TRACE MAPS FROM THE ALGEBRAIC

K-THEORY OF THE INTEGERS

(AFTER MARCEL BÖKSTEDT)

John Rognes

Abstract. Let p be any prime. We consider Bökstedt’s topological refinement K(ℤ) → T (ℤ) = T HH(ℤ) of the Dennis trace map from algebraic K-theory of the integers to topological Hochschild homology of the integers. This trace map is shown to induce a surjection on homotopy in degree 2p − 1, onto the first p-torsion in the target. Furthermore, Bökstedt’s map factors through the S^{1}-homotopy fixed points T (ℤ)^{hS^{1}} of T (ℤ), and it is shown that the first p-torsion element in degree 2p − 3 of the stable homotopy groups of spheres is detected in the homotopy of T (ℤ)^{hS^{1}}. Both results are due to Bökstedt, but have remained unpublished.

1. Introduction

The purpose of this paper is to provide a reference for two theorems due to Marcel Bökstedt.

Let K(ℤ) be the K-theory spectrum, and T (ℤ) = T HH(ℤ) the topological Hochschild homology spectrum of the integers. We write K_{i}(ℤ) = π_{i}K(ℤ) and T_{i}(ℤ) = π_{i}T (ℤ). The trace map tr: K(ℤ) → T (ℤ) is the map constructed by Bökstedt in [B1], which strengthens the Dennis trace map to ordinary Hochschild homology. By the calculations of [B2], reproduced in [HM], T_{0}(ℤ) = ℤ and T_{2i−1}(ℤ) ≅ ℤ/i for all i ∈ ℕ, while the remaining groups are zero.

Theorem 1.1 (Bökstedt). Let p be any prime. The trace map induces a surjection

π_{2p−1}(tr): K_{2p−1}(ℤ) ⟶ T_{2p−1}(ℤ) ≅ ℤ/p

onto the first p-torsion in T_{∗}(ℤ).

Bökstedt’s proof appears in an unpublished Bielefeld preprint [B3]. Another proof is given in Section 10 of [BM], but that proof apparently assumes p is odd. We give a proof in Section 2 below, taking special care to cover the case p = 2.

The topological Hochschild homology spectrum admits the structure of an S^{1}- spectrum, and there is a compatible family of factorizations of tr

K(ℤ) ⟶^{tr_{p^{n}}} T (ℤ)^{C_{p^{n}}} ⊆ T (ℤ),

Typeset by AMS-TEX

for a fixed prime p and for all n ≥ 0. Here C_{p^{n}} is the cyclic subgroup of S^{1} with p^{n} elements. See [BHM] or [HM] for more on this and the following material. These factorizations, composed with the standard maps

Γ: T (ℤ)^{C_{p^{n}}} ⟶ T (ℤ)^{hC_{p^{n}}}

from fixed points to homotopy fixed points, induce a map of homotopy limits

K(ℤ) ⟶ holim_{n} T (ℤ)^{C_{p^{n}}} ⟶ holim_{n} T (ℤ)^{hC_{p^{n}}} .

After p-adic completion (denoted in this paper by a subscript p) there is a natural homotopy equivalence

T (ℤ)_{p}^{hS^{1}} ⟶^{≃} holim_{n} T (ℤ)_{p}^{hC_{p^{n}}}

determining a map

tr_{S^{1}} : K(ℤ)_{p} ⟶ T (ℤ)_{p}^{hS^{1}}

which we call the circle trace map. The cyclotomic trace map trc: K(ℤ)_{p} → T C(ℤ, p) of [BHM] is a further refinement of this map.

There is a second quadrant spectral sequence E_{∗∗}^{r} with d^{r}: E_{s,t}^{r} → E_{s−r,t+r−1}^{r}, converging to

π_{s+t}T (ℤ)_{p}^{hS^{1}} = π_{s+t} Map(ES_{+}^{1}, T (ℤ))_{p}^{S^{1}}

and having

(1.2)  E_{s,t}^{2} = H^{−s}(BS^{1}; T_{t}(ℤ)_{p}).

