≥ 0Σ^q-s+qαH_MU_*MU^s,2q

at least after inverting the exponential characteristic of the
base field.


We refer to <cit.> for basic facts about _MU_*MU, and we use its naming conventions for elements.  Importantly, there is a single nonzero class α_1∈_MU_*MU^1,2 that represents η.  Multiplication by α_1 induces a map of spectral sequences α_1:^q,m+nα→^q+1,m+(n+1)α.  Taking the colimit of the tower given by iterating this map produces the α_1-periodic slice spectral sequence, α_1^-1.  We will analyze the target and convergence properties of α_1^-1 momentarily, but it certainly appears that this construction ought to say something about η^-1 or a related object.

There is another obvious spectral sequence we could consider, namely the slice spectral sequence for η^-1, but it turns out that the two spectral sequences are the same.  For a motivic spectrum X, recall from <cit.> that (X) is the slice completion of X.


The slice spectral sequence for η^-1 and α_1^-1 are isomorphic as spectral sequences.  They both have first page additively isomorphic to π_⋆ H_2[α_1^± 1,α_3,α_4]/(α_4^2) and conditionally converge to π_⋆(η^-1) in the sense of <cit.>.  If <∞, then (η^-1)≃η^-1 and the isomorphic spectral sequences conditionally converge to π_⋆η^-1.



In cor:fdconv and sec:comp, we will see that convergence is in fact strong if  has odd characteristic or _2<∞.  In sec:char0, we will see that convergence over characteristic 0 fields to π_⋆(η^-1) is strong.



As a ring object, s_*η^-1 is not an H_2-algebra <cit.>, and our identification of α_1^-1_1 in thm:etaSlices is not multiplicative.  By a bi-degree argument and the general properties of slice multiplicativity given in <cit.>, the multiplication on α_1^-1_1 agrees with the “naive” multiplication up to addition of some terms involving ^1.  Our determination of α_1^-1_2 does not depend on the precise multiplicative structure, and we will see in thm:d1 that the multiplication on α_1^-1_2 is fairly simple.



Let E denote the slice spectral sequence for η^-1.  Then

    E_1≅π_⋆ H_2[α_1^± 1,α_3,α_4]/(α_4^2)

by <cit.>.[Although thm:slices holds
after inverting the exponential characteristic, the slices s_*η^-1
are known without inverting the exponential characteristic. The reason is that
if k is a field of odd characteristic p, then multiplication with p
is an isomorphism on the Witt ring of k, and hence on η^-1.] In particular, the canonical map → E takes α_1 to a unit and hence induces a map α_1^-1→ E.  By <cit.>, α_1^-1_MU_*MU≅_2[α_1^± 1,α_3,α_4]/(α_4^2).  Given this result and the form of _1 in thm:slices, we conclude that α_1^-1_1→ E_1 is an isomorphism, and it follows that α_1^-1≅ E.

Our first convergence statement is formal given the construction of the slice spectral sequence (see <cit.>).  For the second convergence statement, thm:slices tells us that ()≃ when <∞.  Given the conditional convergence conditions of <cit.>, our result follows as long as the sequential colimit that inverts α_1 commutes with the limit defining slice completion.  Our assumption on cohomological dimension implies a vanishing line parallel to α_1-multiplication, and hence the limit in question is finite and commutes with sequential colimits.


This leads us to the main theorem of this section, a determination of the first slice differentials and α_1^-1_2:


Over any field  of characteristic different from 2, the first slice differential for η^-1 has the form

    [      ^2       0       τ       0       0       0       0       ⋯;       0       0       0       0       0       0       0       ⋯;    ^3^1       0  ^2+ρ^1       0       0       0       0       ⋯;       0       0       0      ^2       0       τ       0       ⋯;       0       0    ^3^1 ^2^1+^3      ^2       ρ       τ       ⋯;       0       0       0    ^3^1       0  ^2+ρ^1       0       ⋯;       0       0       0       0    ^3^1 ^2^1+^3  ^2+ρ^1       ⋯;       0       0       0       0       0    ^3^1       0       ⋯;       0       0       0       0       0       0    ^3^1       ⋯;       ⋮       ⋮       ⋮       ⋮       ⋮       ⋮       ⋮       ⋱ ].

