TWO-PRIMARY ALGEBRAIC

K-THEORY OF POINTED SPACES

John Rognes

Abstract. We compute the mod 2 cohomology of Waldhausen’s algebraic K-theory spectrum A(∗) of the category of finite pointed spaces, as a module over the Steenrod algebra. This also computes the mod 2 cohomology of the smooth Whitehead spectrum of a point, denoted Wh^{Diff}(∗). Using an Adams spectral sequence we compute the 2-primary homotopy groups of these spectra in dimensions ∗ ≤ 18, and up to extensions in dimensions 19 ≤ ∗ ≤ 21. As applications we show that the linearization map L: A(∗) → K(ℤ) induces the zero homomorphism in mod 2 spectrum cohomology in positive dimensions, the space level Hatcher–Waldhausen map hw: G/O → ΩWh^{Diff}(∗) does not admit a four-fold delooping, and there is a 2-complete spectrum map M: Wh^{Diff}(∗) → Σg/o_{⊕} which is precisely 9-connected. Here g/o_{⊕} is a spectrum whose underlying space has the 2-complete homotopy type of G/O.

Introduction

Let A(X) be Waldhausen’s algebraic K-theory of spaces functor evaluated on the space X, see [Wa1]. When X is a manifold, A(X) provides the fundamental link between algebraic K-theory and the geometric topology of X — in particular with the concordance space, the h-cobordism space and the automorphism space of X, see [Wa3]. We are therefore interested in evaluating its homotopy type. It is the aim of this paper to compute the 2-primary homotopy type of A(X) in the case when X = ∗ is the one-point space. We achieve this by computing the mod 2 spectrum cohomology of A(∗) as a module over the mod 2 Steenrod algebra. The result is a complete calculation valid in all dimensions; we also compute the homotopy groups of A(∗) modulo odd torsion in dimensions ∗ ≤ 18, and up to extensions in dimensions 19 ≤ ∗ ≤ 21.

We begin by discussing some definitions and interpretations of A(X), in order to explain why this is an important homotopy type.

One way to define A(X) is as the algebraic K-theory of a category with cofibrations and weak equivalences R_{f}(X), whose objects are retractive spaces over X subject to a relative finiteness condition, see [Wa5]. When X = ∗ this category R_{f}(∗) is the category of finite pointed CW-complexes and pointed cellular maps, and is the category of pointed spaces alluded to in the title. The cofibrations are the cellular embeddings, and the weak equivalences are the homotopy equivalences.

Let hR_{f}(X) be the subcategory of R_{f}(X) obtained by restricting the morphisms to be homotopy equivalences, and let |hR_{f}(X)| denote its geometric realization. As a space, A(X) is defined as the loop space Ω|hS_{•}R_{f}(X)|, where S_{•} is Waldhausen’s simplicial construction of the same name. This construction can be iterated, and

Typeset by AMS-TEX

in fact A(X) is an infinite loop space with nth delooping |hS_{•}^{(n)}R_{f}(X)| for each n ≥ 1. There is a canonical map

e: |hR_{f}(X)| ⟶ A(X)

from the geometric realization of the category of finite retractive spaces over X and homotopy equivalences to the infinite loop space A(X).

There is a natural isomorphism π_{0}A(X) ≅ ℤ, and for every object Y ∈ hR_{f}(X) the image under π_{0}(e) of the corresponding point in |hR_{f}(X)| is the Euler characteristic χ(Y, X) = χ̃(Y/X) of Y relative to X. From this point of view the map e is a lift of the usual Euler characteristic that takes values in the integers, to a map which takes values in the infinite loop space A(X). Furthermore, a diagram of spaces and homotopy equivalences given as a functor F: C → hR_{f}(X) gives rise to a map e ∘ |F |: |C| → A(∗), which will detect more information than just the Euler characteristics of the individual spaces in the diagram. For example a finite pointed G-CW complex Y gives rise to a map BG → A(∗) whose homotopy class is a refined invariant of Y. We think of e as a homotopy theoretic improvement on the Euler characteristic, able also to detect information about diagrams of spaces and homotopy equivalences, rather than just individual spaces, and A(X) is the receptacle for this improved Euler characteristic.

In fact A(∗) is a kind of universal receptacle for homotopy invariants of finite pointed spaces that take values in infinite loop spaces and are subject to the following additivity condition: for each cofiber sequence Y′ → Y → Y′′ we have [Y′] + [Y′′] = [Y] where [Y] ∈ π_{0}A(∗) denotes the path component in A(∗) of the invariant applied to Y. Of course, the corresponding universal invariant taking values in an abelian group is just the reduced Euler characteristic. We shall not make the universality claim more precise in this introduction, but note that a similar discussion applies for A(X) and suitably additive homotopy invariants of finite retractive spaces over X.

Hereafter it will be more convenient to work with spectra than with infinite loop spaces. The given deloopings of the infinite loop space A(X) determine a unique connective spectrum, and from now on A(X) will refer to this spectrum. The body of this paper is also written in terms of spectra rather than infinite loop spaces, partly because a few non-connective spectra will appear.

Suspension of retractive spaces over X induces an equivalence on the level of algebraic K-theory, and so A(X) can also be considered as the algebraic K-theory of a category of spectra over X. It is simplest to make this precise for X = ∗, when A(∗) is equivalent to the algebraic K-theory of the category of finite CW-spectra, with respect to suitable notions of cofibrations and stable equivalences, see [Wa4].

Let 𝕊 be the sphere spectrum in some good closed symmetric monoidal category of spectra and spectrum maps, for example the S-modules of [EKMM] or the Γspaces of [Se] and [Ly]. In either case the ring spectrum 𝕊 is a monoid object with respect to the internal smash product, and a spectrum is a module over 𝕊, so we can sensibly refer to spectra as 𝕊-modules. Then A(∗) can be described as the algebraic K-theory of a category of 𝕊-modules subject to suitable finiteness conditions, and briefly A(∗) is the algebraic K-theory of the ring spectrum 𝕊. See [BHM] for a discussion in terms of FSPs.

More generally, for a unital and associative ring spectrum A we may consider a category of finitely generated free A-modules, and form its algebraic K-theory, see [D2]. These ring spectra are the monoids in one of the symmetric monoidal categories of spectra considered above, and may conveniently be called 𝕊-algebras. For each ring R in the algebraic sense, the Eilenberg–Mac Lane spectrum HR is an 𝕊-algebra whose algebraic K-theory agrees with Quillen’s K(R), see [Q2]. For a simplicial monoid G the unreduced suspension spectrum Σ^{∞}(G_{+}) is an 𝕊-algebra whose algebraic K-theory agrees with Waldhausen’s A(X) for X = BG. Thus 𝕊algebras encompass the previous examples of inputs for algebraic K-theory. Now 𝕊 is a commutative 𝕊-algebra, so its algebraic K-theory K(𝕊) = A(∗) is itself a ring spectrum, and furthermore the algebraic K-theory K(A) of any 𝕊-algebra is a module spectrum over A(∗). Hence every algebraic K-theory spectrum considered so far is a module spectrum over A(∗), which further emphasizes the special role played by A(∗).

The relationship of A(X) to geometric topology is based on the splitting of spectra A(X) ≃ Σ^{∞}(X_{+}) ∨ Wh^{Diff}(X) for the smooth category, and the cofiber sequence of spectra

A(∗) ∧ X_{+} ⟶^{α} A(X) ⟶ Wh^{PL}(X)

for the piecewise linear category, see [Wa3] and [Wa6]. Here α is the assembly map, one construction of which uses that A(X) is a homotopy functor in X, see [WW2].

The spectra Wh^{Diff}(X) and Wh^{PL}(X) are the smooth and PL Whitehead spectra, respectively. The topological Whitehead spectrum Wh^{Top}(X) is equivalent to the PL one by [KS] and [BuLa]. Thus knowledge of A(∗) determines Wh^{Diff}(∗) and is the ingredient needed to pass from A(X) to Wh^{PL}(X) ≃ Wh^{Top}(X). The underlying infinite loop spaces of these Whitehead spectra are called Whitehead spaces, and it is perhaps more common to work in terms of these.

When X is a smooth manifold, Ω^{∞}Wh^{Diff}(X) gives the homotopy functor that best approximates the space C^{Diff}(X) of smooth concordances (= pseudoisotopies) of X. By Igusa’s stability theorem [Ig] there is a stabilization map

Σ_{X}^{Diff} : C^{Diff}(X) ⟶ Ω^{2}Ω^{∞}Wh^{Diff}(X)

which is at least roughly n/3-connected where n is the dimension of X. Similar results relate Wh^{PL}(X) and Wh^{Top}(X) to the PL- and topological concordance spaces C^{PL}(X) and C^{Top}(X) when X is a PL- or topological manifold, respectively.

Furthermore there is a geometrically significant involution on A(X), related through the Whitehead spectra to the involution on concordance spaces arising from ‘turning a concordance upside-down’, see [H] and [Vog]. By [WW1] there is a map

Φ_{X}^{Diff} : D̃iff(X)/Diff(X) ⟶ Ω^{∞}(EC_{2+} ∧_{C_{2}} ΩWh^{Diff}(X))

which is at least as connected as the stabilization map considered by Igusa. The C_{2}-action on ΩWh^{Diff}(X) on the right is given by the involution, and the homotopy orbit construction is formed on the spectrum level. This is a space level interpretation of the output of the Hatcher spectral sequence [H], which works on the level of homotopy groups.

The space D̃iff(X)/Diff(X) measures the difference between the topological group Diff(X) of diffeomorphisms of the smooth manifold X and the simplicial group D̃iff(X) of ‘block diffeomorphisms’, which is computable in terms of surgery theory, see [H]. Thus knowledge of the homotopy orbits for the involution acting on the spectrum Wh^{Diff}(X), or equivalently on the spectrum A(X), can be viewed as giving knowledge of the homotopy type of the space of diffeomorphisms Diff(X) in dimensions up to roughly n/3, where n is the dimension of X. Similar results apply for the spaces of PL homeomorphisms of PL manifolds and homeomorphisms of topological manifolds. See [WW3] for a more detailed survey.

In this paper we shall determine the homotopy type of the 2-primary completion of the spectrum Wh^{Diff}(∗). Since the Whitehead spectrum is a homotopy functor and preserves connectivity of maps, for any smooth n-manifold X which is roughly n/3-connected the map Φ_{X}^{Diff} composed with the natural map

Ω^{∞}(EC_{2+} ∧_{C_{2}} ΩWh^{Diff}(X)) ⟶ Ω^{∞}(EC_{2+} ∧_{C_{2}} ΩWh^{Diff}(∗))

is roughly n/3-connected. Thus when our 2-primary calculation is extended to a calculation of the C_{2}-homotopy orbits of Wh^{Diff}(∗), we will have complete information about the 2-primary homotopy type of the space of diffeomorphisms Diff(X) of roughly n/3-connected manifolds up to dimension roughly n/3. We leave these calculations for a future paper.

We now turn to a description of the contents of the present paper.

We are able to access the homotopy type of A(∗) by means of a comparison of algebraic K-theory with the topological cyclic homology theory of Bökstedt, Hsiang and Madsen [BHM], relying on a theorem of Dundas [D1]. In Chapter 1 we review these notions, and are led in Theorem 1.11 to the homotopy cartesian square

(0.1)  \xymatrix{A(∗) \ar[r]^{L} \ar[d]^{_{trc_{∗}}} & K(ℤ) \ar[d]^{_{trc_{ℤ}}} \\ T C(∗) \ar[r]^{L} & T C(ℤ) .}

Here T C denotes the topological cyclic homology functor, and the natural transformation trc is the cyclotomic trace map of [BHM]. After 2-adic completion we can identify the primary homotopy type of A(∗) because the 2-primary homotopy types of the three other spectra in this diagram are known, together with sufficient information about the maps in the diagram. More specifically, the homotopy type of T C(∗) was determined in [BHM], for odd primes p the p-adic completion of T C(ℤ) was computed in [BM], and the 2-adic completion was determined in [R5]. The 2-adic completion of K(ℤ) was found in [RW], by arguments based on Voevodsky’s proof of the Milnor conjecture [Voe] and the Bloch–Lichtenbaum spectral sequence [BlLi]. The 2-adic map trc_{ℤ}: K(ℤ) → T C(ℤ) was also studied in [R5], in sufficient detail that we can describe A(∗) as an extension of T C(∗) by the common homotopy fiber of the maps labelled trc_{∗} and trc_{ℤ} in the diagram above.

At odd primes p, the missing information needed to determine the p-primary homotopy type of A(∗) is the identification of the p-adic completion of K(ℤ), i.e., a proof of the p-primary Lichtenbaum–Quillen conjecture for the integers, and the determination of how A(∗) is an extension of T C(∗) by the homotopy fiber of trc_{ℤ}, after p-adic completion. Since A(∗) has finite type [Dw], and is rationally equivalent to K(ℤ), this would suffice to determine the integral homotopy type of A(∗).

Also in Chapter 1 we make precise a part of the calculation of T C(∗) from [BHM], relating its p-adic completion to the Thom spectrum ℂP_{−1}^{∞} = T h(−γ^{1}) of minus the canonical complex line bundle over ℂP^{∞}. See Theorem 1.16 and Corollary 1.21, which when combined yield a homotopy equivalence T C(∗) ≃ Σ^{∞}S^{0} ∨ΣℂP_{−1}^{∞} after p-adic completion.