The spectral sequence arises from the skeleton filtration of a standard model for ES^{1}, a contractible space with a free action of S^{1}, and the cohomology groups arise as the cohomology of the topological group S^{1} acting on T_{∗}(ℤ). Since S^{1} is a path-connected group the action is trivial, and hence

E_{s,t}^{2} = \{ \begin{matrix}T_{t}(ℤ)_{p} & when s ≤ 0 is even, \\ 0 & otherwise. \end{matrix}

The edge homomorphism

π_{∗}T (ℤ)_{p}^{hS^{1}} ⟶ E_{0,∗}^{∞} ⟶ T_{∗}(ℤ)_{p}

is induced by the natural map

T (ℤ)_{p}^{hS^{1}} = Map(ES_{+}^{1}, T (ℤ))_{p}^{S^{1}} ⟶ Map(S_{+}^{1}, T (ℤ))_{p}^{S^{1}} ≅ T (ℤ)_{p}

given by restriction over any choice of S^{1}-equivariant imbedding S_{+}^{1} ⊂ ES_{+}^{1}. This spectral sequence may be derived from the spectral sequence of a tower of fibrations constructed by Bousfield and Kan in [BK], following page 258.

Hence Theorem 1.1 has the following corollary.

TRACE MAPS FROM K-THEORY 3

Corollary 1.3. Let p be any prime. There is a class λ_{2p−1} ∈ K_{2p−1}(ℤ)_{p} such that tr_{S^{1}}(λ_{2p−1}) ∈ π_{2p−1}T (ℤ)_{p}^{hS^{1}} is detected on a permanent cycle surviving to E^{∞} in bidegree (0, 2p − 1) of the spectral sequence (1.2). When p = 2, the class λ = λ_{3} ∈ K_{3}(ℤ)_{2} ≅ ℤ/16 is a generator. □

The second theorem concerns the class α_{1} ∈ π_{2p−3}Q(S^{0})_{p} generating the first p-torsion in the stable homotopy groups of spheres. When p = 2 this is the stable class of the Hopf map η : S^{3} → S^{2}.

Theorem 1.4 (Bökstedt). The composite

Q(S^{0})_{p} ⟶ K(ℤ)_{p} ⟶^{tr_{S^{1}}} T (ℤ)_{p}^{hS^{1}}

maps α_{1} ∈ π_{2p−3}Q(S^{0})_{p} to an element of π_{2p−3}T (ℤ)_{p}^{hS^{1}} which is detected on a permanent cycle that survives to E^{∞} in bidegree (−2, 2p − 1) of the spectral sequence (1.2).

We give a proof in Section 3.

I thank Marcel Bökstedt for explaining these results, and many others, to me.

2. The trace map K(ℤ) ⟶ T (ℤ)

The proof of Theorem 1.1 depends on Waldhausen’s Corollary 3.7 of [W], and on Bökstedt and Madsen’s Lemma 10.5 of [BM].

Let F be a functor with smash product (FSP). See [B1] or [BHM] for the definition of this notion, and for the construction of the K-theory K(F) and topological Hochschild homology T (F) of such a functor, together with the trace map tr : K(F) → T (F).

Let F^{s} be the underlying ring spectrum of F, associated to the prespectrum {F (S^{n})}_{n}, and let M_{1}(F) be its zeroth space. π_{0}M_{1}(F) = π_{0}F^{s} is a ring, and GL_{1}(F) ⊂ M_{1}(F) is defined as the union of the components corresponding to units in π_{0}M_{1}(F). Then GL_{1}(F) is an associative topological monoid. Let F_{(k)} be the k × k matrix FSP with

F_{(k)}(X) = Map([k], [k] ∧ F (X))

(based maps) where [k] = {0, 1, …, k}. Indeed, π_{∗}F_{(k)}^{s} is the k × k matrix algebra over π_{∗}F^{s}. Write M_{k}(F) = M_{1}(F_{(k)}) and GL_{k}(F) = GL_{1}(F_{(k)}).

Let BGL_{k}(F) and N^{cy}GL_{k}(F) be the classifying space and the cyclic nerve of GL_{k}(F), respectively. There is a natural projection π: N^{cy}GL_{k}(F) → BGL_{k}(F), with a (weak homotopy) section i: BGL_{k}(F) → N^{cy}GL_{k}(F). The K-theory K(F) is constructed as a group completion of the topological monoid ∐_{k≥0} BGL_{k}(F). Let the cyclic K-theory K^{cy}(F) be likewise constructed from the topological monoid ∐_{k≥0} N^{cy}GL_{k}(F).