Here the i-th column, i≥ 0, gives the first slice differential restricted to the summand Σ^i+qαH_2 of s_qη^-1 (the summand is 0 if i=1).  The j-th row, j≥ 0, describes the incoming first slice differential for the summand Σ^j+1+(q+1)αH_2 of Σ s_q+1η^-1 (the summand is 0 if j=1).

This results in an isomorphism of k^M_*-algebras

    α_1^-1_2 ≅ k^M_*[α_1^± 1,α_4,α_5]/(α_4^2)

where |α_4| = (4,3+4α) and α_5 = α_3^2α_1^-1 has degree (5,4+5α).



The factor of α_1^-1 in the definition of α_5 is not strictly necessary, but is there so that α̅_5↦α_5 under the localization map →α_1^-1.



The determination
of the first slice differential for η^-1 complements the
occurrences of multiplications with τ, which were used on p.11
of [17] to deduce vanishing columns in the in the Andrews-Miller range 
of the unlocalized slice spectral sequence.


The pattern of differentials indicated by thm:d1 is represented graphically in fig:E1.  The form of α_1^-1_2 also implies an important convergence result, which we state presently.




If _2()=r<∞, then α_1^-1() collapses at its (r+1)-th page and converges strongly to π_⋆(η^-1).


The form of α_1^-1_2 (which is presented graphically in fig:E2) and the fact that k^M_>r()=0 imply that d_>r=0.  This collapse along with the conditional convergence of thm:sliceCond imply the strong convergence portion of the corollary.


Our proof of thm:d1 requires a fascinating detour through connective Witt K-theory.  Let  denote the motivic spectrum representing Hermitian K-theory,[The Q is for Quadratic.] let = η^-1 denote the η-periodization of , which is the motivic spectrum representing Balmer's higher Witt groups, and let  denote the connective cover of  (in the sense of Morel's homotopy t-structure).

The slices of , the d_1-differentials of its slice spectral sequence, and the effect of s_* on the unit map η^-1→ are known.


Suppose the base scheme S is a scheme over ([1/2]).  Then

    s_*≃ H[η^± 1,√(β)]/(2η,2√(β),η^2√(β))

where |η| = α, |√(β)| = 2+2α, and the first slice differential takes the form

    [     ^2      0      τ      0      0      0      0      ⋯;      0      0      0      0      0      0      0      ⋯;   ^3^1      0 ^2+ρ^1      0      0      0      0      ⋯;      0      0      0      0      0      0      0      ⋯;      0      0   ^3^1      0     ^2      0      τ      ⋯;      0      0      0      0      0      0      0      ⋯;      0      0      0      0   ^3^1      0 ^2+ρ^1      ⋯;      0      0      0      0      0      0      0      ⋯;      0      0      0      0      0      0   ^3^1      ⋯;      ⋮      ⋮      ⋮      ⋮      ⋮      ⋮      ⋮      ⋱ ]

(with the same conventions as thm:d1).  Moreover, there is a splitting of s_0η^-1 such that the unit η^-1→ induces an inclusion on every even summand, and ^1 on every odd summand.



  The description of the slices, as well as their multiplicative
  structure, is given in <cit.>. The
  behaviour of the unit map follows from <cit.>.


Note that s_* and the pattern of d_1 differentials is precisely the black portion of fig:E1.  The remaining portion of s_*η^-1S (the red part of fig:E1) is handled by the following theorem.


Over =, there is a unique homotopy class σ_∞:Σ^3 →η^-1 inducing an isomorphism on Π_3.  This map induces 
(1,^1) on every summand of a slice.



Fix =.  Recall from <cit.> that there is a cell presentation of  over  of the following form.  Namely, there is a sequence of cellular motivic spectra factoring the unit of  as

    η^-1 = _1_2⋯→_n⋯→

such that for every n the map _n→ is (4n-1)-connective and the composition η^-1→_n→ induces isomorphisms on Π_4k.  For every n≥ 1, there is a unique nontrivial class a_n:Σ^4n-1η^-1→_n in π_4n-1_n≅π_4n-1η^-1 such that

    Σ^4n-1η^-1_n_n+1Σ^4nη^-1

is a homotopy cofiber sequence with c_n inducing an isomorphism on Π_4k+3 whenever k≥ n.  Taking the colimit as n→∞ gives a cell presentation of .