In Chapter 2 we analyze the 2-primary homotopy type of ℂP_{−1}^{∞} by classical methods. We obtain its homotopy groups in dimensions ∗ ≤ 20 in Theorem 2.13, by use of the Atiyah–Hirzebruch spectral sequence for stable homotopy associated to the skeleton filtration of ℂP_{−1}^{∞} by the subspectra ℂP_{−1}^{s} for s ≥ −1. The E^{1}-term in this spectral sequence is given in terms of the stable homotopy groups of spheres, π_{∗}^{S}, and the differentials depend on the attaching maps for the cells in ℂP_{−1}^{∞}. This involves primary and secondary operations in homotopy, somewhat along the lines of Toda’s book [To], and we build on previous work for ℂP^{∞} by Mosher [Mo] and Mukai [Mu1], [Mu2] and [Mu3].

It is much easier to describe ℂP_{−1}^{∞} cohomologically, and in Proposition 2.15 we find that the mod 2 spectrum cohomology of ℂP_{−1}^{∞} is cyclic as an A-module, where A is the mod 2 Steenrod algebra, and we describe the annihilator ideal C of the generator in Definition 2.14. The squaring operations Sq^{i} with i odd together with the admissible monomials Sq^{I} of length ≥ 2 form a basis for C as an 𝔽_{2}-vector space. Thus H_{spec}^{∗}(ℂP_{−1}^{∞}; 𝔽_{2}) ≅ Σ^{−2}A/C as left graded A-modules. This allows us to describe the E_{2}-term of the Adams spectral sequence for the 2-adically completed homotopy of ℂP_{−1}^{∞} in a range in Tables 2.18(a) and (b). Combined with the results from the Atiyah–Hirzebruch spectral sequence, we are also able to determine the differentials that land in homotopical degree t − s ≤ 20 in this spectral sequence. The details of this computation will be applied in Chapter 5, where Adams filtration and sparseness in the Adams spectral sequence will make it easier for us to study the homotopy type of A(∗) (and Wh^{Diff}(∗)) in terms of its spectrum cohomology and the differentials in its Adams spectral sequence, rather than by means of the long exact sequences in homotopy arising from Dundas’ homotopy cartesian square (0.1).

In Chapter 3 we familiarize ourselves with the spectrum hofib(trc) defined as the homotopy fiber of the (implicitly 2-completed) map

trc_{ℤ}: K(ℤ) → T C(ℤ) .

By Dundas’ theorem this is also the homotopy fiber of the map trc_{∗}: A(∗) → T C(∗). The principal result is Theorem 3.13, which expresses this common homotopy fiber as the homotopy fiber of the spectrum map δ: Σ^{−2}ku ⟶ Σ^{4}ko given as a suitably connected cover of the explicit composite map

Σ^{4}r ∘ β^{−2} ∘ (ψ^{3} − 1) ∘ β^{−1} : Σ^{−2}KU ⟶ Σ^{4}KO .

From this description it is easy to extract other homotopical information about hofib(trc), such as its homotopy groups (Corollary 3.16), its spectrum cohomology (Theorem 4.4), or its Adams spectral sequence (Tables 3.18(a) and (b)).

The calculations in Chapter 3 are based on the spectrum level description of K(ℤ[\frac{1}{2}]) given in Theorem 3.4, and of K(ℚ_{2}) given in Theorem 3.6, which were obtained in [RW] and [R5, 8.1] respectively. The calculation of K(ℤ[\frac{1}{2}]) relied on the proven Lichtenbaum–Quillen conjecture in this case [RW], using essential inputs from algebraic geometry [Voe] and [BlLi], while the identification of K(ℚ_{2}) in [R5] amounted to the calculation of T C(ℤ) completed at 2, which used topological cyclic homology and calculational spectral sequence techniques from stable homotopy theory. The results in Chapter 3 also rely on knowing how the natural map j′: K(ℤ[\frac{1}{2}]) → K(ℚ_{2}) acts on the level of homotopy groups, which was determined in [R5, 7.7 and 9.1]. Those results depended on knowing the structure of the Ktheory spectra involved, not just their homotopy groups, and were feasible because the prime 2 is so small, or perhaps because it is regular.

These inputs allow us to obtain a spectrum level description of the homotopy fiber of j′ in Propositions 3.10 and 3.11, with a more convenient reformulation given in Proposition 3.12. The arguments rely on knowing the endomorphism algebras of the 2-completed connective topological K-theory spectra ko and ku, as well as all the maps between them, which stems from [MST]. Using Quillen’s localization sequence in algebraic K-theory, and Hesselholt and Madsen’s link between K(ℤ_{2}) and T C(ℤ) from [HM, Thm. D], we rework the description of hofib(j′) into a spectrum level description of hofib(trc) in Theorem 3.13, as desired.

In Chapter 4 we use the cofiber sequence (3.14)

ℂP_{−1}^{∞} ⟶^{i} hofib(trc) ⟶^{j} Wh^{Diff}(∗) ,

and the splitting A(∗) ≃ Σ^{∞}S^{0} ∨ Wh^{Diff}(∗), to reduce the identification of A(∗) to that of ℂP_{−1}^{∞}, which was studied in Chapter 2, to that of hofib(trc), which was settled in Chapter 3, and the map i between the two. At the prime 2 we are in the fortunate situation that the mod 2 spectrum cohomology of ℂP_{−1}^{∞} is cyclic as an A-module on a generator in degree −2, so because Wh^{Diff}(∗) is 2-connected it follows that i induces a surjection on cohomology in all degrees. Thus we can omit any discussion of the linearization map L: T C(∗) → T C(ℤ) in Dundas’ homotopy cartesian square, and still obtain a complete cohomological description of Wh^{Diff}(∗).

This is achieved in the main Theorem 4.5. We have an isomorphism of left graded A-modules

H_{spec}^{∗}(A(∗); 𝔽_{2}) ≅ H_{spec}^{∗}(Σ^{∞}S^{0}; 𝔽_{2}) ⊕ H_{spec}^{∗}(Wh^{Diff}(∗); 𝔽_{2})

where H_{spec}^{∗}(Σ^{∞}S^{0}; 𝔽_{2}) = 𝔽_{2} is the trivial A-module in dimension zero, and there is a unique nontrivial extension of left graded A-modules

Σ^{−2}C/A(Sq^{1}, Sq^{3}) ⟶ H_{spec}^{∗}(Wh^{Diff}(∗); 𝔽_{2}) ⟶ Σ^{3}A/A(Sq^{1}, Sq^{2})

characterizing H_{spec}^{∗}(Wh^{Diff}(∗); 𝔽_{2}). Here C ⊂ A is the annihilator ideal of the generator for H_{spec}^{∗}(ℂP_{−1}^{∞}; 𝔽_{2}), introduced in Definition 2.14. The assertion of the theorem is that algebraically there are precisely two such extensions of left graded A-modules, and H_{spec}^{∗}(Wh^{Diff}(∗); 𝔽_{2}) is the one that does not split.

In Chapter 5 we turn to a homotopical analysis of the smooth Whitehead spectrum Wh^{Diff}(∗), and thus also of A(∗). Our approach is to study the Adams spectral sequence (5.5)

E_{2}^{s,t} = Ext_{A}^{s,t}(H_{spec}^{∗}(Wh^{Diff}(∗); 𝔽_{2}), 𝔽_{2}) ⟹ π_{t−s}(Wh^{Diff}(∗))_{2}^{∧} .

Here we can in principle compute the E_{2}-term in a large range of bidegrees, but there will be many families of differentials and a complete determination of the homotopy groups of Wh^{Diff}(∗) is out of reach.

The cofiber sequence (3.14) displayed above has the special property that its connecting map induces the zero map in mod 2 spectrum cohomology, so its associated long exact sequence in cohomology breaks up into short exact sequences, which in turn induce long exact sequences of Ext_{A}-groups. Thus the E_{2}-terms of the Adams spectral sequences for ℂP_{−1}^{∞}, hofib(trc) and Wh^{Diff}(∗) are linked in a long exact sequence (5.6). The spectral sequence for hofib(trc) was completely described in Chapter 3, and in Chapter 5 we use the long exact sequence of E_{2}-terms to translate the information from Chapter 2 about differentials in the Adams spectral sequence for ℂP_{−1}^{∞} to information about differentials in the Adams spectral sequence (5.5) for Wh^{Diff}(∗). This is a convenient approach, because the Adams spectral sequence of hofib(trc) is concentrated above the line t − s = 2s + 3, while the differentials in the spectral sequence for ℂP_{−1}^{∞} mostly originate below this line. The only subtle point concerns whether certain h_{1}-divisible classes in bidegrees (s, t) = (4k, 12k+3) of (5.5) are hit by differentials, but a comparison with [R5, 9.1] reveals that they indeed do survive to the E_{∞}-term. Thus the complexity of determining the homotopy groups of Wh^{Diff}(∗) is in practice equivalent to that of determining the homotopy groups of ℂP_{−1}^{∞}, which is a well-explored but not exhaustively analyzed problem.

The Adams E_{2}-term for Wh^{Diff}(∗) is displayed in part in Tables 5.7(a) and (b), and the nonzero differentials landing in homotopical dimension t − s ≤ 21 are listed in Proposition 5.9. This leads to a calculational conclusion in Theorem 5.10, where the 2-completed homotopy groups of Wh^{Diff}(∗) are listed in dimensions ∗ ≤ 18, and up to group extensions in dimensions 19 ≤ ∗ ≤ 21. Previously only the homotopy groups in dimensions ≤ 3 were known, see [BW]. We do not give names to the classes identified in π_{∗}(Wh^{Diff}(∗)), but in Theorem 7.5 we show that the (space level) Hatcher–Waldhausen map hw: G/O → ΩWh^{Diff}(∗) constructed in [Wa3, §3] induces an isomorphism on 2-primary homotopy groups in dimensions ∗ ≤ 8, and an injection on 2-primary homotopy groups in dimensions ∗ ≤ 13. Thus the better known homotopy groups of G/O ≃ BSO × CokJ account for much of the low-dimensional homotopy of Wh^{Diff}(∗).

In Chapter 6 we use the known spectrum level description of K(ℤ) completed at 2 to compute its mod 2 spectrum cohomology in Theorem 6.4, and to show in Corollary 6.8 that the linearization map L: A(∗) → K(ℤ) induces the zero map in mod 2 spectrum cohomology in positive dimensions. Thus the linearization map does not itself provide a good cohomological approximation to A(∗). In Remark 6.9 we explain why the Hatcher–Waldhausen map hw does not admit a four-fold delooping, using that multiplication by the Hopf map σ ∈ π_{7}^{S} is nonzero on π_{4}(ΩWh^{Diff}(∗)), but is zero on π_{4}(G/O). We also explain how this relates to the results of [R1], where an infinite loop map from G/O to a different infinite loop space structure on ΩWh^{Diff}(∗) was obtained.

Following Miller and Priddy [MP], we describe in (6.3) a spectrum g/o_{⊕} as the homotopy fiber of the 2-completed unit map Σ^{∞}S^{0} → K(ℤ). Its underlying space G/O_{⊕} has the same 2-adic homotopy type as the usual G/O. Although there is no spectrum map Σg/o_{⊕} → Wh^{Diff}(∗) inducing a π_{3}-isomorphism, we construct in Chapter 7 a 2-complete spectrum map M: Wh^{Diff}(∗) → Σg/o_{⊕} which induces an isomorphism on mod 2 spectrum cohomology in all dimensions ∗ ≤ 9. This is a best possible approximation, since the cohomology groups differ in dimension 10. The comparison of Wh^{Diff}(∗) with Σg/o_{⊕} finally allows us to evaluate the Hatcher– Waldhausen map on 2-completed homotopy groups in dimensions ∗ ≤ 13, leading to the previously cited Theorem 7.5.

Acknowledgement. The main part of this work was done in December 1997 during visits to Aarhus and Bielefeld. The author thanks M. Bökstedt, I. Madsen, J. Tornehave and F. Waldhausen for helpful discussions and hospitality.

1. Algebraic K-theory and topological cyclic homology

We commence by discussing the cyclotomic trace map from algebraic K-theory to topological cyclic homology, and a special case of Dundas’ theorem comparing relative algebraic K-theory to relative topological cyclic homology.

1.1. Γ-spaces and 𝕊-algebras. Let S_{∗} be the category of pointed simplicial sets, and let Γ^{op} be the category of finite pointed sets k_{+} = {0, 1, …, k} based at 0, and base-point preserving functions. This is the opposite of Segal’s category Γ from [Se]. Let ΓS_{∗} be the category of Γ-spaces, i.e., functors F: Γ^{op} → S_{∗} with F (0_{+}) = ∗. Each Γ-space F naturally extends to a functor F: S_{∗} → S_{∗}, which when evaluated on spheres determines a connective (pre-)spectrum {n ↦ F (S^{n})}. We write π_{∗}(F) for the homotopy groups of this spectrum. The natural inclusion Γ^{op} → S_{∗} is a Γ-space denoted 𝕊, whose associated spectrum is the sphere spectrum. The groups π_{∗}(𝕊) are the stable homotopy groups of spheres.