There is a natural projection π: K^{cy}(F) → K(F), with a section i: K(F) → K^{cy}(F). The trace map tr: K(F) → T (F) factors though i by construction. A standard inclusion GL_{1}(F) → GL_{k}(F) induces maps BGL_{1}(F) → K(F) and N^{cy}GL_{1}(F) → K^{cy}(F), compatible with the projections and sections π and i.

The composite

s: N^{cy}GL_{1}(F) ⟶ K^{cy}(F) ⟶ T (F)

is given in simplicial degree q by

(f_{0}, …, f_{q}) ⟼ f_{0} ∧ ⋯ ∧ f_{q}.

Here each f_{i}: S^{n_{i}} → F (S^{n_{i}}) is assumed to stabilize to a class in π_{0}GL_{1}(F) ⊂ π_{0}M_{1}(F) as n_{i} → ∞. The map s may clearly also be factorized as

N^{cy}GL_{1}(F) ⟶ N^{cy}M_{1}(F) ⟶ T (F).

Let λ: S_{+}^{1} ∧ M_{1}(F) → T (F) be given by the S^{1}-action on T (F) combined with the inclusion of M_{1}(F) as the zero–simplices T (F)_{0} = hocolim_{n∈I} Map(S^{n}, F (S^{n})) into T (F). In simplicial degree q the map λ identifies (C_{q+1})_{+} ∧ M_{1}(F) with the maps

f_{0} ∧ ⋯ ∧ f_{q}: S^{n_{0}} ∧ ⋯ ∧ S^{n_{q}} → F (S^{n_{0}}) ∧ ⋯ ∧ F (S^{n_{q}})

in T (F)_{q} where all but one of the f_{i} equal a unit map 1_{S^{n_{i}}} : S^{n_{i}} → F (S^{n_{i}}). Here C_{q+1} is the cyclic group with (q + 1) elements, viewed as the q-simplices in a simplicial model for S^{1}.

Restricting λ over S_{+}^{1} ∧ GL_{1}(F) → S_{+}^{1} ∧ M_{1}(F) we get a factorization through s: N^{cy}(GL_{1}(F)) → T (F):

(C_{q+1})_{+} ∧ GL_{1}(F) ⟶ N^{cy}(GL_{1}(F))_{q}
τ_{q+1}^{i} ∧ f ⟼ (1, …, 1, f, 1, …, 1)

with f in the ith position, for i ∈ [q]. Here τ_{q+1} is a generator of C_{q+1}. Hence we have the following commutative diagram, natural in F:

[figure: BGL_{1}(F); i; N^{cy}GL_{1}(F); s; S_{+}^{1} ∧ GL_{1}(F); K(F); i; K^{cy}(F); T (F); S_{+}^{1} ∧ M_{1}(F); λ]

Let F_{1} be the identity FSP with F_{1}(X) = X, and let F_{2} be the Eilenberg– Mac Lane FSP of the integers, with F_{2}(S^{n}) = K(ℤ, n). There is a linearization morphism ℓ: F_{1} → F_{2} of FSPs, inducing a π_{0}-isomorphism on underlying ring spectra

ℓ: F_{1}^{s} = S^{0} ⟶ F_{2}^{s} = Hℤ.

Let SG ⊂ G be the identity component and the homotopy units of Q(S^{0}), respectively. We have M_{1}(F_{1}) ≃ Q(S^{0}), M_{1}(F_{2}) ≃ ℤ, GL_{1}(F_{1}) ≃ G and GL_{1}(F_{2}) ≃ {±1} ≅ ℤ/2. We identify N^{cy}SG with the free loop space ΛBSG as usual. Consider the diagram of homotopy fibers of maps induced by ℓ in the diagram above:

(2.1)  [figure: BSG; i; ΛBSG; s; S_{+}^{1} ∧ SG; ≃; K(F_{1} → F_{2}); i; K^{cy}(F_{1} → F_{2}); T (F_{1} → F_{2}); S_{+}^{1} ∧ SG; λ]

Here K(F_{1} → F_{2}) = hofib(ℓ : K(F_{1}) → K(F_{2})), and so on.