We now construct σ_∞.  Consider the map σ_1=η^-4σ:Σ^3η^-1 = Σ^3_1→η^-1.  Assume for induction that for some n≥ 1 a map σ_n:Σ^3_n→η^-1 is given such that


  * σ_n i_n-1 = σ_n-1 and

  * [Σ^4a_n-1,η^-1]:[Σ^4_n-1,η^-1]→ [Σ^4n-1η^-1,η^-1] is an isomorphism.

Then the cofiber sequence above induces a long exact sequence

    [Σ^4n+2η^-1,η^-1] [Σ^3 _n, η^-1] [Σ_3_n+1,η^-1] [Σ^4n+3η^-1,η^-1] [Σ^4 _n, η^-1]←⋯.


The Andrews-Miller theorem on π_⋆η^-1 <cit.> implies

    0 = [Σ^4n+2,η^-1] = [Σ^4n+2η^-1,η^-1],

showing that σ_n lifts to a map σ_n+1 such that σ_n+1i_n=σ_n.

Now note that assumption (2) implies that

    [Σ^4 c_n-1,η^-1]:[Σ^4nη^-1,η^-1]→ [Σ^4_n,η^-1]

is surjective; furthermore, the composition

    [Σ^4nη^-1,η^-1] [Σ^4_n,η^-1] [Σ^4n+3η^-1,η^-1]

is the map sending η^-5nμ_9^n to η^-5n-4σμ_9^n, hence an isomorphism.  It follows that the map [Σ^4a_n,η^-1] is an isomorphism, as desired.  (In fact, we also get that there is a unique σ_n+1:Σ^3_n+1→η^-1 such that σ_n+1i_n=σ_n.)

Induction and the universal property of colimits now produces a map σ_∞:Σ^3_∞≃Σ^3→η^-1 sending 1∈π_3Σ^3 to η^-4σ∈π_3η^-1.  The uniqueness of σ_∞ follows from the Milnor exact sequence and the vanishing of lim_n^1 [Σ^4_n,η^-1] (every group [Σ^4_n,η^-1] beging finite of order 2).

Since σ_∞ is a map of η^-1-modules, it induces isomorphisms on Π_4m+3 for every integer m.  The statement on slices follows from
the behaviour of the unit map η^-1→ on slices given
in thm:kw_slice.



As η^-1 and s_*η^-1 are invariant under base change, it suffices to determine the first slice differential d_1 over [1/2]. 
On a summand Σ^nH_2, it is of the form

    (x_nτ,a_nτ^1+β_n,b_n^2+γ_n,c_n^2^1+d_n^3,e_n^3^1)

with x_n,a_n,b_n,c_n,d_n,e_n elements in /2, and 
β_n,γ_n square classes
of units in [1/2]. 
The behavior of the unit map η^-1→ on slices from 
thm:kw_slice provides immediate restrictions:

    x_4n   =0,    a_4nρ+β_4n   =γ_4n,    b_4n   =1,    e_4n   =1+c_4n,
     
    x_4n+1+a_4n+1   =1,     (a_4n+1+1)ρ   =β_4n+1,     c_4n+1   =1,     b_4n+1   =d_4n+1,
     
     x_4n+2   =1,     (a_4n+2+1)ρ+β_4n+2   =γ_4n+2,     b_4n+2   =1,     e_4n+2   =1+c_4n+2,
     
     x_4n+3   =a_4n+3,     a_4n+3ρ   =β_4n+3,     c_4n+3   =1,     b_4n+3   =d_4n+3.

The map σ_∞:Σ^3_→η^-1_ on 
slices from thm:sigma_infty imposes further restrictions:

    x_4n+1   =1,    a_4n+1   =0,    b_4n+1   =c_4n+1=d_4n+1=1, 
    e_4n+1   =1+d_4n+2,
       x_4n+3   =0=a_4n+3,    b_4n+3   =c_4n+3=d_4n+3=1,    e_4n+3   =1+d_4n+4.