There is a smash product ∧ of Γ-spaces defined by Lydakis in [Ly], making (ΓS_{∗}, ∧, 𝕊) a symmetric monoidal category. A monoid A in this symmetric monoidal category will be called an 𝕊-algebra. Its associated spectrum is an associative ring spectrum, conveniently thought of as an algebra over the sphere spectrum.

1.2. Examples of 𝕊-algebras. When G is a simplicial group the functor Σ^{∞}(G_{+}) given by Σ^{∞}(G_{+})(k_{+}) = G_{+} ∧ k_{+} is a Γ-space. The group multiplication and unit define the structure maps

μ: Σ^{∞}(G_{+}) ∧ Σ^{∞}(G_{+}) ⟶ Σ^{∞}(G_{+})

and η: 𝕊 → Σ^{∞}(G_{+}) making Σ^{∞}(G_{+}) an 𝕊-algebra. Its associated ring spectrum is the unreduced suspension spectrum on G, with product map induced by the multiplication on G.

When R is a (discrete) ring the functor HR given by HR(k_{+}) = R{1, …, k} (the free R-module on the non-basepoint elements in k_{+}) is a Γ-space. The ring multiplication and unit define the structure maps

μ: HR ∧ HR ⟶ HR

and η: 𝕊 → HR making HR an 𝕊-algebra. Its associated ring spectrum is the Eilenberg–Mac Lane spectrum representing ordinary cohomology with coefficients in R.

Let G be a simplicial group, with group of path components π_{0}(G), and let R = ℤ[π_{0}(G)] be its integral group ring. The linearization map is the map of 𝕊algebras L: Σ^{∞}(G_{+}) → HR taking g ∧ i ∈ G_{+} ∧ k_{+} to [g] · i ∈ R{1, … k}, where g ∈ G, i ∈ {1, … k} and [g] denotes the path component of g viewed as an element of π_{0}(G) ⊂ R.

1.3. Algebraic K-theory, topological Hochschild homology and topological cyclic homology. Let A be an 𝕊-algebra. The extended functor A: S_{∗} → S_{∗} comes equipped with a product and unit map making it an FSP (functor with smash product) in the sense of [B2]. In [BHM] Bökstedt, Hsiang and Madsen functorially define the algebraic K-theory spectrum K(A), topological Hochschild homology spectrum T HH(A) and topological cyclic homology spectrum T C(A, p) of an FSP A. Here p is any prime. An integral functor A ↦ T C(A) has been defined by Goodwillie (unpublished), together with a natural p-adic equivalence T C(A) → T C(A, p) for each prime p.

When G is a simplicial group with classifying space X = BG, we write A(X) = K(Σ^{∞}(G_{+})), T HH(X) = T HH(Σ^{∞}(G_{+})) and T C(X, p) = T C(Σ^{∞}(G_{+}), p). Here A(X) is naturally homotopy equivalent to Waldhausen’s algebraic K-theory spectrum A(X) of the space X [Wa1], i.e., the algebraic K-theory of the category of finite retractive spaces over X.

When R is a ring we write K(R) = K(HR), T HH(R) = T HH(HR) and T C(R, p) = T C(HR, p). Here K(R) is naturally homotopy equivalent to Quillen’s algebraic K-theory spectrum K(R) of the ring R [Q2], i.e., the algebraic K-theory of the category of finitely generated projective R-modules.

We recall from [BHM, 3.7] that there are C-equivariant homotopy equivalences

(1.4)  T HH(X) ≃_{C} Σ_{C}^{∞}(ΛX_{+})

for each finite subgroup C ⊂ S^{1}. Here Σ_{C}^{∞} denotes the C-equivariant suspension spectrum, and C ⊂ S^{1} acts on the free loop space ΛX by rotating the loops.

1.5. Trace maps. A trace map tr_{X}: A(X) → T HH(X) was defined by Waldhausen in [Wa2], and Bökstedt defined a trace map tr_{A}: K(A) → T HH(A) in [B2], as a natural transformation of functors from FSPs to spectra. The cyclotomic trace map trc_{A} of [BHM] gives a factorization

K(A) ⟶^{trc_{A}} T C(A, p) ⟶^{β_{A}} T HH(A)

of tr_{A}, although the map to T C(A, p) was initially only defined up to homotopy. The map β_{A} is a projection map from the homotopy limit defining T C(A, p). When A = Σ^{∞}(G_{+}) with X = BG or A = HR we substitute X or R, respectively, for A in the notations trc_{A}, β_{A} and tr_{A}. Thus trc_{X}: A(X) → T C(X, p), etc.

In the case A = Σ^{∞}(G_{+}) with X = BG the six authors of [6A] gave a model for the cyclotomic trace map trc_{X} as a natural transformation in X. When A = HR, Dundas and McCarthy [DuMc] gave models for K(R) and T C(R) such that trc_{R} is a natural transformation. Finally Dundas [D2] has provided a construction of functors K, T HH and T C from 𝕊-algebras to spectra, and natural transformations trc: K → T C, β: T C → T HH and tr: K → T HH with tr = β ∘ trc, which agree up to natural homotopy equivalence with the preceding definitions.

1.6. Dundas’ theorem. The following theorem of Dundas [D1] generalizes to maps of 𝕊-algebras a theorem of McCarthy [Mc] valid for maps of simplicial rings. Both results are analogous to an older theorem about rational algebraic K-theory due to Goodwillie [Go].

Theorem 1.7 (Dundas). Let ϕ: A → B be a map of 𝕊-algebras, such that the ring homomorphism π_{0}(ϕ): π_{0}(A) → π_{0}(B) is a surjection with nilpotent kernel. Then the commutative square of spectra

\xymatrix{K(A) \ar[r]^{ϕ} \ar[d]^{_{trc_{A}}} & K(B) \ar[d]^{_{trc_{B}}} \\ T C(A) \ar[r]^{ϕ} & T C(B)}

is homotopy cartesian.

Corollary 1.8 (Dundas). Let G be a simplicial group, and write X = BG and R = ℤ[π_{1}(X)] = ℤ[π_{0}(G)]. The linearization map L: Σ^{∞}(G_{+}) → HR induces a homotopy cartesian square

\xymatrix{A(X) \ar[r]^{L} \ar[d]^{_{trc_{X}}} & K(R) \ar[d]^{_{trc_{R}}} \\ T C(X) \ar[r]^{L} & T C(R) .}

In particular, the vertical homotopy fiber hofib(trc_{X}) only depends on the fundamental group π_{1}(X), for a pointed connected space X.

For the last claim we used that any pointed connected space X is homotopy equivalent to BG for a simplicial group G, e.g. the Kan loop group of X. See [Wa7].

1.9. Whitehead spectra. There are natural cofiber sequences of spectra

Σ^{∞}(X_{+}) ⟶^{i} A(X) ⟶ Wh^{Diff}(X)

and

A(∗) ∧ X_{+} ⟶^{α} A(X) ⟶ Wh^{PL}(X)

where Wh^{Diff}(X) is the smooth Whitehead spectrum of X, and Wh^{PL}(X) is the piecewise linear Whitehead spectrum of X. The sequences are constructed geometrically in [Wa3], where Wh^{Diff}(X) is interpreted in terms of stabilized smooth concordance spaces and stabilized spaces of smooth h-cobordisms, and similarly in the piecewise linear case. The identification of the upper left hand homology theory in X with Σ^{∞}(X_{+}) uses the ‘vanishing of the mystery homology theory’ established in [Wa6].

The composite

Σ^{∞}(X_{+}) ⟶^{i} A(X) ⟶^{tr_{X}} T HH(X) ≃ Σ^{∞}(ΛX_{+}) ⟶^{ev} Σ^{∞}(X_{+})

is homotopic to the identity. Here ev: ΛX → X is the map evaluating a free loop S^{1} → X at the identity 1 ∈ S^{1}. Hence ev ∘ tr_{X} provides a natural splitting for the first cofiber sequence above, as in

A(X) ≃ Σ^{∞}(X_{+}) ∨ Wh^{Diff}(X) .

We can therefore identify Wh^{Diff}(X) with the homotopy fiber of the splitting map ev ∘ tr_{X}.

1.10. The smooth Whitehead spectrum of a point. Suppose G = 1, so that X = ∗. Then ev: ΛX → X is the identity map, T HH(∗) ≃ Σ^{∞}S^{0}, and the splitting above identifies Wh^{Diff}(∗) with the homotopy fiber of tr_{∗}. We obtain a map of horizontal (co-)fiber sequences of spectra:

[figure: Wh^{Diff}(∗); t̃rc; A(∗); ^{tr^{∗}}; _{trc_{∗}}; T HH(∗); T̃ C(∗); T C(∗); β_{∗}; T HH(∗) .]

Here T̃ C(∗) is defined as the homotopy fiber of β_{∗}, and t̃rc is the induced map of homotopy fibers over trc_{∗} and the identity map on T HH(∗). The unit map Σ^{∞}S^{0} → A(∗) → T C(∗) and β_{∗} yield a splitting

T C(∗) ≃ Σ^{∞}S^{0} ∨ T̃ C(∗) .

Theorem 1.11. The two squares

\xymatrix{Wh^{Diff}(∗) \ar[r] \ar[d]^{t̃rc} & A(∗) \ar[r]^{L} \ar[d]^{_{trc_{∗}}} & K(ℤ) \ar[d]^{_{trc_{ℤ}}} \\ T̃ C(∗) \ar[r] & T C(∗) \ar[r]^{L} & T C(ℤ)}

are homotopy cartesian, and induce homotopy equivalences of vertical homotopy fibers

hofib(t̃rc) ⟶^{≃} hofib(trc_{∗}) ⟶^{≃} hofib(trc_{ℤ}) .

We denote either of these by hofib(trc). □

1.12. The topological cyclic homology of a point. The topological cyclic homology T C(X, p) of a pointed connected space X was computed by Bökstedt, Hsiang and Madsen in [BHM]. We recall their result, making precise a point that was omitted in the published argument. See [Ma, §4.4] for more details about the following review.

Fix a prime p. From (1.4) there is an equivalence T HH(X)^{C_{p^{n}}} ≃ Σ_{C_{p^{n}}}^{∞} (ΛX_{+})^{C_{p^{n}}} for each n ≥ 0. The Segal–tom Dieck splitting

Σ_{C_{p^{n}}}^{∞} (ΛX_{+})^{C_{p^{n}}} ≃ ∏_{k=0}^{n} Σ^{∞}(EC_{p^{k}} ×_{C_{p^{k}}} ΛX^{C_{p^{n−k}}})_{+}

and the power map homeomorphisms Δ_{p}^{n−k}: ΛX ≅ ΛX^{C_{p^{n−k}}} combine to give an equivalence

(1.13)  T HH(X)^{C_{p^{n}}} ≃ ∏_{k=0}^{n} Σ^{∞}(EC_{p^{k}} ×_{C_{p^{k}}} ΛX)_{+} .

The pth power map Δ_{p}: Σ^{∞}ΛX_{+} → Σ^{∞}ΛX_{+} is induced by taking a free loop S^{1} → X to its precomposition by the standard degree p map S^{1} → S^{1}. Let

t_{p}: Σ^{∞}(EC_{p^{n}} ×_{C_{p^{n}}} ΛX)_{+} → Σ^{∞}(EC_{p^{n−1}} ×_{C_{p^{n−1}}} ΛX)_{+}

be the Becker–Gottlieb transfer for the principal C_{p}-bundle EC_{p^{n−1}} ×_{C_{p^{n−1}}} ΛX → EC_{p^{n}} ×_{C_{p^{n}}} ΛX. There are restriction and Frobenius maps R, F: T HH(X)^{C_{p^{n}}} → T HH(X)^{C_{p^{n−1}}} . Up to homotopy these are given by the formulas:

R(x_{0}, x_{1}, …, x_{n}) = (x_{0}, x_{1}, …, x_{n−1})
F (x_{0}, x_{1}, …, x_{n}) = (Δ_{p}(x_{0}) + t_{p}(x_{1}), t_{p}(x_{2}), …, t_{p}(x_{n})) .

Here x_{k} refers to the factor in Σ^{∞}(EC_{p^{k}} ×_{C_{p^{k}}} ΛX)_{+} in the equivalence (1.13), and the formulas must be interpreted as giving maps defined in terms of this splitting.

Writing

(1.14)  T R(X, p) = holim_{n,R} T HH(X)^{C_{p^{n}}} ≃ ∏_{n=0}^{∞} Σ^{∞}(EC_{p^{n}} ×_{C_{p^{n}}} ΛX)_{+}

we have R(x_{0}, x_{1}, x_{2}, …) = (x_{0}, x_{1}, x_{2}, …) and F (x_{0}, x_{1}, x_{2}, …) = (Δ_{p}(x_{0}) + t_{p}(x_{1}), t_{p}(x_{2}), t_{p}(x_{3}), …) up to homotopy. The topological cyclic homology spectrum T C(X, p) is defined as the homotopy equalizer

\xymatrix{T C(x, p) \ar[r]^{π} & T R(X, p) \ar[r]^{R} \ar[r]_{F} & T R(X, p) ,}

and is homotopy equivalent to the homotopy fiber of 1−F: T R(X, p) → T R(X, p). Let T, D: T R(X, p) → T R(X, p) be given up to homotopy by the formulas:

T (x_{0}, x_{1}, x_{2}, …) = (t_{p}(x_{1}), t_{p}(x_{2}), t_{p}(x_{3}), …)
D(x_{0}, x_{1}, x_{2}, …) = (Δ_{p}(x_{0}), 0, 0, …) .