The map ℓ: F_{1}^{s} → F_{2}^{s} is r = (2p − 3)-connected when localized at p. We need the following two lemmas.

Lemma 2.2. Let F_{1} be the identity FSP, and F_{2} the Eilenberg–Mac Lane FSP of the integers, as above. Then

λ: S_{+}^{1} ∧ SG_{(p)} ⟶ T (F_{1} → F_{2})_{(p)}

is (2r + 1) = (4p − 5)-connected.

Proof. Let F_{0}(X) = hofib(ℓ: F_{1}(X) → F_{2}(X)) for all X. Then F_{0} is a F_{1}–F_{1}- bimodule FSP. Let T (F_{1}, F_{0}) be the topological Hochschild homology space of F_{1} with coefficients in F_{0}, as defined in Section 10 of [BM]. T (F_{1}, F_{0}) is the geometric realization of a simplicial space with q-simplices

T (F_{1}, F_{0})_{q} = hocolim_{(n_{i})_{i}∈I^{q+1}} Map(S^{n_{0}} ∧ ⋯ ∧ S^{n_{q}}, F_{0}(S^{n_{0}}) ∧ S^{n_{1}} ∧ ⋯ ∧ S^{n_{q}}).

Here we are using the assumption that F_{1} is the identity FSP.

The inclusion of the zero–simplices

T (F_{1}, F_{0})_{0} = hocolim_{n∈I} Map(S^{n}, F_{0}(S^{n})) → T (F_{1}, F_{0})

is a weak homotopy equivalence, because for n ∈ ℕ the map

Ω^{n}F_{0}(S^{n}) ⟶ Ω^{n}Q(F_{0}(S^{n}))

is (n+1)-connected. Thus, if we identify M_{1}(F_{0}) with the zero–simplices T (F_{1}, F_{0})_{0}, we obtain a homotopy equivalence

SG = M_{1}(F_{0}) → T (F_{1}, F_{0}).

In [BM], pages 130–134, there is constructed a map S_{+}^{1} ∧ T (F_{1}, F_{0}) → T (F_{1} → F_{2}), and it is easy to see that there is a factorization of λ as

S_{+}^{1} ∧ M_{1}(F_{0}) ⟶ S_{+}^{1} ∧ T (F_{1}, F_{0}) ⟶ T (F_{1} → F_{2}).

Lemma 10.5 of [BM] states that the second map in this factorization is (2r)- connected, and in fact their proof shows that the map is (2r + 1)-connected. (The map S_{+}^{1} ∧ T (F_{1}, F_{0}) → T (F_{1} → F_{2}) is the geometric realization of a map of simplicial spaces which is a homotopy equivalence in simplicial degree zero, and (2r)- connected in all other degrees. The result follows.)

Thus λ is the composite of a weak homotopy equivalence and a (2r+1)-connected map. This completes the proof of Lemma 2.2. □

Lemma 2.3. Let F_{1} and F_{2} be as above. Localized at p,

π_{2p−2}T (F_{1} → F_{2})_{(p)} ≅ \{ \begin{matrix}ℤ/p & if p is odd, \\ ℤ/2 ⊕ ℤ/2 & if p = 2, \end{matrix}

and likewise

π_{2p−2}ΛBSG_{(p)} ≅ \{ \begin{matrix}ℤ/p & if p is odd, \\ ℤ/2 ⊕ ℤ/2 & if p = 2. \end{matrix}

Proof. The inclusion of the zero–simplices Q(S^{0}) ≃ T (F_{1})_{0} → T (F_{1}) is a homotopy equivalence, so the map ℓ: T (F_{1}) → T (F_{2}) factors up to homotopy through the zero–simplices ℤ ≃ T (F_{2})_{0} → T (F_{2}). Thus ℓ induces an inessential map on connected components, and so π_{2p−2}T (F_{1} → F_{2}) ≅ T_{2p−1}(ℤ) ⊕ π_{2p−2}Q(S^{0}).