Base change and the previous equations then provide the following equations:

    β_4n+1=ρ,   γ_4n+1+β_4n+2=ρ,   β_4n+3= 
    γ_4n+3=γ_4n =β_4n=β_4n+2=0.

Since the composition d_1∘ d_1=0, Adem relations
imply further coefficients. Considering the component

    d_1^2Σ^4n+2H_2 →Σ^4n+1+2αH_2

implies that a_4n+2=a_4n=0. 
Considering the component

    d_1^2Σ^4nH_2 →Σ^4n+2+2αH_2

provides that c_4n=d_4n=0, and hence e_4n=1=e_4n+3. 
The similar component 

    d_1^2Σ^4n+2H_2 →Σ^4n+4+2αH_2

gives only c_4n+2=d_4n+2, and hence e_4n+1=e_4n+2.
Resorting to s_∗ provides the solution c_4n+2=d_4n+2=0
and e_4n+1=e_4n+2=1 as follows. Consider the summand
Σ^2n+(2n+1)αH_2 in s_2n+1 generated by α_2n+1. 
The first slice differential maps it via inc^2_y_2n+2^2^1
to the top degree summand Σ^2n+1+(2n+2)α H/(y_2n+2) in
s_2n+2 by <cit.>; here y_2n+2 is the
order of a cyclic group and divisible by four.
The map 

    Σ^αΣ^2n+1+(2n+2)α H/(y_2n+2) 
    →Σ^2n+1+(2n+3)α H_2

induced by η is the projection pr^y_2n+2_2. Hence
already after one multiplication with η, the 
degree 2+α part of the first differential is zero on that
summand. It follows 
that c_2n=0. 

Given the form of the differentials, the additive calculation of α_1^-1_2 is nearly the same as the proof of <cit.>.  The exotic multiplication on α_1^-1_1 mentioned in rmk:mult reduces to k^M_*[α_1^± 1,α_4,α_5]/(α_4^2) in the subquotient α_1^-1_2 since ^1 is only potentially nonzero on terms involving an odd power of τ, and there are no τ's in α_1^-1_2.




§ COMPUTATIONS FOR FIELDS WITH ODD CHARACTERISTIC OR COHOMOLOGICAL DIMENSION AT MOST 1


Given the form of α_1^-1_2 and the spectral sequence's convergence properties determined in the previous section, we can now make short work of the following computations.


If _2≤ 1, then the α_1-periodic slice spectral sequence for  collapses with α_1^-1_2=α_1^-1_∞ and converges strongly to π_⋆η^-1.


This is a specialization of cor:fdconv.





If  has odd characteristic, then the α_1-periodic slice spectral sequence for  collapses with α_1^-1_2=α_1^-1_∞.


Suppose  has characteristic p and write i for the map →_p.  The essentially smooth base change functor i^* induces a map of spectral sequences i^*:α_1^-1_2(_p)→α_1^-1_2() which is given by the extension of scalars map on k^M_* and the identity on α_i for i=1,4,5.  Given the form of α_1^-1_2(), it suffices to show that d_r^α_5=0 for all r≥ 2, but d_r^α_5 = i^*d_r^_pα_5, and d_r^_pα_5=0 by thm:collapse.  


At this point, we know that if  has odd characteristic or if _2 ≤ 1, then the α_1-periodic slice spectral sequence collapses with α_1^-1_2=α_1^-1_∞≅ k^M_*[α_1^± 1,α_4,α_5]/α_4^2.  Paired with the conditional convergence portion of thm:etaSlices, this implies that the spectral sequence in fact converges strongly to π_⋆η^-1. In order to completely determine π_⋆η^-1 for such , we must resolve extension problems and understand the multiplicative structure. 

