FIXED POINTS OF TOPOLOGICAL HOCHSCHILD HOMOLOGY

AND K-THEORY OF THE TWO-ADIC INTEGERS

John Rognes

I will discuss a program to determine the algebraic K-theory of the ring of twoadic integers which originated with Marcel Bökstedt’s ideas regarding the trace map

tr: K(ℤ) ⟶ T (ℤ) = T HH(ℤ)

In degree zero, this map takes an idempotent matrix (whose image is a projective module) to its trace (which is the rank of the module). Hochschild homology is the global object designed to receive higher–dimensional analogs of such traces, somewhat like K-theory globalizes the category of projective modules. Topological Hochschild homology is a refinement of this idea based on working with rings up to homotopy (algebras over the sphere spectrum) instead of ordinary rings (algebras over the integers). It turns out that T (ℤ) is a product of Eilenberg–Mac Lane spaces, and so is determined by its homotopy groups which are

π_{∗}T (ℤ) = \{ \begin{matrix}ℤ & for ∗ = 0, \\ ℤ/i & for ∗ = 2i − 1, \\ 0 & otherwise. \end{matrix}

In particular π_{3}T (ℤ) = ℤ/2 and the trace map K_{3}(ℤ) → π_{3}T (ℤ) is the surjection ℤ/48 → ℤ/2 detecting the interesting part of K_{3}(ℤ) not coming from stable homotopy.

A further refinement of this approach notes that the circle S^{1} acts on T (ℤ), and the trace map factors over the corresponding homotopy fixed points giving a circle trace map

tr_{S^{1}} : K(ℤ) ⟶ T (ℤ)^{hS^{1}}.

This will be a map of ring spectra, and if the Lichtenbaum–Quillen conjecture is true there should be a ring map from the model JK(ℤ) for K(ℤ) based on étale K-theory to T (ℤ)^{hS^{1}}. In particular there should be an algebra map linking the mod two spectrum homology algebras H_{∗}^{spec}(JK(ℤ); ℤ/2) and H_{∗}^{spec}(T (ℤ)^{hS^{1}}; ℤ/2) compatible with the (co-)action of the dual Steenrod algebra A_{∗}.

Here the source is relatively well known, and in the eighties Bökstedt began to make calculations of the target. He did not get complete answers, but thought for a while that although there were unresolved extension questions, none of the possible answers were compatible with the existence of such a ring map. This would contradict the Lichtenbaum–Quillen conjecture at two (in the form of Dwyer and Friedlander), but so far the details of such an argument remain unfinished.

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My subjective impression is that there is no contradiction, and that at two K(ℤ) and JK(ℤ) are the same, but I cannot prove this.

For the record, the Lichtenbaum–Quillen conjecture at two for the integers amounts to the assertion that the square of rings

[figure: ℤ[\frac{1}{2}]; ℝ; c; ℤ_{3}; ℂ]

(for a suitable imbedding of ℤ_{3} into ℂ) induces a homotopy cartesian square in Ktheory after completion at two, so that K(ℤ[\frac{1}{2}]) would be the homotopy pullback of the maps

K(ℤ_{3}) ⟶ K(ℂ) ⟵^{c} K(ℝ).

Here we may identify K(ℤ_{3}) ≃ K(ℤ/3) ≃ Im J_{ℂ}, K(ℝ) ≃ ℤ × BO and K(ℂ) ≃ ℤ × BU after completion. So the candidate for K(ℤ[\frac{1}{2}]) is also the homotopy pullback of certain maps

Im J_{ℂ} ⟶ ℤ × BU ⟵^{c} ℤ × BO.

Then there should be a localization fiber sequence

K(ℤ/2) ⟶ K(ℤ) ⟶ K(ℤ[\frac{1}{2}])

with K(ℤ/2) ≃ Hℤ at two, so in the end the candidate JK(ℤ) for K(ℤ) at two fits into several different fiber sequences

Im J_{ℝ} ⟶ JK(ℤ) ⟶ BBSO
SU ⟶ JK(ℤ) ⟶ ℤ × BO
BBO ⟶ JK(ℤ) ⟶ Im J_{ℂ}.

Here Im J_{ℝ} is the real image of J, defined by a fibration

Im J_{ℝ} ⟶ ℤ × BO ⟶^{ψ^{3}−1} BSpin

while Im J_{ℂ} is the complex image of J, defined by the fibration

Im J_{ℂ} ⟶ ℤ × BU ⟶^{ψ^{3}−1} BU.