The fiber sequence ΩBSG → ΛBSG → BSG has a section, so π_{2p−2}ΛBSG ≅ π_{2p−2}ΩBSG ⊕ π_{2p−2}BSG. Now π_{2p−3}SG_{(p)} ≅ ℤ/p for all p, while π_{2p−2}SG_{(p)} = 0 for p odd and π_{2}SG ≅ ℤ/2. □

We return to the proof of the theorem. Consider the diagram of homotopy fibers (2.1). We implicitly localize at the prime p. The map λ is (4p − 5) > (2p − 2)- connected, so π_{2p−2}(λ) is a surjection (in fact an isomorphism). Thus π_{2p−2}(s) is a split surjection of isomorphic finite groups, and therefore injective. π_{2p−2}(i) is a split injection, so the composite ℤ/p ≅ π_{2p−2}BSG → π_{2p−2}T (F_{1} → F_{2}) is also injective, and is in particular nonzero. Hence the relative trace map π_{2p−2}K(F_{1} → F_{2}) → π_{2p−2}T (F_{1} → F_{2}) is nonzero.

Now consider the following diagram, where the vertical maps are boundary maps in the fiber sequences induced by ℓ: F_{1} → F_{2}, and the top horizontal map is the map we wish to show induces a surjection on π_{2p−2}.

[figure: ΩK(F_{2}); Ωtr; ΩT (F_{2}); K(F_{1} → F_{2}); T (F_{1} → F_{2})]

By Waldhausen’s Corollary 3.7 of [W], the map ΩK(F_{2}) → K(F_{1} → F_{2}) induces a surjection on π_{2p−2}. Hence the composite ΩK(F_{2}) → T (F_{1} → F_{2}) induces a nonzero map on π_{2p−2}, and it follows that

π_{2p−2}(Ωtr): π_{2p−2}ΩK(F_{2}) → π_{2p−2}ΩT (F_{2}) ≅ ℤ/p

is nonzero, and thus surjective. This completes the proof of Bökstedt’s Theorem 1.1. □

3. The circle trace map

We now turn to the proof of Theorem 1.4.

Let E = ES^{1} be a contractible S^{1}-space with free S^{1}-action. We will use as a concrete model for E the (thin) geometric realization of the usual simplicial space [q] ↦ (S^{1})^{q+1}. Let Ē be the corresponding thick realization, where the degenerate simplices are not collapsed. There is a natural S^{1}-homotopy equivalence Ē → E induced by collapsing degenerate simplices. Let Ē^{(k)} and E^{(k)} denote the respective k-skeleta.

Then Ē^{(0)} = E^{(0)} = S^{1}. Ē^{(1)} can be described as the quotient space

S^{1} ∪ (S^{1} × S^{1} × I)/ ∼

with (g_{0}, g_{1}, 0) ∼ g_{0} and (g_{0}, g_{1}, 1) ∼ g_{1}. E^{(1)} is the further quotient space where we also identify (g, g, t) ∼ g for all t ∈ I.

So Ē^{(1)} is the equalizer of the two projection maps pr_{1}, pr_{2} : S^{1} × S^{1} → S^{1}. The map Ē^{(1)} → E^{(1)} identifies a diagonal torus to a circle by a projection map ΔS^{1} × (I/∂I) → S^{1} onto the first factor. Here ΔS^{1} ⊂ S^{1} × S^{1} is the diagonal circle.

We remark that E^{(1)} ≅ S^{3}, and the skeleton filtration E^{(0)} ⊂ E^{(1)} ⊂ … of E = ES^{1} agrees with the unit sphere filtration S^{1} = S(ℂ^{1}) ⊂ S^{3} = S(ℂ^{2}) ⊂ … of S^{∞} = S(ℂ^{∞}) ≅ ES^{1}. Let Σ_{+}(X) = Σ(X_{+}) = X_{+} ∧ S^{1}.

Lemma 3.1. There is a map of Puppe cofibration sequences

[figure: (S^{1} × S^{1})_{+}; pr_{1}; pr_{2}; S_{+}^{1}; Ē_{+}^{(1)}; Σ_{+}(S^{1} × S^{1}); a; c; Σ_{+}S^{1}; S_{+}^{1}; E_{+}^{(1)}; Σ(S^{1} × S^{1}/ΔS^{1}); b; Σ_{+}S^{1}]

where a is homotopic to Σ_{+}(pr_{2}) − Σ_{+}(pr_{1}), and c is the suspension of the collapse map (S^{1} × S^{1})_{+} → S^{1} × S^{1}/ΔS^{1}.