Suppose that s≡ 0 or 3 4, and consider the short exact sequences

    0 → f_q+1π_sη^-1→ f_qπ_sη^-1→ k^M_q → 0

obtained from the slice filtration and the determination of α_1^-1_∞
for a field of odd characteristic.
Choose a lift g_s∈ f_0π_sη^-1=π_sη^-1 
of the nontrivial element in k^M_0, compatible with field extensions
from the prime field. If s=0,
π_0η^-1 is known to be the Witt ring by Morel's theorem,
and g_0 should be chosen as the unit. The slice filtration
on π_0η^-1 coincides with the filtration by powers
of the fundamental ideal I, as one deduces for example 
from <cit.>. The
multiplicative structure on the slice filtration then 
supplies a natural
transformation to the sequence (<ref>)
from the short exact sequence

    0 → I^q+1→ I^q → h^q,q→ 0

solving Milnor's conjecture on quadratic forms <cit.>.
The convergence statement cor:fdconv shows that this
natural transformation is an isomorphism for fields of
finite cohomological dimension. Since the constructions
involved commute with filtered colimits of fields, it is
thus an isomorphism for any field of odd characteristic.
In particular, the slice filtration is Hausdorff by
the main result of <cit.>.


If  has odd characteristic or if _2≤ 1, then, as a ring,

    π_⋆(η^-1) ≅ W()[η^± 1,σ,μ]/(σ^2)

where |η|=α, |σ|=3+4α, and |μ|=4+5α.  If additionally <∞, then (η^-1)≃η^-1 and this is a computation of the η-periodic homotopy groups of the motivic sphere spectrum.


The additive structure (which is simply a copy of W() in nonnegative simiplicial degrees congruent to 0 or 3 mod 4) follows from the above filtration considerations.  There is no room for hidden extensions, so the result follows.




§ CHARACTERISTIC 0 FIELDS


We now consider the α_1-periodic slice spectral sequence over a general field  of characteristic 0.  We prove that for any  this spectral sequence converges strongly to π_⋆(η^-1).  Moreover, the spectral sequence over  completely determines the spectral sequence over  in a manner that we make precise in thm:profile.  This allows us to extend the conclusion of prop:main to fields with _2≤ 2 and to extensions of (√(-1)), resulting in thm:main.  We conclude with a conjectural description of the differentials which we hope will inspire further work on this problem.

The structure of our argument is somewhat surprising.  After proving that d_2^=0, we are able to put strong restrictions on the form of the differentials which may appear in α_1^-1().  We then employ a theorem of Orlov-Vishik-Voevodsky <cit.> to show that for arbitrary /, the differentials in α_1^-1() are of the same form.  The proscribed form of the differentials guarantees that Boardman's RE_∞=0, whence strong convergence follows.  The primary obstruction to computing the differentials seems to be the lack of a good description of (η^-1).

We make some preliminary definitions in order to start our arguments.  Recall that α_1^-1_1≅π_⋆ H_2[α_1^± 1,α_3,α_4]/α_4^2.  For k≥ 0, set α_2k+1:=α_3^kα_1^1-k, and for k≥ 2 set α_2k:=α_4α_3^k-2α_1^2-k.  These classes are chosen so that α̅_ℓ↦α_ℓ under the localization map _1→α_1^-1_1 for all ℓ 2.  Note that as a π_⋆ H_2[α_1^± 1]-module, α_1^-1_1 is generated by 1,α_3,α_4,α_5,α_6,….  Also note (for the purposes of applying the Leibniz rule) that, up to multiplication by a unit, α_4k+1 is the square of α_2k+1.


The d_2 differential in α_1^-1() is trivial.


It suffices to prove that d_2^α_5=0.  We know that d_2^α_5∈ k^M_2(){α_1^3α_4}.  Base change to _p provides a comparison map α_1^-1()→α_1^-1(_p).  Since _2(_p)=2, cor:fdconv implies that α_1^-1(_p) converges strongly to π_⋆η^-1.  Furthermore, every class in k^M_2() is detected in some k^M_2(_p).[Indeed, <cit.> tells us that the map k^M_2()→ k^M_2()⊕⊕_p k^M_2(_p) is injective and computed on components by quadratic Hilbert symbols.  Hilbert reciprocity then implies that k^M_2()→⊕_p k^M_2(_p) is injective as well.]  As such, the computations of Wilson <cit.> over _p imply that d_2^α_5=0.