The homotopy groups of these spaces are well known, as are the homotopy groups of the model JK(ℤ) for K(ℤ) at two.

Returning to Bökstedt’s idea, there is now a refined version of the circle trace map due to Bökstedt, Hsiang and Madsen, called the cyclotomic trace map. Its construction runs as follows.

K-THEORY OF ℤ_{2} 3

The circle action on T (ℤ) determines spaces of fixed points T (ℤ)^{C_{p^{n}}} for each subgroup C_{p^{n}} ⊂ S^{1}. Here p can be any prime; we will specialize to the case p = 2 a little later. There are natural inclusions

F: T (ℤ)^{C_{p^{n}}} ⟶ T (ℤ)^{C_{p^{n−1}}}

which we will call Frobenius maps. These were called D in the old notation. In addition topological Hochschild homology admit other maps

R: T (ℤ)^{C_{p^{n}}} ⟶ T (ℤ)^{C_{p^{n−1}}}

called restriction maps. These were called Φ in the old notation, and arise by restricting a C_{p^{n}}-equivariant map to its C_{p}-fixed points, giving a C_{p^{n−1}}-equivariant map. Both maps are interesting, and together give T (ℤ) the structure of a cyclotomic spectrum.

The trace map tr admits lifts

tr_{p^{n}}: K(ℤ) ⟶ T (ℤ)^{C_{p^{n}}}

compatible with both F and R. In fact

F ∘ tr_{p^{n}} ≃ R ∘ tr_{p^{n}} ≃ tr_{p^{n−1}}.

The cyclotomic trace map combines these lifts into a map to the topological cyclic homology T C(ℤ) at p.

[figure: K(ℤ); trc; tr_{p^{n}}; tr_{p^{n−1}}; T C(ℤ); T (ℤ)^{C_{p^{n}}}; F; R; T (ℤ)^{C_{p^{n−1}}}; …; T (ℤ).]

So by definition

T C(ℤ) = holim_{F,R} T (ℤ)^{C_{p^{n}}}

and trc: K(ℤ) → T C(ℤ) is the cyclotomic trace map. Here implicitly all spaces are completed at our chosen prime p. We could also consider the homotopy limit over only the Frobenius maps

T F (ℤ) = holim_{F} T (ℤ)^{C_{p^{n}}}

and since F and R commute there is a fiber sequence

… ⟶^{∂} T C(ℤ) ⟶^{π} T F (ℤ) ⟶^{R−1} T F (ℤ) ⟶ …

The ring map ℤ → ℤ_{p} to the ring of p-adic integers induces a nontrivial map K(ℤ) → K(ℤ_{p}), but after p-adic completion the natural map T C(ℤ) → T C(ℤ_{p}) is a homotopy equivalence. Hence the cyclotomic trace map K(ℤ) → T C(ℤ) ≃ T C(ℤ_{p}) naturally factors through K(ℤ_{p}), and it is in fact as an invariant of the K-theory of the p-adic integers, rather than of the rational integers, that T C(ℤ) is an interesting invariant.

Theorem (McCarthy, Hesselholt–Madsen). The cyclotomic trace map

trc: K(ℤ_{p}) ⟶ T C(ℤ)

induces a homotopy equivalence on p-adically completed connective covers.

We say that trc is a connective p-adic equivalence. When p is an odd prime, we have the following calculation:

Theorem (Bökstedt–Madsen). There is a homotopy equivalence

T C(ℤ) ≃ Im J × B Im J × BBU

on p-adically completed connective covers.

Here Im J is defined by the fiber sequence

Im J ⟶ ℤ × BU ⟶^{ψ^{k}−1} BU

where k is a topological generator of the p-adic units. Such a complete description of the K-theory of a ring was previously essentially only known in the cases of finite fields (Quillen) and algebraically closed fields (Suslin).

What about the case p = 2 ? For multiplicative reasons the formula above cannot hold, neither with Im J replaced by Im J_{ℝ} or by Im J_{ℂ}. Instead we have the following chain of results. Hereafter suppose all spaces are implicitly completed at two.

The main technical work goes into the following calculation.