Proof. The diagram is induced by the skeleton–preserving map Ē → E. The claim about a follows from making the obvious choice of homotopy inverse to the collapse map

Ē_{+}^{(1)} ∪ C(S_{+}^{1}) ⟶^{≃} Ē_{+}^{(1)}/S_{+}^{1} ≅ Σ_{+}(S^{1} × S^{1}). □

There is an S^{1}-homeomorphism h : S_{+}^{1} ∧ S_{+}^{1} → (S^{1} × S^{1})_{+} given by h(g, s) = (g, gs), which descends over c to another S^{1}-homeomorphism S_{+}^{1} ∧ S^{1} → (S^{1} × S^{1})/ΔS^{1}. Hence we can make compatible identifications

Map(S_{+}^{1}, T (ℤ))^{S^{1}} ≅ T (ℤ)
(3.2)  Map(Σ_{+}(S^{1} × S^{1}), T (ℤ))^{S^{1}} ≅ ΩΛT (ℤ)
Map(Σ(S^{1} × S^{1}/ΔS^{1}), T (ℤ))^{S^{1}} ≅ Ω^{2}T (ℤ).

For example, an S^{1}-map f: S_{+}^{1} → T (ℤ) is identified with f(1) ∈ T (ℤ).

Lemma 3.3. There is a map of Puppe fiber sequences

[figure: ΩT (ℤ); α; ΩΛT (ℤ); Map(Ē_{+}^{(1)}, T (ℤ))^{S^{1}}; T (ℤ); ΩT (ℤ); β; Ω^{2}T (ℤ); γ; Map(E_{+}^{(1)}, T (ℤ))^{S^{1}}; T (ℤ)]

where α is the looped difference of the adjoints to the circle action map μ: S_{+}^{1} ∧ T (ℤ) → T (ℤ) and the trivial action map ν: S_{+}^{1} ∧ T (ℤ) → T (ℤ). γ is the usual looped inclusion Ω(ΩT (Z)) → Ω(ΛT (ℤ)).

Proof. We apply Map(−, T (ℤ))^{S^{1}} to the map of Puppe cofibration sequences in Lemma 3.1, and make the identifications of (3.2). Then γ is induced by the collapse map S_{+}^{1} → S^{1} taking 1_{+} ⊂ S_{+}^{1} to the base point. Finally pr_{2} corresponds under (3.2) to the circle action map μ, and pr_{1} to the trivial action map ν which forgets the S_{+}^{1}-factor. The lemma follows. □

We momentarily change to spectrum level notation. Recall the splitting from [B2]

T (ℤ) ≃ Hℤ ∨ ⋁_{i≥2} Σ^{2i−1}Hℤ/i.

Here the inclusion of the zero-simplices ι: Hℤ → T (ℤ) gives the map to the first summand.

Let A_{∗} = H_{∗}(Hℤ/p; ℤ/p) be the dual of the Steenrod algebra, with polynomial generators (ξ_{i})_{i≥1} and exterior generators (τ_{i})_{i≥0} when p is odd, and polynomial generators (ζ_{i})_{i≥1} when p = 2. Let χ denote the canonical anti–involution on A_{∗}. Then H_{∗}(Hℤ; ℤ/p) is the subalgebra of A_{∗} generated by (ξ_{i})_{i≥1} and (χτ_{i})_{i≥1} when p is odd, and by (ζ_{1}^{2}, χζ_{2}, χζ_{3}, …) when p = 2. For p odd, ξ_{1} ∈ A_{2p−2} is dual to the Steenrod pth power operation P^{1}, while for p = 2 the class ζ_{1}^{2} ∈ A_{2} is dual to Sq^{2}. (We are following Milnor in writing ζ_{i} rather than ξ_{i} for the polynomial generators in the case p = 2, to better distinguish between the even and odd cases.)

Let X = Map(E_{+}^{(1)}, T (ℤ))_{p}^{S^{1}} be the p-completed mapping spectrum, and let X[0, ∞) be its connective cover. From the bottom fibration sequence in Lemma 3.3 it is clear that the first nonzero homotopy groups of X[0, ∞) are π_{0}X = ℤ̂_{p}, and π_{2p−3}X ≅ ℤ/p.