There is a nondecreasing[In fact, the sequence is strictly increasing unless it is eventually constant at ∞.] sequence of extended integers r_k∈_≥ 3∪{∞} for k≥ 2 such that if r_k<∞ then d_r_k^α_2^k+1 = ρ^r_kα_2^kα_1^r_k+1, and if r_k=∞ then α_2^k+1 is a permanent cycle in α_1^-1().  The rest of the differentials in α_1^-1() are determined by the Leibniz rule.



The above theorem may be thought of in the following terms.  In the weight n α_1-periodic slice spectral sequence, the 4k-column of α_1^-1_2 is, up to multiplication by some power of the unit α_1, generated by α_4k+1, and the 4k-1-column is generated by α_4k in the same sense.  These columns are connected by d_r_ν_2(4k) = ·ρ^r_ν_2(4k) on k^M_*, where ν_2 is 2-adic valuation.



By lemma:d2 and thm:d1, α_1^-1_3≅ k^M_*[α_1^± 1,α_4,α_5]/α_4^2.  If the spectral sequence does not collapse, then the first nonzero differential is necessarily of the form d_rα_5 = xα_4α_1^r+1 for some r≥ 3 and x∈ k^M_r().  Since k^M_r() = /2{ρ^r} for r≥ 3, we in fact have d_rα_5=ρ^rα_4α_1^r+1.  Set r_2 equal to this r.  The α_1^-1_r_2+1-page then has k^M_*[α_1^± 1]/ρ^r_2 in positive stems congruent to 3 mod 8 and _ρ^r_2k^M_*[α_1^± 1] in positive stems congruent to 4 mod 8 (where _xk^M_* = {y∈ k^M_*| xy=0} is the x-torsion in k^M_*); the (r_2+1)-page also continues to have k^M_*[α_1^± 1] in nonnegative stems congruent to 0 or 7 mod 8, and is 0 otherwise.

The potential targets of the _ρ^r_2k^M_*[α_1^± 1] terms are all 0, hence these classes are permanent.  Thus the next nonzero differential in the spectral sequence (if one exists) is necessarily of the form d_r_3α_9 = ρ^r_3α_8α_1^r_3+1.  The ρ^r_3-torsion terms in the (r_3+1)-page are again permanent, and the next differential is of the form d_r_4α_2^4+1=ρ^r_4α_2^4α_1^r_4+1.  Proceeding inductively proves the theorem.


We now abstract the behavior observed in thm:Qprofile and show that it is in fact generic.


For a given field , suppose that there is a nondecreasing sequence of extended integers r_k∈_≥ 3∪{∞} for k≥ 2 such that the differentials d_r_kα_2^k+1=ρ^r_kα_2^kα_1^r_k+1 and the Leibniz rule determine α_1^-1().  In this case, we call {r_2,r_3,…} the profile of α_1^-1() and say that α_1^-1() is determined by the profile {r_k}.
  


Let {r_k} denote the profile of α_1^-1() (guaranteed to exist by thm:Qprofile).  Then for any characteristic 0 field , α_1^-1() is also determined by the profile {r_k}.



Consider the map of map of spectral sequences i^*:α_1^-1()→α_1^-1() induced by essentially smooth base change along →.  We have

    d_r_k^α_2^k+1 = i^*d_r_k^α_2^k+1 = i^*ρ^r_kα_2^kα_1^r_k+1 = ρ^r_kα_2^kα_1^r_k+1.


It remains to show that _ρ^r_kk^M_*[α_1^± 1]{α_2^k+1} supports no higher differentials. Invoking <cit.>, we see that _ρ^r_kk^M_* is generated in degree 1 as a k^M_*-module, so it suffices to show that d_r^ [u]α_2^k+1=0 for all r>r_k and u∈^× such that [u]ρ^r_k=0∈ k^M_r_k+1.  Fix such a u and consider the subextension /(u)/.  Let j:(u)→ denote the corresponding map with associated map of spectral sequences j^*:α_1^-1((u))→α_1^-1().  Our argument now splits into two cases: u algebraic, and u transcendental.