The following observation lets us calculate T C(X, p).

Lemma 1.15. The composite (1−T)∘(1−D): T R(X, p) → T R(X, p) is homotopic to (1 − F).

Proof. In terms of the splitting (1.14) it is clear that (1 − D)(x_{0}, x_{1}, x_{2}, …) = (x_{0} − Δ_{p}(x_{0}), x_{1}, x_{2}, …) is mapped by (1 − T) to (x_{0} − Δ_{p}(x_{0}) − t_{p}(x_{1}), x_{1} − t_{p}(x_{2}), x_{2} − t_{p}(x_{3}), …), which is homotopic to (1 − F)(x_{0}, x_{1}, x_{2}, …). □

Given such a choice of commuting homotopy for the right hand square below, there is an induced map of horizontal (co-)fiber sequences

[figure: T C(X, p); π; α_{X}; T R(X, p); 1−F; 1−D; T R(X, p); C(X, p); T R(X, p); 1−T; T R(X, p) .]

Here we have written C(X, p) for the homotopy limit holim_{n,t_{p}} Σ^{∞}(EC_{p^{n}} ×_{C_{p^{n}}} ΛX)_{+}, which is homotopy equivalent to the homotopy fiber of 1−T in view of (1.14). When α_{X} is determined by the right hand commuting homotopy, the left hand square is homotopy cartesian (with respect a suitable commuting homotopy). Let pr: T R(X, p) → T HH(X) ≃ Σ^{∞}ΛX_{+} denote projection to the zeroth term in the homotopy limit defining T R(X, p). Then there is clearly a homotopy cartesian square

\xymatrix{T R(X, p) \ar[r]^{pr} \ar[d]^{1−D} & Σ^{∞}ΛX_{+} \ar[d]^{1−Δ_{p}} \\ T R(X, p) \ar[r]^{pr} & Σ^{∞}ΛX_{+} ,}

again with respect to a suitable commuting homotopy. We can combine these two homotopy cartesian squares horizontally. Then the upper composite β_{X} = pr ∘ π: T C(X, p) → T R(X, p) → T HH(X) ≃ Σ^{∞}ΛX_{+} agrees with the natural transformation β of 1.5. The lower composite is the projection pr_{0} : C(X, p) → Σ^{∞}ΛX_{+} from the homotopy limit system over the Becker–Gottlieb transfer maps to its zeroth term.

Theorem 1.16. [BHM, §5] Let X be a pointed connected space and write C(X, p) = holim_{n,t_{p}} Σ^{∞}(EC_{p^{n}} ×_{C_{p^{n}}} ΛX)_{+}. The diagram

\xymatrix{T C(X, p) \ar[r]^{α_{X}} \ar[d]^{β_{X}} & C(X, p) \ar[d]^{pr_{0}} \\ Σ^{∞}ΛX_{+} \ar[r]^{1−Δ_{p}} & Σ^{∞}ΛX_{+}}

homotopy commutes, and there exists a commuting homotopy making the diagram homotopy cartesian.

This is now clear. (The proofs in [BHM] and [Ma] only show that the horizontal homotopy fibers in this diagram are homotopy equivalent, not necessarily by the map induced by β_{X} and pr_{0}.) Specializing to X = ∗ we have the following corollary, which is what we will use in the rest of the present paper.

Corollary 1.17. There is a (co-)fiber sequence of spectra

T̃ C(∗, p) ⟶ holim_{n,t_{p}} Σ^{∞}(BC_{p^{n}+}) ⟶^{pr_{0}} Σ^{∞}S^{0} .

For each n ≥ 0 there is a dimension-shifting S^{1}-transfer map

trf_{S^{1}}^{n}: Σ^{∞}(Σ(ℂP_{+}^{∞})) ⟶ Σ^{∞}(BC_{p^{n}+})

associated to the S^{1}-bundle BC_{p^{n}} → BS^{1} ≃ ℂP^{∞}. See [K], [LMS] or [Mu1]. These induce a map

Σ^{∞}(Σ(ℂP_{+}^{∞})) ⟶ holim_{n,t_{p}} Σ^{∞}(BC_{p^{n}+})

which is a homotopy equivalence after p-adic completion. Hence we can identify the map pr_{0} above with the S^{1}-transfer map trf_{S^{1}}^{0}, briefly denoted trf_{S^{1}}, after p-adic completion. Combined with the p-adic equivalence T C(∗) → T C(∗, p) we obtain:

Corollary 1.18. [BHM, 5.15] There is a homotopy equivalence

T̃ C(∗) ≃ hofib(trf_{S^{1}} : Σ^{∞}(Σ(ℂP_{+}^{∞})) → Σ^{∞}S^{0})

after p-adic completion, for each prime p.

1.19. A Thom spectrum. Let ℂP_{k}^{∞} denote the truncated complex projective space with one cell in each even dimension greater than or equal to 2k, interpreted as a spectrum when k < 0. There is a homotopy equivalence

ℂP_{k}^{∞} ≃ T h(kγ^{1})

where the right hand side is the Thom spectrum of k times the canonical complex line bundle over ℂP^{∞}, see [At]. We shall be concerned with the case k = −1, i.e., with the spectrum ℂP_{−1}^{∞}, which can be thought of as the Thom spectrum of minus the canonical line bundle on ℂP^{∞}.

Theorem 1.20 (Knapp). There is a homotopy equivalence

ΣℂP_{−1}^{∞} ≃ hofib(trf_{S^{1}} : Σ^{∞}(Σ(ℂP_{+}^{∞})) → Σ^{∞}S^{0}) .

See [K, 2.9] for a proof. Bringing these results together we have shown:

Corollary 1.21. There is a homotopy equivalence

(ΣℂP_{−1}^{∞})_{p}^{∧} ≃ T̃ C(∗)_{p}^{∧}

of p-adically completed spectra. □

2. Two-primary homotopy of ℂP_{−1}^{∞}

In this chapter we study the 2-primary homotopy type of the Thom spectrum ℂP_{−1}^{∞} of minus the canonical complex line bundle over ℂP^{∞}. We first use a reindexed Atiyah–Hirzebruch spectral sequence for stable homotopy to compute the 2-completed homotopy groups π_{∗}(ℂP_{−1}^{∞})_{2}^{∧} in dimensions ∗ ≤ 20, and next compare with the Adams spectral sequence with the same abutment to determine the differentials in the latter spectral sequence in the same range of dimensions.

Only the simple cohomology calculations in Proposition 2.15 and Lemma 2.16 are needed for Chapters 3 and 4, including Theorem 4.5. The harder homotopy calculations in this chapter will first be applied in Chapter 5 and onwards. The reader who is primarily interested in the spectrum cohomology of A(∗), rather than its homotopy groups, can therefore read 2.14 – 2.16 and then skip ahead to Chapter 3.

The reindexed Atiyah–Hirzebruch spectral sequence in question is derived from the stable homotopy exact couple associated to the filtration of ℂP_{−1}^{∞} by the subspectra ℂP_{−1}^{s}, for s ≥ −1. Its E^{1}-term is

(2.1)  E_{s,t}^{1} = π_{s+t}(ℂP_{−1}^{s}/ℂP_{−1}^{s−1}) ≅ π_{t−s}^{S}

for s ≥ −1, and zero elsewhere. Here π_{k}^{S} = π_{k}(Σ^{∞}S^{0}) is the kth stable stem.

To determine the differentials in the reindexed Atiyah–Hirzebruch spectral sequence, we compare with the computation by Mosher [Mo] of the differentials in the corresponding spectral sequence for the stable homotopy of ℂP^{∞}. The E^{1}-term of the latter spectral sequence is obtained from (2.1) by restricting to filtrations s ≥ 1, i.e., by omitting the columns s = −1 and s = 0, and the collapse map j: ℂP_{−1}^{∞} → ℂP^{∞} induces a map of spectral sequences. From here on we often use the same notation for a based space and its suspension spectrum, such as writing S^{0} for Σ^{∞}S^{0}.

The differentials in (2.1) landing in filtration s = 0 are always zero, due to the splitting ℂP_{0}^{∞} = ℂP_{+}^{∞} ≃ ℂP^{∞} ∨ S^{0}. The differentials in (2.1) landing in filtration s = −1 arise from the connecting map in the cofiber sequence S^{−2} → ℂP_{−1}^{∞} → ℂP_{+}^{∞}. This is the wedge sum of the (desuspended) S^{1}-transfer map ℂP^{∞} → S^{−1}, and the (desuspended) multiplication by η map S^{0} → S^{−1}. The image of the S^{1}- transfer map was computed in dimensions ∗ ≤ 20 by Mukai in [Mu1], [Mu2] and [Mu3], and we use these results to determine the differentials in (2.1) landing in filtration s = −1 in the same range of dimensions.

For ease of reference we use similar notation for classes in our spectral sequence (2.1) as in [Mo]. Thus we write β_{s} ∈ E_{s,s+t}^{1} for the class corresponding to β ∈ π_{t}^{S}, and write ℤ/n(β) for a cyclic group of order n with generator β. In Tables 2.5 and 2.12 we briefly write n(β) for ℤ/n(β) and (β) for ℤ(β), to save some space. Hereafter we concentrate on the 2-primary components, and all spectra and groups are implicitly 2-completed. Differentials are mostly given only up to multiplication by a 2-adic unit.

In dimensions ∗ ≤ 22, we will use the following presentation for the stable stems π_{∗}^{S}, following the tables in [To, XIV] and [Ra, A3.3].

π_{0}^{S} = ℤ(ι), π_{1}^{S} = ℤ/2(η), π_{2}^{S} = ℤ/2(η^{2}), π_{3}^{S} = ℤ/8(ν), π_{4}^{S} = 0, π_{5}^{S} = 0, π_{6}^{S} = ℤ/2(ν^{2}), π_{7}^{S} = ℤ/16(σ), π_{8}^{S} = ℤ/2(ν̄) ⊕ ℤ/2(ϵ), π_{9}^{S} = ℤ/2(ν^{3}) ⊕ ℤ/2(ηϵ) ⊕ ℤ/2(μ), π_{10}^{S} = ℤ/2(ημ), π_{11}^{S} = ℤ/8(ζ), π_{12}^{S} = 0, π_{13}^{S} = 0, π_{14}^{S} = ℤ/2(σ^{2}) ⊕ ℤ/2(κ), π_{15}^{S} = ℤ/32(ρ) ⊕ ℤ/2(ηκ), π_{16}^{S} = ℤ/2(η^{∗}) ⊕ ℤ/2(ηρ), π_{17}^{S} = ℤ/2(ηη^{∗}) ⊕ ℤ/2(νκ) ⊕ ℤ/2(η^{2}ρ)⊕ℤ/2(μ̄), π_{18}^{S} = ℤ/8(ν^{∗})⊕ℤ/2(ημ̄), π_{19}^{S} = ℤ/2(σ̄)⊕ℤ/8(ζ̄), π_{20}^{S} = ℤ/8(κ̄), π_{21}^{S} = ℤ/2(νν^{∗}) ⊕ ℤ/2(ηκ̄) and π_{22}^{S} = ℤ/2(νσ̄) ⊕ ℤ/2(η^{2}κ̄).

For a fixed r, the d^{r}-differentials in the spectral sequence for π_{∗}^{S}(ℂP^{∞}) are periodic in the filtration degree s, see [Mo, 4.4], and this periodicity propagates to the spectral sequence (2.1). Hence Mosher’s description of the d^{1}-, d^{2}- and d^{3}- differentials for ℂP^{∞} in [Mo, 5.1, 5.2, and 5.4] extends to give the formulas 2.2, 2.3 and 2.4 for the corresponding differentials in (2.1). Let β ∈ π_{∗}^{S}.

Proposition 2.2. d^{1}(β_{s}) = 0 for s odd and d^{1}(β_{s}) = ηβ_{s−1} for s even.

Proposition 2.3. d^{2}(β_{s}) = νβ_{s−2} for s ≡ 0, 1, 4, 5 mod 8, d^{2}(β_{s}) = 2νβ_{s−2} for s ≡ 3, 6 mod 8 and d^{2}(β_{s}) = 0 for s ≡ 2, 7 mod 8.

Proposition 2.4. d^{3}(β_{s}) = 0 for s odd. If s is even then d^{3}(β_{s}) = γ_{s−3}, where γ ∈ ⟨η, ν, β⟩ for s ≡ 0 mod 8, γ ∈ ⟨ν, η, β⟩ for s ≡ 2 mod 8, γ ∈ ⟨2ν, η, β⟩ + ⟨η, ν, β⟩ for s ≡ 4 mod 8 and γ ∈ ⟨ν, η, β⟩ + ⟨η, 2ν, β⟩ for s ≡ 6 mod 8.

The d^{1}-differentials in (2.1) are given by the following multiplicative relations in π_{∗}^{S}, see [Ra] and [To].