Claim 1. The nontrivial mod two homotopy groups of T C(ℤ) have orders

#π_{∗}(T C(ℤ); ℤ/2) = \{ \begin{matrix}2 & for ∗ = −1 or 0, \\ 4 & for ∗ = 1 or ∗ ≥ 2 even, \\ 8 & for ∗ ≥ 3 odd. \end{matrix}

Hence these are also the orders of π_{∗}(K(ℤ_{2}); ℤ/2) for ∗ ≥ 0.

The degree eight Adams map A = v_{1}^{4} acts injectively on the mod two homotopy groups above. Inverting A only changes the homotopy groups below degree 1, so K(ℤ_{2}) is essentially K-local, in the sense that the localization map K(ℤ_{2}) → L_{K}K(ℤ_{2}) is an equivalence above degree 1.

Let us review the construction of the Galois reduction map,

red: K(ℤ_{2}) → Im J_{ℂ}

by adapting ideas of Dwyer and Friedlander to the two-primary case.

We have inclusions

ℤ_{2} ⊂ ℚ_{2} ⊂ ℚ_{2}(μ_{2^{∞}}) ⊂ Q̄_{2}

and choose an imbedding Q̄_{2} → ℂ. Here ℚ_{2}(μ_{2^{∞}}) is the field obtained by adjoining all 2^{n}th roots of unity to ℚ_{2}, for all n, while Q̄_{2} is the algebraic closure. There is a Galois automorphism θ^{3} of ℚ_{2}(μ_{2^{∞}}) fixing ℚ_{2} and given by θ^{3}(ζ) = ζ^{3} for each 2^{n}th root of unity ζ. Next we may choose an extension of θ^{3} to a Galois automorphism ϕ^{3} of Q̄_{2}. This involves a choice, but we will only require that ϕ^{3}(\sqrt{3}) = +\sqrt{3}, rather than −\sqrt{3}. We have this choice because \sqrt{3} ∉ ℚ_{2}(μ_{2^{∞}}).

Then ϕ^{3} induces a self–map of K(Q̄_{2}) compatible under Suslin’s equivalence K(Q̄_{2}) ≃ K(ℂ) ≃ ℤ × BU with the Adams operation ψ^{3}. Hence we can arrange to have maps of ring spectra

K(ℤ_{2}) ⟶ K(ℚ_{2}) ⟶ K(Q̄_{2})^{hϕ^{3}} ⟶^{≃} (ℤ × BU)^{hψ^{3}}

where the superscript ( )^{hf} means the homotopy fiber of f −1. Passing to connective covers we get the reduction map

red: K(ℤ_{2}) ⟶ Im J_{ℂ}.

We define the reduced K-theory K^{red}(ℤ_{2}) as the homotopy fiber of red.

Claim 2. The reduction map induces surjections on mod two homotopy in all degrees, and split surjections on integral homotopy in all degrees. The nontrivial mod two homotopy groups of K^{red}(ℤ_{2}) have orders

#π_{∗}(K^{red}(ℤ_{2}); ℤ/2) = \{ \begin{matrix}2 & for ∗ = 1 or ∗ ≥ 2 even, \\ 4 & for ∗ ≥ 3 odd. \end{matrix}

Note that the two-adic unit 3 in Q̄_{2} has a square root which is invariant under ϕ^{3}. Hence its symbol {3} in K_{1} is divisible by two, and maps to zero in π_{1} Im J_{ℂ} ≅ ℤ/2. Thus π_{1}K^{red}(ℤ_{2}) ≅ ℤ_{2} is free on one generator, namely {3}. This class is represented by an infinite loop map {3}: Q(S^{1}) → K^{red}(ℤ_{2}), and since K(ℤ_{2}) and thus K^{red}(ℤ_{2}) is essentially K-local, there is a factorization in the diagram

[figure: Q(S^{1}); {3}; K^{red}(ℤ_{2}); B Im J_{ℝ}; L_{K}Q(S^{1}); L_{K}K^{red}(ℤ_{2})]

giving a map B Im J_{ℝ} → K^{red}(ℤ_{2}) inducing an isomorphism on π_{1}. There is a complexification map Bc: B Im J_{ℝ} → B Im J_{ℂ}. Briefly let W denote its homotopy fiber, as in the diagram below:

[figure: W; B Im J_{ℝ}; Bc; B Im J_{ℂ}; K^{red}(ℤ_{2})]

Claim 3. The composite W → B Im J_{ℝ} → K^{red}(ℤ_{2}) is null homotopic. Hence there is an extension B Im J_{ℂ} → K^{red}(ℤ_{2}) of the map {3}, which induces an isomorphism on π_{1}.