First suppose that u is algebraic, in which case (u) is a number field.  Tate's theorem <cit.> implies that k^M_n((u)) = /2{ρ^n} or 0 for n≥ 3 and we have already seen that d_r_k^(u)α_2^k+1 = ρ^r_kα_2^kα_1^r_k+1.  Recall that r_k≥ 3, so this differential kills k^M_* classes at and above degree r_k.  In particular, for r>r_k the target group for d_r^(u)[u]α_2^k+1 is 0 and hence the differential is 0.  Finally, we see that

    d_r^[u]α_2^k+1 = j^*d_r^(u)[u]α_2^k+1 = 0

as well, as desired.

Now suppose that u is transcendental, in which case <cit.> implies that there is a split short exact sequence

    0→ k^M_*→ k^M_*(u)⊕_π k^M_*-1[u]/(π)→ 0

where π ranges over monic irreducible polynomials in [u] and ∂_π:k^M_*→ k^M_*-1[u]/(π) is the residue map taking [π,u_2,u_3,…,u_n] to [u_2,…,u_n].  In particular, for n≥ 4, k^M_n(u) has _2-basis consisting of ρ^n and [π]ρ^n-1 for π∈[u] monic irreducible.  Thus the differential d_r_k^(u)α_2^k+1=ρ^r_kα_2^kα_1^r_k+1 kills k^M_*((u))[α_1^± 1]{α_2^k} in Milnor-degree r_k+1 and above.  It follows that d_r^(u)[u]α_2^k+1=0 for r>r_k and the same base change trick as in the previous paragraph implies that d_r^ [u]α_2^k+1=0.  We conclude that α_1^-1() is determined by the profile {r_k}.



Let  be any field of characteristic different from 2.  Then α_1^-1() converges strongly to π_⋆(η^-1).  If <∞, this target is isomorphic to π_⋆η^-1.


We have already verified this result for odd characteristic fields and fields with finite cohomological dimension.  It remains to check characteristic 0 fields of arbitrary cohomological dimension.  By thm:etaSlices, we have weak convergence to (η^-1), so it suffices to check vanishing of Boardman's RE_∞ term (for E=α_1^-1()).  By <cit.>, it in turn suffices to check that for each tri-degree (s,m+nα) there are at most finitely many nonzero differentials d_r:α_1^-1_r^s,m+nα→α_1^-1_r^s+r,m-1+nα.  By thm:profile, α_1^-1() has profile {r_k} where {r_k} is the profile of α_1^-1().  In particular, the finiteness condition on nonzero differentials is met, and we may conclude that we indeed have strong convergence.



Suppose  is a field of characteristic 0 which has profile {r_k}.  Let ν_2 denote 2-adic valuation.  If r_k<∞ for all , then

    π_m (η^-1) ≅
        W()   if m=0;
    
        W()/2^r_k   if m>0, m=4ℓ-1, and k=ν_2(4ℓ);
    _2^r_kW()   if m>0, m=4ℓ, and k=ν_2(4ℓ);
    
        0    otherwise.

If {r_k} eventually takes the value ∞ with first instance r_K=∞, then

    π_m (η^-1) ≅
        W()   if m=0;
    
        W()/2^r_k   if m>0, m=4ℓ-1, and k=ν_2(4ℓ)<K;
    _2^r_kW()   if m>0, m=4ℓ, and k=ν_2(4ℓ)<K;
    
        W()   if m>0, m=4ℓ-1 or 4ℓ, and ν_2(4ℓ)≥ K;
    
        0    otherwise.



This all follows from the slice filtration being the I-adic filtration, ρ representing 2 in W(), the structure of the differentials in thm:profile, and strong convergence in thm:conv.



Suppose that  is not of characteristic 2 and that -1 is a sum of four squares in . Then, as a ring,

    π_⋆(η^-1) ≅ W()[η^± 1,σ,μ]/(σ^2)

where |η|=α, |σ|=3+4α, and |μ| = 4+5α.  If additionally <∞, then (η^-1)≃η^-1 and this is a computation of the η-periodic homotopy groups of the motivic sphere spectrum.


By prop:main, we may assume without loss of generality that =0 and that -1 is a sum of four squares in .  It is standard that the latter condition is equivalent to ρ^3=0∈ k^M_3() (see <cit.>).  By thm:profile, we see that the spectral sequence collapses (regardless of the profile of ).  By thm:conv, this proves the theorem.