η · ι = η, η · η = η^{2}, η · η^{2} = 4ν, η · ν = 0, η · ν^{2} = 0, η · σ = ν̄ + ϵ, η · ν̄ = ν^{3}, η · ϵ = ηϵ, η · ν^{3} = 0, η · ηϵ = 0, η · μ = ημ, η · ημ = 4ζ, η · ζ = 0, η · σ^{2} = 0, η · κ = ηκ, η · ρ = ηρ, η · ηκ = 0, η · η^{∗} = ηη^{∗}, η · ηρ = η^{2}ρ, η · ηη^{∗} = 4ν^{∗}, η · νκ = 0, η · η^{2}ρ = 0, η · μ̄ = ημ̄, η · ν^{∗} = 0, η · ημ̄ = 4ζ̄, η · σ̄ = 0, η · ζ̄ = 0, η · κ̄ = ηκ̄, η · νν^{∗} = 0 and η · ηκ̄ = η^{2}κ̄.

For example, σ̄ = ⟨ν, ησ, σ⟩, so η · σ̄ = −⟨η, ν, ησ⟩σ = 0 with zero indeterminacy. The d^{2}-differentials in (2.1) are given by the following multiplicative relations in

π_{∗}^{S}, see [Ra] and [To].

ν · ι = ν, ν · ν = ν^{2}, ν · ν^{2} = ν^{3}, ν · σ = 0, ν · ν̄ = 0, ν · ν^{3} = 0, ν · ηϵ = 0, ν · μ = 0, ν · ζ = 0, ν · σ^{2} = 0, ν · κ = νκ, ν · ρ = 0, ν · ηκ = 0, ν · η^{∗} = 0, ν · νκ = 4κ̄, ν · η^{2}ρ = 0, ν · μ̄ = 0, ν · ν^{∗} = νν^{∗}, ν · ημ̄ = 0, ν · σ̄ = νσ̄ and ν · ζ̄ = 0.

The d^{3}-differentials are given by the following secondary compositions, from [MT], [Mo, 10.1] and [To].

⟨ν, η, ν⟩ = ν̄, ⟨η, ν, 2ν⟩ = ⟨η, 2ν, ν⟩ = {ϵ, ν̄}, ⟨ν, η, ζ⟩ ⊆ {0, ηρ}, ⟨η, ν, ζ⟩ = {0, ηρ}, ⟨ν, η, σ^{2}⟩ = σ̄, ⟨ν, η, 2ρ⟩ = {0, 4κ̄} and ⟨ν, η, ηκ⟩ = ±2κ̄ by [MT].

The resulting E^{4}-term is shown in Table 2.5, accounting for all differentials landing in total degree s + t ≤ 20.

In lemmas 2.6 to 2.11, we only consider differentials landing in total degree s + t ≤ 20.

Lemma 2.6. The nonzero d^{4}-differentials in (2.1) are d^{4}(2ι_{3}) = 2σ_{−1}, d^{4}(4ι_{5}) = 8σ_{1}, d^{4}(4ι_{6}) = 8σ_{2}, d^{4}(ι_{7}) = 2σ_{3}, d^{4}(8ι_{8}) = 8σ_{4}, d^{4}(4ι_{9}) = 4σ_{5}, d^{4}(2ι_{10}) = 2σ_{6} and d^{4}(σ_{3}) = σ_{−1}^{2}.

Proof. The d^{4}-differentials landing in filtration s ≥ 1 and total degree s + t ≤ 19 are determined by those in the spectral sequence for π_{∗}^{S}(ℂP^{∞}), and are given in [Mo, 5.6 and 6.4].

In total degree 20, d^{4}(ζ_{5}) = 0 by the computation of π_{20}^{S}(ℂP^{5}) following [Mu3, 4.2], and d^{4}(σ_{7}) = 0 by the proof of [Mu3, 4.3] (the formula γ_{6}σ = 2iσ̃′σ).

The differentials landing in filtration s = 0 are always zero, as noted above. The differentials landing in filtration s = −1 are determined by the computation of the S^{1}-transfer in [Mu1] and [Mu2]. Thus d^{4}(2ι_{3}) = 2σ_{−1} by [Mu1, 13.1(iii)], d^{4}(σ_{3}) = σ_{−1}^{2} by the proof of [Mu2, 5.3] (the formula g_{4}σ̃′ = σ^{2}), d^{4}(ν̄_{3}) = 0 by the proof of [Mu2, 5.3] (the formula g_{8}i˜ν̄ = ηκ), d^{4}(μ_{3}) = 0 by the proof of [Mu2, 5.4] (the formula g_{4}μ̃ = 0), and d^{4}(ζ_{3}) = 0 by the proof of [Mu2, 5.5] (the formula g_{4}π_{17}^{S}(ℂP^{3}) = 0). □

Lemma 2.7. The nonzero d^{5}-differentials in (2.1) are d^{5}(8ι_{6}) = μ_{1}, d^{5}(16ι_{8}) = μ_{3} and d^{5}(16ι_{10}) = μ_{5}.

Proof. The d^{5}-differentials landing in filtration s ≥ 1 and total degree s + t ≤ 19 are determined by those in the spectral sequence for π_{∗}^{S}(ℂP^{∞}), and are given in [Mo, 6.5].

In total degree 20, d^{5}(ηϵ_{6}) = 0 by the calculation of π_{20}^{S}(ℂP^{6}) following [Mu3, 4.2].

The differentials landing in filtration s = −1 are d^{5}(8ι_{4}) = 0 by [Mu1, 13.1(iv)], d^{5}(2σ_{4}) = 0 by the proof of [Mu2, 5.4] (the formula g_{5}˜2σ′ ≡ 0 mod μσ), d^{5}(ν_{4}^{3}) = 0 and d^{5}(ηϵ_{4}) = 0 by the proof of [Mu2, 5.5] (the formulas g_{5}ν̃^{3} ≡ 0 mod {4ν^{∗}, ημ̄} and g_{5}λ = 0, where λ was chosen as a coextension of η^{2}σ before [Mu2, 4.7]), and d^{5}(ζ_{4}) = 0 by [Mu3, 5.1]. □

Lemma 2.8. The nonzero d^{6}-differentials in (2.1) are d^{6}(8ι_{5}) = ζ_{−1}, d^{6}(8ι_{7}) = 2ζ_{1}, d^{6}(32ι_{8}) = 2ζ_{2}, d^{6}(16ι_{9}) = ζ_{3}, d^{6}(32ι_{10}) = 4ζ_{4}, d^{6}(2ν_{5}) = κ_{−1}, d^{6}(σ_{5}) = ν_{−1}^{∗} and d^{6}(σ_{7}) = 2ν_{1}^{∗}.

Proof. The differentials landing in filtration s ≥ 1 and total degree s + t ≤ 19 come from [Mo, 6.6], and d^{6}(σ_{7}) = 2ν_{1}^{∗} by [Mu3, 4.3] and its proof (namely, γ_{6}σ =

TWO-PRIMARY ALGEBRAIC K-THEORY OF POINTED SPACES 17

0

0 4(2κ̄_{0})
2(κ̄_{−1}) 2(σ̄_{0}) 2(2ν_{1}^{∗})
⊕
8(ζ̄_{0})
4(ζ̄_{−1}) 8(ν_{0}^{∗}) 2(μ̄_{1}) 0 32(ρ_{3})
4(ν_{−1}^{∗}) 2(νκ_{0}) 2(η_{1}^{∗}) 16(2ρ_{2}) 2(σ_{3}^{2}) 0
⊕  ⊕
2(η^{2}ρ_{0})  2(κ_{3})
2(μ̄_{−1}) 0 32(ρ_{1}) 0 0 0 4(ζ_{5})
2(η_{−1}^{∗}) 16(2ρ_{0}) 2(σ_{1}^{2}) 0  0 8(ζ_{4}) 0 2(ηϵ_{6})
⊕
2(ηκ_{0})
32(ρ_{−1}) 2(σ_{0}^{2}) 0  0 4(ζ_{3}) 0 2(μ_{5}) 0 16(σ_{7})
2(σ_{−1}^{2}) 0 0 8(ζ_{2}) 0 2(ν_{4}^{3}) 0 8(2σ_{6}) 0 0
⊕  ⊕
2(κ_{−1})  2(ηϵ_{4})
0  0 4(ζ_{1}) 0 2(μ_{3}) 0 16(σ_{5}) 0  0  0 0
0  8(ζ_{0}) 0 2(ηϵ_{2}) 2(ν̄_{3}) 8(2σ_{4}) 2(ν_{5}^{2}) 0  0 2(4ν_{8}) 0 0
4(ζ_{−1}) 0 2(μ_{1}) 0 16(σ_{3}) 0  0  0  0  0 0 (2ι_{10})
0 2(ν_{0}^{3}) 0 8(2σ_{2}) 0  0  0 2(ν_{6}) 0  0 (4ι_{9})
⊕
2(ηϵ_{0})
2(μ_{−1}) 0 16(σ_{1}) 0  0  0 2(2ν_{5}) 0  0 (8ι_{8})
0 8(2σ_{0}) 2(ν_{1}^{2}) 0  0  0  0  0 (ι_{7})
16(σ_{−1}) 2(ν_{0}^{2}) 0  0  0  0  0 (4ι_{6})
0  0  0  0  0  0 (4ι_{5})
0  0  0  0  0 (8ι_{4})
0  8(ν_{0}) 0  0 (2ι_{3})
0  0  0 (2ι_{2})
0  0 (4ι_{1})
0  (2ι_{0})

(ι_{−1})

Table 2.5. E^{4} in total degrees s + t ≤ 20.

2iσ̃′σ = 2iν^{∗}).

Also d^{6}(8ι_{5}) = ζ_{−1} by [Mu1, 13.1(v)], d^{6}(2ν_{5}) = κ_{−1} by the proof of [Mu2, 5.3] (the formula g_{8}(i˜2ν′′′) ≡ κ mod σ^{2}), d^{6}(ν_{5}^{2}) = 0 by the proof of [Mu2, 5.4] (the formula g_{9}iν̃^{2} ≡ ω^{∗}η mod iνκ), and d^{6}(σ_{5}) = ±ν_{−1}^{∗} by the proof of [Mu2, 5.5] (the formula g_{6}σ̃′′ ≡ xν^{∗} mod ημ̄ where x is odd). □

Lemma 2.9. The only nonzero d^{7}-differential in (2.1) is d^{7}(ν_{6}) = η_{−1}^{∗}.

Proof. We have d^{7}(ν_{6}) = η_{−1}^{∗} by [Mu2, 5.4] and its proof (the formula g_{7}ν̃′′ = ω^{∗}). All other d^{7}-differentials are zero by [Mo, 6.7] or bidegree reasons. □

Lemma 2.10. The nonzero d^{8}-differentials in (2.1) are d^{8}(16ι_{7}) = 2ρ_{−1}, d^{8}(64ι_{9}) = 16ρ_{1} and d^{8}(64ι_{10}) = 16ρ_{2}.

Proof. These follow from [Mu1, 4.3] since 2ρ generates the complex image of J in dimension 15, and from [Mo, 6.8]. □

Lemma 2.11. The remaining nonzero differentials in (2.1) are d^{9}(2^{7}ι_{10}) = μ̄_{1} and d^{10}(2^{7}ι_{9}) = ζ̄_{−1}.

Proof. These follow from [Mo, 6.9] and [Mu1, 4.3], since ζ̄ generates the complex image of J in dimension 19. □

This leaves us with the E^{∞}-term shown in Table 2.12, in total degrees s+t ≤ 20. Recall the convention that n(β) denotes a cyclic group of order n, generated by β.