The proof uses the description of W = hofib(Bc) as an invertible spectrum in the K-local category at two, representing an element of order 4 in the Picard group P ic_{1} (chromatic type 1) of Hopkins, Mahowald and Sadofsky. These authors give an inductive procedure for how to construct W from ∗ by successively attaching mapping cones over L_{K}(S^{i}/2) where S^{i}/2 is a mod two Moore spectrum. So to check that W → K^{red}(ℤ_{2}) is null homotopic it suffices to check that certain maps S^{i}/2 → L_{K}S^{i}/2 → K^{red}(ℤ_{2}) can be arranged to be null homotopic, which amounts to questions about π_{i+1}(K^{red}(ℤ_{2}); ℤ/2). This is precisely the kind of information the T C calculations supplied, and the result can be proved in this way.

Claim 4. The map B Im J_{ℂ} → K^{red}(ℤ_{2}) induces an injection on mod two homotopy groups in all degrees, and split injections on all integral homotopy groups. Hence its (infinite loop space) cofiber X has the mod two homotopy groups of BBU, and in fact X ≃ BBU.

Hence we have the following theorem.

Theorem (Rognes). The two-completed homotopy type of K(ℤ_{2}) as an infinite loop space is determined by the following diagram of infinite loop space fiber sequences

[figure: B Im J_{ℂ}; K^{red}(ℤ_{2}); K(ℤ_{2}); red; Im J_{ℂ}; BBU.]

The vertical connecting map f: BU → B Im J_{ℂ} represents a generator of the group of such infinite loop maps which induce trivial homomorphisms on homotopy. This group is free of rank one as a module over the two-adic integers.

The horizontal connecting map g: Im J_{ℂ} → BK^{red}(ℤ_{2}) also represents a generator of the group of such infinite loop maps, which again is free of rank one as a module over the two-adic integers.

Both fibrations may thus be viewed as maximally twisted extensions. But looking at K-groups we have

K_{∗}(ℤ_{2}) ≅ π_{∗} Im J_{ℂ} × π_{∗}B Im J_{ℂ} × π_{∗}BBU.

How do we access π_{∗}(T C(Z); Z/2) ? Through the fiber sequence

T C(Z) ⟶^{π} T F (Z) ⟶^{R−1} T F (Z).

Here

T F (Z) = holim T (Z)^{C_{2^{n}}} ⟶ holim T (Z)^{hC_{2^{n}}} ⟵ T (Z)^{hS^{1}}.

The last map is a two-complete homotopy equivalence.

Theorem (Tsalidis). If the natural map Γ_{1} : T (Z)^{C_{p}} → T (Z)^{hC_{p}} is a connective p-adic equivalence, then so is Γ_{n}: T (Z)^{C_{p^{n}}} → T (Z)^{hC_{p^{n}}} for all n ≥ 1.

The proof is a reduction to the rank one elementary abelian p-group case, along the lines of Carlsson’s reduction of the Segal conjecture.

Hence once we check that Γ_{1} is a connective two-adic equivalence, then the composite map

K(Z_{2}) ⟶^{trc} T C(Z) ⟶^{π} T F (Z)

is equivalent to the circle trace map tr_{S^{1}} studied by Bökstedt. The cyclotomic trace map merely takes into account the additional coherence required by the Rmaps. So we want to understand π_{∗}(T F (Z); Z/2), which equals π_{∗}(T (Z)^{hS^{1}}; Z/2) in nonnegative degrees, and the limiting case of the reduction map R: T F (Z) → T F (Z) compared to the identity.

We will do this in inductive steps, proceding from the mod two homotopy of T (Z) to T (Z)^{C_{2}}, through to that of T (Z)^{C_{2^{n}}} for all n, simultaneously keeping an eye on R: T (Z)^{C_{2^{n}}} → T (Z)^{C_{2^{n−1}}} . The inductive approach is made possible by the following norm–restriction fiber sequence

T (Z)_{C_{2^{n}}} ⟶^{N} T (Z)^{C_{2^{n}}} ⟶^{R} T (Z)^{C_{2^{n−1}}} .