Lemma 3.4. The first k-invariant of the connective cover of Map(E_{+}^{(1)}, T (ℤ))_{p}^{S^{1}} is the Steenrod pth power operation

P^{1} : Hℤ̂_{p} ⟶^{ι} T (ℤ)_{p} ⟶^{β} Σ^{−1}T (ℤ)_{p} ⟶ Σ^{2p−2}Hℤ/p

when p is odd, respectively the Steenrod squaring operation Sq^{2} : Hℤ̂_{2} → Σ^{2}Hℤ/2 when p = 2.

Proof. The maps μ and ν: S_{+}^{1} ∧ T (ℤ) → T (ℤ) restrict over ι: Hℤ → T (ℤ) to give maps λ and ν ∘ ι: S_{+}^{1} ∧ Hℤ → T (ℤ), which agree on 1_{+} ∧ Hℤ ⊂ S_{+}^{1} ∧ Hℤ. Their difference thus extends over S^{1} ∧ Hℤ → T (ℤ), and induces the derivation

σ: H_{∗}(Hℤ; ℤ/p) ⟶ H_{∗+1}(T (ℤ); ℤ/p)

given by σ(x) = λ_{∗}([S^{1}] ⊗ x), where [S^{1}] ∈ H_{1}(S_{+}^{1}; ℤ/p) is the fundamental class.

By the calculations of [B2], σ maps ξ_{1} ∈ H_{2p−2}(Hℤ; ℤ/p) to the spherical element e_{2p−1} ∈ H_{2p−1}(T (ℤ); ℤ/p) for p odd, while σ maps ζ_{1}^{2} ∈ H_{2}(Hℤ; ℤ/2) to the spherical element e_{3} ∈ H_{3}(T (ℤ); ℤ/2) when p = 2. So the k-invariant Hℤ̂_{p} → Σ^{2p−2}Hℤ/p maps ξ_{1} or ζ_{1}^{2} to the fundamental class of Σ^{2p−2}Hℤ/p, and is therefore equal to the dual cohomology operation, namely P^{1} or Sq^{2} respectively. □

We may now prove Bökstedt’s Theorem 1.4. We return to space level notation.

Consider the diagram

(3.5)  [figure: Ω^{2}T (ℤ)_{p}; Q(S^{0})_{p}; K(ℤ)_{p}; tr_{S^{1}}; tr; T (ℤ)_{p}^{hS^{1}}; ρ; X; T (ℤ)_{p}.]

Here the vertical maps are part of the bottom fiber sequence of Lemma 3.3, and ρ is given by restriction over the S^{1}-inclusion E_{+}^{(1)} ⊂ E_{+} = ES_{+}^{1}. On the level of spectral sequences, ρ induces the natural map from (1.2) to its two rightmost nonzero columns, where s = 0 or s = −2. The resulting two–column spectral sequence is simply the long exact homotopy sequence of the cited fiber sequence.

Recall that the first nonzero homotopy groups of Q(S^{0})_{p} are π_{0}Q(S^{0})_{p} ≅ ℤ̂_{p} and π_{2p−3}Q(S^{0})_{p} ≅ ℤ/p, and the first k-invariant is P^{1} detecting α_{1} in the odd primary case, and Sq^{2} detecting η in the case p = 2.

The composite Q(S^{0})_{p} → X[0, ∞) induces a π_{0}-isomorphism, and by Lemma 3.4 the first k-invariants of these spaces agree. Hence the induced map on connected components induces a π_{2p−3}-isomorphism, taking α_{1} to the generator of π_{2p−3}X.

Thus α_{1} ∈ π_{2p−3}Q(S^{0}) is detected in the rightmost two nonzero columns of the spectral sequence (1.2), where the only nonzero summand in total degree 2p − 3 is in bidegree (−2, 2p − 1). Thus a generator in this bidegree is hit. This completes the proof of Theorem 1.4. □

References

[B1] M. Bökstedt, Topological Hochschild homology, Topology (to appear).

[B2] , The topological Hochschild homology of ℤ and ℤ/p, Ann. of Math. (to appear).

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Department of Mathematics, University of Oslo, Norway

E-mail address: rognes@math.uio.no