Theorem 2.13. The 2-primary homotopy groups of ℂP_{−1}^{∞} in dimensions ∗ ≤ 20 are as follows:

π_{−2}(ℂP_{−1}^{∞}) = ℤ(ι_{−1})
π_{−1}(ℂP_{−1}^{∞}) = 0
π_{0}(ℂP_{−1}^{∞}) = ℤ(2ι_{0})
π_{1}(ℂP_{−1}^{∞}) = 0
π_{2}(ℂP_{−1}^{∞}) = ℤ(4ι_{1})
π_{3}(ℂP_{−1}^{∞}) = ℤ/8(ν_{0})
π_{4}(ℂP_{−1}^{∞}) = ℤ(2ι_{2})
π_{5}(ℂP_{−1}^{∞}) = ℤ/2(σ_{−1})
π_{6}(ℂP_{−1}^{∞}) = ℤ/2(ν_{0}^{2}) ⊕ ℤ(16ι_{3})
π_{7}(ℂP_{−1}^{∞}) = ℤ/2(μ_{−1}) ⋊ ℤ/8(2σ_{0})
≅ ℤ/16(2σ_{0})
π_{8}(ℂP_{−1}^{∞}) = ℤ/2(ν_{1}^{2}) ⊕ ℤ(8ι_{4})
π_{9}(ℂP_{−1}^{∞}) = ℤ/2(ν_{0}^{3}) ⊕ ℤ/2(ηϵ_{0}) ⊕ ℤ/8(σ_{1})
π_{10}(ℂP_{−1}^{∞}) = ℤ(32ι_{5})
π_{11}(ℂP_{−1}^{∞}) = ℤ/8(ζ_{0}) ⊕ ℤ/4(2σ_{2})
π_{12}(ℂP_{−1}^{∞}) = ℤ(16ι_{6})
π_{13}(ℂP_{−1}^{∞}) = ℤ/2(ρ_{−1}) ⋊ ℤ/2(ζ_{1}) ⋊ ℤ/2(ηϵ_{2})
≅ ℤ/2(ρ_{−1}) ⋊ ℤ/4(ηϵ_{2})
π_{14}(ℂP_{−1}^{∞}) = ℤ/2(σ_{0}^{2}) ⊕ ℤ/2(ν̄_{3}) ⊕ ℤ(2^{8}ι_{7})
π_{15}(ℂP_{−1}^{∞}) = ℤ/2(μ̄_{−1}) ⋊ ℤ/16(2ρ_{0}) ⊕ ℤ/2(ηκ_{0}) ⋊ ℤ/2(ζ_{2}) ⋊ ℤ/4(2σ_{4})
≅ ℤ/32(2ρ_{0}) ⊕ ℤ/2(ηκ_{0}) ⋊ ℤ/2(ζ_{2}) ⋊ ℤ/4(2σ_{4})

TWO-PRIMARY ALGEBRAIC K-THEORY OF POINTED SPACES 19

0

0 4(2κ̄_{0})
2(κ̄_{−1}) 2(σ̄_{0}) 0
⊕
8(ζ̄_{0})
0 8(ν_{0}^{∗}) 0  0
0 2(νκ_{0}) 2(η_{1}^{∗}) 8(2ρ_{2}) 2(σ_{3}^{2})
⊕  ⊕
2(η^{2}ρ_{0})  2(κ_{3})
2(μ̄_{−1}) 0 16(ρ_{1}) 0 0 0
0 16(2ρ_{0}) 2(σ_{1}^{2}) 0  0 4(ζ_{4}) 0
⊕
2(ηκ_{0})
2(ρ_{−1}) 2(σ_{0}^{2}) 0  0  0  0  0  0
0  0  0 2(ζ_{2}) 0 2(ν_{4}^{3}) 0  0  0
⊕
2(ηϵ_{4})
0 0 2(ζ_{1}) 0 0 0 0 0 0 0
0 8(ζ_{0}) 0 2(ηϵ_{2}) 2(ν̄_{3}) 4(2σ_{4}) 2(ν_{5}^{2}) 0  0 2(4ν_{8}) 0
0  0  0  0  0  0  0  0  0  0  0 (2^{8}ι_{10})
0 2(ν_{0}^{3}) 0 4(2σ_{2}) 0  0  0  0  0  0 (2^{9}ι_{9})
⊕
2(ηϵ_{0})
2(μ_{−1}) 0 8(σ_{1}) 0  0  0  0  0  0 (2^{7}ι_{8})
0 8(2σ_{0}) 2(ν_{1}^{2}) 0  0  0  0  0 (2^{8}ι_{7})
2(σ_{−1}) 2(ν_{0}^{2}) 0  0  0  0  0 (16ι_{6})
0  0  0  0  0  0 (32ι_{5})
0  0  0  0  0 (8ι_{4})
0 8(ν_{0}) 0  0 (16ι_{3})
0 0 0 (2ι_{2})
0  0 (4ι_{1})
0 (2ι_{0})

(ι_{−1})

Table 2.12. E^{∞} in total degrees s + t ≤ 20.

π_{16}(ℂP_{−1}^{∞}) = ℤ/2(σ_{1}^{2}) ⊕ ℤ/2(ν_{5}^{2}) ⊕ ℤ(2^{7}ι_{8})
π_{17}(ℂP_{−1}^{∞}) = ℤ/2(νκ_{0}) ⊕ ℤ/2(η^{2}ρ_{0}) ⊕ ℤ/16(ρ_{1}) ⋊ ℤ/2(ν_{4}^{3}) ⊕ ℤ/2(ηϵ_{4})

≅ ℤ/2(νκ_{0}) ⊕ ℤ/2(η^{2}ρ_{0}) ⊕ ℤ/32(ηϵ_{4}) ⊕ ℤ/2(ν_{4}^{3})
π_{18}(ℂP_{−1}^{∞}) = ℤ/2(κ̄_{−1}) ⋊ ℤ/8(ν_{0}^{∗}) ⋊ ℤ/2(η_{1}^{∗}) ⊕ ℤ(2^{9}ι_{9})
π_{19}(ℂP_{−1}^{∞}) = ℤ/2(σ̄_{0}) ⊕ ℤ/8(ζ̄_{0}) ⊕ ℤ/8(2ρ_{2}) ⋊ ℤ/4(ζ_{4}) ⋊ ℤ/2(4ν_{8})
≅ ℤ/2(σ̄_{0}) ⊕ ℤ/8(ζ̄_{0}) ⊕ ℤ/64(4ν_{8})
π_{20}(ℂP_{−1}^{∞}) = ℤ/4(2κ̄_{0}) ⋊ ℤ/2(σ_{3}^{2}) ⊕ ℤ/2(κ_{3}) ⊕ ℤ(2^{8}ι_{10}) .

Proof. Up to extensions, this can be read off from the E^{∞}-term above.

In dimensions ∗ = 9, 11, 14, 17, 19 the subgroup in filtration s = 0 is split off by the composite map ℂP_{−1}^{∞} → ℂP_{+}^{∞} → S^{0}, followed by a retraction of π_{∗}^{S} onto the kernel of η: π_{∗}^{S} → π_{∗+1}^{S}.

The extension in dimension 7 will follow from the proof of 2.21 below, in view of h_{0}-multiplications in the Adams spectral sequence for π_{∗}(ℂP_{−1}^{∞}).

The right hand extension in dimension 13 can be read off from

π_{13}^{S}(ℂP^{2}) ≅ ℤ/8(ηϵ_{2}) ⊕ ℤ/2(ν_{2}^{3}),

see [Mu2, p.197].

The left hand extension in dimension 15 can be read off from π_{19}^{S}(ℂP^{2}) ≅ π_{15}^{S}(ℂP_{−1}^{0}), see [Mu3, p.133].

The splitting in dimension 16 can be deduced from the injection π_{16}(ℂP_{−1}^{∞}) → π_{16}(ℂP^{∞}) ≅ (ℤ/2)^{3} ⊕ ℤ, see [Mu2, 1(ii)].

The right hand extension in dimension 17 can be read off from π_{17}^{S}(ℂP^{4}), see [Mu2, 4.7 and 4.8]. Note that η^{2}σ = ν^{3} + ηϵ, so twice the coextension λ of η^{2}σ is twice a coextension of ηϵ.

The middle and right hand extensions in dimension 19 follow from [Mu3, 3.2]. □

We proceed to compare these results with the Adams spectral sequence for π_{∗}^{S}(ℂP_{−1}^{∞})_{2}^{∧}. Let A = A(2) be the mod 2 Steenrod algebra, generated by the Steenrod squaring operations Sq^{i}. For each sequence of natural numbers I = (i_{1}, …, i_{n}) let Sq^{I} = Sq^{i_{1}} ∘ ⋯ ∘ Sq^{i_{n}} be the composite operation. The sequence I, or the operation Sq^{I}, is said to be admissible if i_{s} ≥ 2i_{s+1} for all 0 ≤ s < n. The set of admissible Sq^{I} form a vector space basis for A.

Definition 2.14. Let C be the left ideal in A with vector space basis the set of admissible Sq^{I} such that I = (i_{1}, …, i_{n}) has length n ≥ 2, or I = (i) with i odd. Then A/C is a cyclic left A-module, with vector space basis the set of Sq^{i} with i ≥ 0 even.

Let us briefly write H^{∗}(X) for the mod 2 spectrum cohomology H_{spec}^{∗}(X; 𝔽_{2}) of a spectrum X. It is naturally a graded left A-module.

Proposition 2.15.

H^{∗}(ℂP_{−1}^{∞}) ≅ Σ^{−2}A/C

as graded left A-modules.

Proof. It is clear that H^{n}(ℂP_{−1}^{∞}) ≅ 𝔽_{2} for n ≥ −2 even, and 0 otherwise. In H^{∗}(Σ^{∞}ℂP_{+}^{∞}) ≅ 𝔽_{2}{y^{j} | j ≥ 0} with deg(y) = 2 the squaring operations are given by Sq^{2i−1}(y^{j}) = 0 and Sq^{2i}(y^{j}) = \binom{j}{i}y^{i+j}. By James periodicity and stability of the squaring operations the same formulas apply in

H^{∗}(ℂP_{−1}^{∞}) ≅ 𝔽_{2}{y^{j} | j ≥ −1} ,

including the case j = −1. Then Sq^{2i}(y^{−1}) = y^{i−1} since \binom{−1}{i} ≡ 1 mod 2. To prove the proposition it remains to show that Sq^{I}(y^{−1}) = 0 when I = (i_{1}, …, i_{n}) is admissible of length ≥ 2. Let z = Sq^{i_{n}}(y^{−1}). Then z has dimension (i_{n}−2) and lifts to the ordinary cohomology H^{∗}(ℂP_{+}^{∞}; 𝔽_{2}) of the space ℂP_{+}^{∞}, which is an unstable A-module. Thus Sq^{i_{n−1}}(z) = 0 since i_{n−1} > i_{n} − 2, and so Sq^{I}(y^{−1}) = 0. □

Lemma 2.16. In ℂP_{−1}^{∞}, the lowest k-invariant

k^{1} : Σ^{−2}Hℤ ⟶ ΣHℤ

is nontrivial, and has mod 2 reduction the class of Sq^{3} mod ASq^{1}.

Proof. The lowest homotopy group of ℂP_{−1}^{∞} is detected by a map ℂP_{−1}^{∞} → Σ^{−2}Hℤ. On cohomology it induces a surjection Σ^{−2}A/ASq^{1} → Σ^{−2}A/C, whose kernel Σ^{−2}C/ASq^{1} begins with Σ^{−2}Sq^{3} mod ASq^{1} in degree 1. This is the cohomology operation represented by the lowest k-invariant k^{1}. □

Consider the Adams spectral sequence

(2.17)  E_{2}^{s,t} = Ext_{A}^{s,t}(H^{∗}(ℂP_{−1}^{∞}), 𝔽_{2}) ⟹ π_{t−s}(ℂP_{−1}^{∞})_{2}^{∧} .

Its E_{2}-term can be computed in a range from a (minimal) resolution of Σ^{−2}A/C, either by hand or by Bruner’s Ext-calculator program [Br]. The E_{2}-term in homotopical degrees t − s ≤ 20 is shown in Tables 2.18(a) and (b). The notation _{s}x represents a class arising in the Adams E_{2}-term for ℂP_{−1}^{s}, mapping to the class usually named x in the Adams E_{2}-term for ℂP_{−1}^{s}/ℂP_{−1}^{s−1} ≅ Σ^{2s}S^{0}. The distinction between classes marked as ‘•’ or as ‘∘’ will be explained in Chapter 5, following the long exact sequence (5.6). Classes supporting or hit by differentials are shaded gray. In bidegree (3, 14) a d_{2}-differential hits the sum of the two given generators, which are therefore both half-shaded.

The cofiber sequence of spectra

ℂP_{−1}^{0} ⟶^{i} ℂP_{−1}^{∞} ⟶^{j} ℂP^{∞}

induces a short exact sequence in mod 2 spectrum cohomology, and thus gives a long exact sequence of Ext-groups relating the Adams E_{2}-term (2.17) to the Adams E_{2}-terms

(2.19)  ′E_{2}^{s,t} = Ext_{A}^{s,t}(H^{∗}(ℂP_{−1}^{0}), 𝔽_{2}) ⟹ π_{t−s}(ℂP_{−1}^{0})_{2}^{∧}

and

(2.20)  ′′E_{2}^{s,t} = Ext_{A}^{s,t}(H^{∗}(ℂP^{∞}), 𝔽_{2}) ⟹ π_{t−s}(ℂP^{∞})_{2}^{∧} .

Knowledge of the stable homotopy of ℂP_{−1}^{0} ≃ Σ^{−4}ℂP^{2} and ℂP^{∞} in a range allows us to determine the differentials in the spectral sequences ′E_{∗} and ′′E_{∗} in a similar range. This is comparatively easy for ℂP_{−1}^{0}, and was done for ℂP^{∞} by Mosher in [Mo]. Using the long exact sequence of E_{2}-terms

… ⟶ ′E_{2}^{s,t} ⟶^{i_{∗}} E_{2}^{s,t} ⟶^{j_{∗}} ′′E_{2}^{s,t} ⟶^{∂} ′E_{2}^{s+1,t} ⟶ …

Table 2.18(a). The Adams E_{2}-term for ℂP_{−1}^{∞}

and the geometric boundary theorem [Ra, 2.3.4] we can transfer some of these differentials to (2.17). (The careful reader should come equipped with the Ext charts for ℂP_{−1}^{0} and ℂP^{∞} to check the details in the following proof.)

Proposition 2.21. In the Adams spectral sequence (2.17) the nonzero differentials landing in homotopical degree ≤ 20 are:

(i) d_{2}^{1,8}(_{3}h_{1}) = _{−1}c_{0}.

(ii) d_{2}^{2,12}(_{5}h_{0}^{2}) = h_{0}^{3} · _{1}h_{3}, d_{2}^{3,13}(_{5}h_{0}^{3}) = h_{0}^{4} · _{1}h_{3} = _{−1}P h_{2} and d_{2}^{4,14}(_{5}h_{0}^{4}) = h_{0}^{5} ·_{1}h_{3} = _{−1}h_{0}P h_{2}.