If <∞ and n≥max{3m+5,4m}, then π_m+nα≅π_m+nαη^-1; if, additionally,  is not of characteristic 2 and -1 is a sum of four squares in , then these groups are 0 or W() according to whether m≡ 1,24 or m≡ 0,34, respectively.


All but the final statement was already observed in <cit.>.


We certainly do not expect that the α_1-periodic slice spectral sequence collapses at its E_2 page in general.  Indeed, inspired by the computations of π_⋆η^-12 by Guillou-Isaksen <cit.> over  and Wilson <cit.> over , we make the following conjecture.


The α_1-periodic slice spectral sequence over  has profile {3,4,5,…}i.e., r_k=k+1 for all k.


If conj holds, then over  of characteristic 0,

    π_m (η^-1)≅
        W()   if m=0,
    
        W()/2^ν_2(4ℓ)+1   if m=4ℓ-1>0,
    _2^ν_2(4ℓ)+1W()   if m=4ℓ>0,
    
        0   otherwise.

Curiously, this makes it appear as if (η^-1) might fit into a “connective image of J fiber sequence” of the form (η^-1)→→Σ^4 where  is the connective cover of the 2-complete Witt K-theory spectrum.  Over =, one may show that the cone on the map σ_∞:Σ^3→η^-1 of thm:sigma_infty coincides with . In fact, the composition Σ^3→η^-1→ is zero, as one may deduce inductively, starting with the triviality of 

    Σ^3η^-1→Σ^3→η^-1→

and continuing along the cell presentation of  given in the proof
of thm:sigma_infty. Hence there is an induced map from the cone 
of Σ^3→η^-1 to . This map induces an isomorphism on homotopy groups, hence is an equivalence by cellularity. In particular, one may express η^-1 over the complex numbers as the fiber of a map →Σ^4. 

The Adams operations on the 2-complete algebraic K-theory spectrum 2 consitute an action of _2^×, the units in the 2-adic integers.  When  has finite virtual cohomological dimension, the results of <cit.> imply that 2≃ (2)^h{± 1} inherits an action of _2^×/{± 1}≅_2 by Adams operations.  Inverting η and taking the connective cover results in Adams operations on 2.  For any such ψ^g, the difference of ring maps ψ^g-1:2→2 lifts to a map ψ^g-1:2→Σ^42.  (This can be seen by observing that Σ^42 is the 4-connective cover of 2 and the cofiber of Σ^42→2 is the Eilenberg-MacLane spectrum associated with the homotopy module W2[η^± 1].)

The η-periodic unit η^-12→2 factors through the fiber _g of ψ^g-1:2→Σ^4 because ψ^g-1 is a difference of ring maps.  This leads to the following conjecture, which is similar in spirit to Mahowald's presentation of the v_1-periodic sphere in topology.  


The map η^-12→_3 induced by the η-periodic unit η^-12→2 is an equivalence.


Work in progress of Tom Bachmann and Mike Hopkins suggests that the action of ψ^3 on HW∧2 is such that the unit map smashed with HW2 induces an equivalence HW2→ HW∧_3.  Since η^-1 is HW-complete, this would immediately prove that η^-12≃_3.  It is presumably also the case that (ψ^3-1)(β^k) = (9^k-1)β^k on π_⋆2, in which case a comparison of slice spectral sequences would prove conj.


The equivalence η^-12≃_3 would also lead to a complete determination of the homotopy type and groups of η^-1.  Let X_ denote the Harrison space of orderings of .  Then π_⋆{η^-1,1/2}≅ H^0(X_;π_m^⊤[1/2]), which can be seen by the results of <cit.>, a descent spectral sequence, and the fact that X_ is a Stone space.  The 2-primary arithmetic fracture square would then imply that

    π_m η^-1≅
        W()   if m=0,
    
        H^0(X_k;π_m^⊤[1/2])   if m>0 and m≡ 1 or 24,
    
        W()/2^ν_2(4ℓ)+1⊕ H^0(X_k;π_m^⊤[1/2])   if m=4ℓ-1>0,
    _2^ν_2(4ℓ)+1W()⊕ H^0(X_k;π_m^⊤[1/2])   if m=4ℓ>0,
    
        0   otherwise.



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