Here the norm map N followed by a forgetful map gives the C_{2^{n}}-equivariant transfer. Mapping T (Z) to the function spectrum F (ES_{+}^{1}, T (Z)) takes the fiber sequence above to the following homotopy norm–restriction sequence

T (Z)_{C_{2^{n}}} ⟶^{N^{h}} T (Z)^{hC_{2^{n}}} ⟶^{R^{h}} ℍ̂(C_{2^{n}}, T (Z)).

Here ℍ̂(C_{2^{n}}, T (Z)) denotes the Tate construction for the action of C_{2^{n}} on T (Z), which merges the homotopy orbit and homotopy fixed point constructions to its left, just like Tate cohomology merges group homology and cohomology.

There is a map of fiber sequences, which is the identity on the left, the natural map Γ_{n} in the middle, and a similar map

Γ̂_{n}: T (Z)^{C_{2^{n−1}}} ⟶ ℍ̂(C_{2^{n}}, T (Z))

on the right. Clearly Γ_{n} is a connective two-adic equivalence if and only if Γ̂_{n} is one. In fact the following theorem

Theorem (Rognes). Γ_{1} is a connective two-adic equivalence.

is proved by showing that Γ̂_{1} : T (Z) → ℍ̂(C_{2}, T (Z)) induces an isomorphism on mod two homotopy in all nonnegative degrees. This is feasible because

π_{∗}(T (Z); Z/2) ≅ ℤ/2[e_{3}, e_{4}]/(e_{3}^{2} = 0)

is known, and there is a spectral sequence Ê^{∗}(C_{2}):

E_{s,t}^{2} = Ĥ^{−s}(C_{2}; π_{t}(T (Z); Z/2)) ⟹ π_{s+t}(ℍ̂(C_{2}, T (Z)); Z/2)

Here Ĥ^{−s} denotes Tate cohomology. The theorem is proved by a comparison between the spectral sequence above, which is not an algebra spectral sequence, and correponding spectral sequences for mod four homotopy and for the action of S^{1} in place of C_{2}.

The whole technical problem at two is that the mod two Moore spectrum M = S^{0}/2 representing mod two homotopy is not a ring spectrum. The obstruction to finding a retraction to the (left) unital map M → M ∧ M factors through the Hopf map η, so because multiplication by η is null homotopic on T (Z), there is an algebra structure on π_{∗}(T (Z); Z/2). But this null homotopy cannot be made even C_{2}-equivariant, and so there is no natural algebra structure on π_{∗}(T (Z)^{C_{2^{n}}}; Z/2) for n ≥ 1. In particular there is no natural algebra structure on the mod two homotopy spectral sequences we are considering.

Hence all Γ_{n} are connective two-adic equivalences, and we may try to compute the mod two homotopy of T (Z)^{C_{2^{n}}} from the spectral sequences E^{∗}(C_{2^{n}}):

E_{s,t}^{2} = H^{−s}(C_{2^{n}}; π_{t}(T (Z); Z/2)) ⟹ π_{s+t}(T (Z)^{hC_{2^{n}}}; Z/2)

Here H^{−s} is ordinary group cohomology, which can be found from Tate cohomology by truncation.

The inductive scheme goes as follows.

Suppose we know the mod two homotopy of T (Z)^{C_{2^{n−1}}} . This is the mod two homotopy of ℍ̂(C_{2^{n}}, T (Z)) in nonnegative degrees. Try to backtrack in the spectral sequence Ê^{∗}(C_{2^{n}}) to determine the differential structure, given the E^{2}-term and the abutment. Then truncate the spectral sequence to give E^{∗}(C_{2^{n}}). This converges to the mod two homotopy of T (Z)^{C_{2^{n}}}. And the map R: T (Z)^{C_{2^{n}}} → T (Z)^{C_{2^{n−1}}} is induced by the map of spectral sequences E^{∗}(C_{2^{n}}) → Ê^{∗}(C_{2^{n}}).

How about the Adams A = v_{1}^{4}-periodicity ? By an explicit symbol calculation, and a theorem of Calvin Moore, η^{3} in π_{3}Q(S^{0}) maps to zero in K_{3}(Z_{2}). Hence ν maps to a four-torsion class in K_{3}(Z_{2}), which lifts over a mod four Bockstein to a class ν̃_{4} ∈ K_{4}(ℤ_{2}; Z/4). Mod four homotopy acts upon mod two homotopy, and multiplication by ν̃_{4} behaves as multiplication by v_{1}^{2}; as closely as possible. At least a detailed look at ν̃_{4}^{2} shows that it acts like A.