(iii) d_{2}^{1,13}(_{6}h_{0}) = h_{0} · _{2}h_{0}h_{3} + h_{1} · _{5}h_{0}^{2}, d_{2}^{2,14}(_{6}h_{0}^{2}) = h_{0}^{2} · _{2}h_{0}h_{3}, d_{2}^{3,15}(_{6}h_{0}^{3}) = h_{0}^{3} ·_{2}h_{0}h_{3} and d_{2}^{2,15}(_{5}h_{0}h_{2}) = _{−1}d_{0}.

(iv) d_{2}^{1,16}(_{6}h_{2}) = h_{0} ·_{0}h_{3}^{2}, d_{3}^{2,17}(h_{0} ·_{6}h_{2}) = _{0}h_{0}d_{0}, d_{3}^{3,18}(h_{0}^{2} ·_{6}h_{2}) = h_{0} ·_{0}h_{0}d_{0} and d_{3}^{4,19}(h_{0}^{3} ·_{6}h_{2}) = h_{0}^{2} ·_{0}h_{0}d_{0} = _{−1}P c_{0}.

(v) d_{2}^{1,18}(_{5}h_{3}) = h_{0} ·_{1}h_{3}^{2}, d_{2}^{2,19}(h_{0} ·_{5}h_{3}) = h_{0}^{2} ·_{1}h_{3}^{2} = _{−1}f_{0} and d_{3}^{3,20}(h_{0}^{2} ·_{5}h_{3}) = 1h_{0}^{2}d_{0}.

(vi) d_{2}^{4,22}(x) = h_{0}^{2} ·_{4}h_{1}c_{0} with h_{0} · x ≠ 0, d_{2}^{5,23}(h_{0} · x) = h_{0}^{3} ·_{4}h_{1}c_{0}, d_{2}^{6,24}(_{9}h_{0}^{6}) = h_{0}^{7}·_{5}h_{3}, d_{2}^{7,25}(h_{0}·_{9}h_{0}^{6}) = h_{0}^{8}·_{5}h_{3} = _{−1}P^{2}h_{2} and d_{2}^{8,26}(h_{0}^{2}·_{9}h_{0}^{6}) = h_{0}^{9}·_{5}h_{3} = _{−1}h_{0}P^{2}h_{2}.

Table 2.18(b). The Adams E_{2}-term for ℂP_{−1}^{∞}

(vii) d_{2}^{4,24} ≠ 0, d_{2}^{5,25} ≠ 0, d_{2}^{6,26} ≠ 0, d_{2}^{7,27} ≠ 0, d_{2}^{1,22}(_{3}h_{4}) ≠ 0 and d_{2}^{6,27} ≠ 0 all have rank 1.

Proof. We compare the Adams E_{2}-term in Table 2.18 with its abutment 2.13. Each h_{0}-torsion class in the E_{∞}-term of (2.17) comes from an h_{0}-torsion class in the E_{2}- term, and so is represented by a 2-torsion class in π_{∗}(ℂP_{−1}^{∞}). (The proof of this assertion goes by induction over the subspectra ℂP_{−1}^{s} of ℂP_{−1}^{∞}.)

In each degree t−s ≤ 5 the order of the 2-torsion in π_{∗}(ℂP_{−1}^{∞}) equals the order of the h_{0}-torsion in Table 2.18, hence there are no nonzero differentials in this range.

(i): In degree t − s = 6 the 2-torsion in the abutment is ℤ/2, while the E_{2}-term has two h_{0}-torsion generators, so one of these must be hit by a differential. For bidegree reasons the only possibility is d_{2}^{1,8}(_{3}h_{1}) = _{−1}c_{0}, and then there is no room for further differentials landing in degrees t − s ≤ 8.

In degree 7 of the E_{∞}-term there is then a nonzero multiplication by h_{0}^{3}, showing that the extension in π_{7}(ℂP_{−1}^{∞}) is cyclic.

(ii) and (iii): We turn to degrees 9 ≤ t − s ≤ 13. The Adams spectral sequence for ℂP^{∞}, denoted ′′E_{∗} in (2.20), has differentials ′′d_{2}(_{5}h_{0}^{2}) = _{1}h_{0}^{3}h_{3} and ′′d_{2}(_{6}h_{0}) = h_{0} ·_{2}h_{0}h_{3} + h_{1} ·_{5}h_{0}^{2}. This uses π_{9}^{S}(ℂP^{∞}) = ℤ/8 and π_{11}^{S}(ℂP^{∞}) = ℤ/4, see [Mo, 7.2].

The map of spectral sequences j_{∗}: E_{2} → ′′E_{2} is an isomorphism in bidegrees (2, 12) and (1, 13), so these differentials lift to E_{2}.

Regarding the first′′d_{2}-differential, both basis elements in E_{2}^{4,13} ≅ 𝔽_{2}{h_{0}^{3}·_{1}h_{3}, h_{1}· 4h_{0}^{3}} map to _{1}h_{0}^{3}h_{3} in ′′E_{2}^{4,13}. Hence d_{2}(_{5}h_{0}^{2}) equals one or the other of these basis elements. It cannot be h_{1} · _{4}h_{0}^{3}, because then d_{2}(_{5}h_{0}^{3}) = 0 by h_{0}-multiplication, and more classes would survive to the E_{∞}-term in degree 9 than the abutment π_{9}(ℂP_{−1}^{∞}) ≅ ℤ/2 ⊕ ℤ/2 ⊕ ℤ/8 allows. Thus d_{2}^{2,12}(_{5}h_{0}^{2}) = h_{0}^{3} ·_{1}h_{3}. Multiplication by h_{0} implies d_{2}^{3,13}(_{5}h_{0}^{3}) = h_{0}^{4} ·_{1}h_{3} and d_{2}^{4,14}(_{5}h_{0}^{4}) = h_{0}^{5} ·_{1}h_{3} in (2.17).

In bidegrees (2, 12), (2, 13) and (3, 14) the map j_{∗} is an isomorphism, so the second ′′d_{2}-differential lifts to d_{2}^{1,13}(_{6}h_{0}) = h_{0} · _{2}h_{0}h_{3} + h_{1} · _{5}h_{0}^{2}. Multiplication by h_{0}, h_{0}^{2} and h_{1} implies d_{2}^{2,14}(_{6}h_{0}^{2}) = h_{0}^{2} · _{2}h_{0}h_{3}, d_{2}^{3,15}(_{6}h_{0}^{3}) = h_{0}^{3} · _{2}h_{0}h_{3} and d_{2}^{2,15}(_{5}h_{0}h_{2}) = _{−1}d_{0}, respectively. There is no room for further differentials landing in degree t − s ≤ 13.

(iv): We turn to degrees 14 ≤ t − s ≤ 15. For bidegree reasons the class 0h_{3}^{2} ∈ E_{2}^{2,16} survives to E_{∞}, and the classes h_{0} · _{0}h_{3}^{2} and _{3}c_{0} in E_{2}^{3,17} can only be affected by a d_{2}-differential from _{6}h_{2} ∈ E_{2}^{1,16}. The 2-torsion in π_{14}(ℂP_{−1}^{∞}) is (ℤ/2)^{2}, so the class h_{0} ·_{0}h_{3}^{2} cannot survive to E_{∞}, i.e., there is a nonzero differential d_{2}(_{6}h_{2}) = h_{0} ·_{0}h_{3}^{2} in E_{∗}.

The Adams spectral sequence for ℂP_{−1}^{0}, denoted ′E_{∗} in (2.19), has a differential ′d_{3}(_{0}h_{0}h_{4}) = _{0}h_{0}d_{0}. (This lifts the usual differential d_{3}(h_{0}h_{4}) = h_{0}d_{0} in the Adams spectral sequence for π_{∗}^{S}. Multiplying this by h_{0}^{2} gives the differential ′d_{3}(_{0}h_{0}^{3}h_{4}) = −1P c_{0}, arising from the hidden multiplicative relation η · {h_{0}^{3}h_{4}} = {P c_{0}} in the stable 16-stem.)

The map of spectral sequences i_{∗}: ′E_{2} → E_{2} is injective in bidegree (2, 17), taking _{0}h_{0}h_{4} to h_{0} ·_{6}h_{2}. For in ′E_{2} we know that h_{2} ·_{0}h_{0}h_{4} = h_{0} ·_{0}h_{2}h_{4}. Thus the image of_{0}h_{0}h_{4} in E_{2} is such that h_{2} times it is divisible by h_{0}, and by inspection this property characterizes h_{0} ·_{6}h_{2} ∈ E_{2}^{2,17}. Thus we have another nonzero differential d_{3}(h_{0} ·_{6}h_{2}) = _{0}h_{0}d_{0} in E_{∗}. Multiplication by h_{0} and h_{0}^{2} leads to the differentials d_{3}(h_{0}^{2} ·_{6}h_{2}) = h_{0} ·_{0}h_{0}d_{0} and d_{3}(h_{0}^{3} ·_{6}h_{2}) = h_{0}^{2} ·_{0}h_{0}d_{0} = _{−1}P c_{0}, respectively. There is no room for further differentials landing in degrees 14 ≤ t − s ≤ 15.

(v): Next we consider differentials landing in degree t − s = 16. For bidegree reasons the two classes h_{1} · _{6}h_{2} and _{1}h_{3}^{2} in E_{2}^{2,18} survive to E_{∞}, and since π_{16}(ℂP_{−1}^{∞}) ≅ ℤ ⊕ (ℤ/2)^{2}, the remaining h_{0}-torsion classes are hit by differentials. Thus d_{2}^{1,18}(_{5}h_{3}) = h_{0} ·_{1}h_{3}^{2} and d_{2}^{2,19}(h_{0} ·_{5}h_{3}) = h_{0}^{2} ·_{1}h_{3}^{2} = _{−1}f_{0}.

To determine the last differential landing in degree 16, we compare once again with the Adams spectral sequence ′′E_{∗} for π_{∗}(ℂP^{∞}). Comparing the ′′E_{2}-term and the ′′E_{∞}-term given in Table 7.2 of [Mo] we deduce that there are differentials ′′d_{2}(_{1}h_{4}) = _{1}h_{0}h_{3}^{2}, ′′d_{3}(h_{0} ·_{1}h_{4}) = _{1}h_{0}d_{0} and ′′d_{3}(h_{0}^{2} ·_{1}h_{4}) = _{1}h_{0}^{2}d_{0}. In particular, the cited table asserts that ′′d_{2}(_{4}h_{1}c_{0}) = 0 does not interfere with the second ′′d_{3}- differential. Also ′′d_{2}(h_{2} ·_{7}1) = h_{2} ·′′d_{2}(_{7}1) = 0, and ′′d_{3}(h_{2} ·_{7}1) = 0 follows from π_{16}^{S}(ℂP^{∞}) ≅ ℤ ⊕ (ℤ/2)^{3}.

The map j_{∗}: E_{2} → ′′E_{2} is an isomorphism in degree t − s = 17 and Adams filtration s ≤ 4, while in degree t − s = 16 the kernel consists of the class h_{0}^{2} · 1h_{3}^{2} = _{−1}f_{0} only. Comparing d_{2}^{1,18} to ′′d_{2}^{1,18} it follows that _{5}h_{3} ∈ E_{2}^{1,18} maps to 1h_{4} mod h_{2} ·_{7}1 ∈ ′′E_{2}^{1,18}. Thus d_{3}(h_{0}^{2} ·_{5}h_{3}) maps under j_{∗} to _{1}h_{0}^{2}d_{0}, and we have proven that d_{3}^{3,20}(h_{0}^{2} ·_{5}h_{3}) = _{1}h_{0}^{2}d_{0} in E_{∗}.

(vi): In degree t − s = 17, the abutment has order 2^{8} and the E_{2}-term has 16 classes. Hence there are five differentials landing in degree 17, in addition to the three differentials we have just found leaving that degree. The 2-torsion in the abutment in degree 18 has order 2^{5}, and the E_{2}-term has seven h_{0}-torsion classes. Hence at most two differentials leave the h_{0}-torsion in degree 18, and at least three differentials leave the h_{0}-periodic part of the E_{2}-term. For bidegree reasons this extreme case is precisely what occurs, so d_{2}^{6,24}(_{9}h_{0}^{6}) ≠ 0, d_{2}^{7,25}(_{9}h_{0}^{7}) = _{−1}P^{2}h_{2} and d_{2}^{8,26}(_{9}h_{0}^{8}) = _{−1}h_{0}P^{2}h_{2}, and there are no nonzero differentials landing in degree t − s = 18.

To precisely pin down the differential d_{2}^{6,24} we use the same argument as for d_{2}^{2,12}. The map j_{∗}: E_{2} → ′′E_{2} is an isomorphism in bidegree (6, 24) and surjective in bidegree (8, 25), so the relation h_{1} ·_{8}h_{0}^{7} = h_{0}^{7} ·_{1}h_{4} in ′′E_{2}^{8,25} and the differential ′′d_{2}(_{9}h_{0}^{6}) = h_{0}^{7} · _{1}h_{4} implies that d_{2}(_{9}h_{0}^{6}) is either h_{1} · _{8}h_{0}^{7} or h_{0}^{7} · _{5}h_{3} in E_{2}^{8,25}. Multiplying with h_{0} and comparing with d_{2}^{7,25} eliminates the first possibility, so in fact d_{2}^{6,24}(_{9}h_{0}^{6}) = h_{0}^{7} ·_{5}h_{3}.

Considering h_{0}- and h_{2}-multiplications in the E_{2}-term, either d_{2} = 0 on all h_{0}- torsion classes in degree t − s = 18, or d_{2}^{4,22}(x) = h_{0}^{2} ·_{4}h_{1}c_{0} on the classes x ∈ E_{2}^{4,22} not divisible by h_{0}, and d_{2}^{5,23}(h_{0} · x) = h_{0}^{3} · _{4}h_{1}c_{0}. In the former case, the d_{3}- differential d_{3}^{2,17} would propagate by h_{2}- and h_{0}-multiplications to three nonzero d_{3}-differentials from the h_{0}-torsion in degree t − s = 18, which is incompatible with the abutment. Thus the two d_{2}-differentials given above are correct, and this accounts for all the differentials from degree t − s = 18.

(vii): The proofs in degrees 19 ≤ t − s ≤ 21 are left as exercises for the reader who needs these results. □

3. The fiber of the cyclotomic trace map

When localized at p = 2, the homotopy type of the spectrum K(ℤ) is known. This involves the Bloch–Lichtenbaum spectral sequence relating motivic cohomology to algebraic K-theory, Voevodsky’s proof of the Milnor conjecture, which relates motivic cohomology to étale cohomology, and knowledge of the étale cohomology of the rational 2-integers ℤ[\frac{1}{2}].

Similarly, the p-adic homotopy type of the spectrum T C(ℤ, p) is known for each prime p. They were determined by Bökstedt and Madsen in [BM1] and [BM2] for p odd, and by the author in [R2], [R3], [R4] and [R5] for p = 2. When p = 2 the homomorphisms induced by trc_{ℤ}: K(ℤ) → T C(ℤ, 2) on homotopy groups are known after 2-adic completion. In this chapter we use this to describe the homotopy fiber of the cyclotomic trace map as a spectrum.

Let all spectra be implicitly completed at 2, throughout this chapter.

3.1. Some two-adic K-theory spectra. We say that a (−1)-connected spectrum is connective, and a 0-connected spectrum is connected. Let KO and KU denote the real and complex topological K-theory spectra, let ko and ku denote their connective covers, and let bo and bu denote their connected covers, respectively. Write bso and bspin for the 1- and 3-connected covers of KO, and bsu for the 3-connected cover of KU, as usual.

Complex Bott periodicity provides a homotopy equivalence β: Σ^{2}KU → KU. There is a complexification map c: KO → KU and a realification map r: KU → KO. Smashing with the Hopf map η: Σ^{∞}S^{1} → Σ^{∞}S^{0} yields a map also denoted η: ΣKO → KO. We use the same notation for the various k-connected covers of these maps. There is a cofiber sequence of spectra

(3.2)  Σko ⟶^{η} ko ⟶^{c} ku ⟶^{Σ^{2}r∘β^{−1}} Σ^{2}ko .

This follows from R. Wood’s theorem KO ∧ ℂP^{2} ≃ KU. See also [MQR, V.5.15]. Here we write Σ^{2}r ∘ β^{−1} for a map ∂: ku → Σ^{2}ko that satisfies ∂ ∘ β = Σ^{2}r. This determines the map up to homotopy, even though β: Σ^{2}ku → ku is not exactly invertible.

Theorem 3.3 (Quillen). There is a cofiber sequence of spectra

K(𝔽_{3}) ⟶^{i_{3}} ku ⟶^{ψ^{3}−1} bu ⟶^{∂_{3}} ΣK(𝔽_{3}) . □

This is the spectrum level statement of Quillen’s computation in [Q1].

The computation in [RW] by Weibel and the author of the 2-primary algebraic Kgroups of rings of 2-integers in number fields relies on Suslin’s motivic cohomology for fields [S2], Voevodsky’s proof of the Milnor conjecture [Voe] and the Bloch– Lichtenbaum spectral sequence [BlLi]. In the case of the 2-integers ℤ[\frac{1}{2}] in ℚ the result implies that there is a 2-adic homotopy equivalence K(ℤ[\frac{1}{2}]) ≃ JK(ℤ[\frac{1}{2}]), where the latter spectrum was defined by Bökstedt in [B1]. This leads to the following statement:

Theorem 3.4 (Rognes–Weibel). There is a cofiber sequence of spectra

Σko ⟶ K(ℤ[\frac{1}{2}]) ⟶^{π_{3}} K(𝔽_{3}) ⟶^{∂} Σ^{2}ko

where π_{3} is induced by the ring surjection π_{3} : ℤ[\frac{1}{2}] → 𝔽_{3}. The connecting map ∂ is homotopic to the composite

K(𝔽_{3}) ⟶^{i_{3}} ku ⟶^{Σ^{2}r∘β^{−1}} Σ^{2}ko .

Proof. Bökstedt’s JK(ℤ[\frac{1}{2}]) can be defined as the homotopy fiber of the composite

ko ⟶^{c} ku ⟶^{ψ^{3}−1} bu .

By [B1, Thm. 2] there is a map Φ: K(ℤ[\frac{1}{2}]) → JK(ℤ[\frac{1}{2}]) inducing a split surjection on homotopy. By [RW], [We] these spectra have isomorphic homotopy groups, hence Φ is a homotopy equivalence. There is a square of horizontal and vertical cofiber sequences:

[figure: JK(ℤ[\frac{1}{2}]); ko; c; bu; K(𝔽_{3}); i_{3}; ∂; ku; ψ^{3}−1; Σ^{2}r∘β^{−1}; bu; Σ^{2}ko; Σ^{2}ko; ∗ .]

The left hand vertical yields the asserted cofiber sequence. Bökstedt’s cited construction of the map Φ identifies the composite K(ℤ[\frac{1}{2}]) → JK(ℤ[\frac{1}{2}]) → K(𝔽_{3}) with that induced by the ring homomorphism π_{3}. □

3.5. The reduction map. Let us recall the Galois reduction map from [DwMi, §13] and [R5, §3]. Let ϕ^{3} ∈ Gal(ℚ̄_{2}/ℚ_{2}) be a Galois automorphism of the algebraic closure ℚ̄_{2} of the field ℚ_{2} of 2-adic numbers, such that ϕ^{3}(ζ) = ζ^{3} when ζ is a 2-power root of unity, i.e., in μ_{2^{∞}} ⊂ ℚ̄_{2}^{×}. We may further assume that ϕ^{3}(+\sqrt{3}) = +\sqrt{3}. Then ϕ^{3} induces a self-map of K(ℚ̄_{2}) which is compatible up to homotopy with ψ^{3} : ku → ku under Suslin’s (implicitly 2-adic) homotopy equivalence K(ℚ̄_{2}) ≃ ku from [S1]. Hence the inclusion K(ℚ_{2}) → K(ℚ̄_{2})^{hϕ^{3}} to the homotopy equalizer of ϕ^{3} and the identity on K(ℚ̄_{2}) yields a spectrum map K(ℚ_{2}) → (ku)^{hψ^{3}}. The connective cover of the target is identified with K(𝔽_{3}) by Quillen’s theorem, which defines the Galois reduction map

red: K(ℚ_{2}) ⟶ K(𝔽_{3}) .

Fix a choice of such a map.

Theorem 3.6 (Rognes). There are cofiber sequences of spectra

K^{red}(ℚ_{2}) ⟶ K(ℚ_{2}) ⟶^{red} K(𝔽_{3}) ⟶^{∂_{2}} ΣK^{red}(ℚ_{2})

and

ΣK(𝔽_{3}) ⟶^{f_{ℂ}} K^{red}(ℚ_{2}) ⟶ Σku ⟶^{∂_{1}} Σ^{2}K(𝔽_{3}) .

The former connecting map ∂_{2} is determined by its composite with ΣK^{red}(ℚ_{2}) ⟶ Σ^{2}ku, which up to a two-adic unit is homotopic to the composite

K(𝔽_{3}) ⟶^{i_{3}} ku ⟶^{1−ψ^{−1}} bu ⟶_{≃}^{β^{−1}} Σ^{2}ku .

The latter connecting map ∂_{1} is homotopic to the composite

Σku ⟶^{Σ(1−ψ^{−1})} Σbu ⟶^{Σ∂_{3}} Σ^{2}K(𝔽_{3}) .

Both connecting maps induce the zero map on homotopy, and the extensions

K_{∗}^{red}(ℚ_{2}) ⟶ K_{∗}(ℚ_{2}) ⟶^{red_{∗}} K_{∗}(𝔽_{3})

and

π_{∗}(ΣK(𝔽_{3})) ⟶^{f_{ℂ∗}} K_{∗}^{red}(ℚ_{2}) ⟶ π_{∗}(Σku)

are both split. □

This is the conclusion of [R5, 8.1]. To go on, we consider the ring homomorphisms j: ℤ → ℤ_{2} and j′: ℤ[\frac{1}{2}] → ℚ_{2}.

Theorem 3.7 (Quillen). There is a map of horizontal cofiber sequences of spectra

[figure: K(𝔽_{2}); K(ℤ); j; K(ℤ[\frac{1}{2}]); j′; K(𝔽_{2}); K(ℤ_{2}); K(ℚ_{2})]

inducing a homotopy equivalence hofib(j) ⟶^{≃} hofib(j′). □

This is the spectrum level statement of the localization sequences in K-theory from [Q2].

Theorem 3.8 (Hesselholt–Madsen). In the commutative square of spectra

\xymatrix{K(ℤ) \ar[r]^{j} \ar[d]^{_{trc_{ℤ}}} & K(ℤ_{2}) \ar[d]^{_{trc_{ℤ_{2}}}} \\ T C(ℤ) \ar[r]^{j}_{≃} & T C(ℤ_{2})}

the right hand map induces a homotopy equivalence on connective covers. The lower map is a homotopy equivalence, and there is a cofiber sequence of spectra

hofib(j) ⟶ hofib(trc_{ℤ}) ⟶ Σ^{−2}Hℤ . □

This is Theorem D of [HM], which uses McCarthy’s theorem [Mc].

Theorem 3.9 (Rognes). The natural map j′: K(ℤ[\frac{1}{2}]) → K(ℚ_{2}) induces an isomorphism of 2-adic homotopy groups modulo torsion, in each positive dimension ∗ ≡ 1 mod 4. □

This is the content of [R5, 7.7]. By a homotopy group modulo torsion we here mean the quotient of the homotopy group by its torsion subgroup. Hence the assertion is stronger than just saying that j′ induces a homomorphism whose kernel and cokernel are torsion groups.

Proposition 3.10. There is a map of horizontal cofiber sequences of spectra

\xymatrix{Σko \ar[r] \ar[d]^{j^{red}} & K(ℤ[\frac{1}{2}]) \ar[d]^{j′} \ar[r]^{π_{3}} & K(𝔽_{3}) \ar[d]^{j̄}_{≃} \\ K^{red}(ℚ_{2}) \ar[r] & K(ℚ_{2}) \ar[r]^{red} & K(𝔽_{3})}

such that the right hand map j̄ is a homotopy equivalence. Hence there is a homotopy equivalence hofib(j^{red}) ⟶^{≃} hofib(j′).

Proof. Suppose we have shown that the composite

Σko ⟶ K(ℤ[\frac{1}{2}]) ⟶^{j′} K(ℚ_{2}) ⟶^{red} K(𝔽_{3})

is null homotopic. Then a choice of null homotopy defines an extension j̄: K(𝔽_{3}) → K(𝔽_{3}) of red ∘j′ over π_{3}, as well as a lifting j^{red} : Σko → K^{red}(ℚ_{2}). By the calculations of [R5, 4.2 and 4.4] the composite red ∘j′ is surjective on homotopy in dimensions 0 ≤ ∗ ≤ 7, hence in all dimensions by v_{1}^{4}-periodicity. Thus j̄ induces surjections on homotopy in all dimensions, and must be a homotopy equivalence. This then completes the proof of the proposition.

To show that the composite map Σko → K(𝔽_{3}) is null homotopic, it suffices to show that precomposition with the connecting map ∂: K(𝔽_{3}) → Σ^{2}ko in 3.4 induces an injection between the groups of homotopy classes of maps to ΣK(𝔽_{3}):

[Σ^{2}ko, ΣK(𝔽_{3})] ⟶^{∂^{#}} [K(𝔽_{3}), ΣK(𝔽_{3})] .

Here ∂ is the composite of i_{3} : K(𝔽_{3}) → ku with the map denoted Σ^{2}r ∘ β^{−1} : ku → Σ^{2}ko. Thus it suffices to show that both homomorphisms i_{3}^{#} and (Σ^{2}r ∘ β^{−1})^{#} are injective.

There is an exact sequence

[bu, ΣK(𝔽_{3})] ⟶^{(ψ^{3}−1)^{#}} [ku, ΣK(𝔽_{3})] ⟶^{i_{3}^{#}} [K(𝔽_{3}), ΣK(𝔽_{3})] .

Any map bu → ΣK(𝔽_{3}) has the form ∂_{3} ∘ ϕ, for some operation ϕ: bu → bu. Thus its precomposition with (ψ^{3} −1) is null homotopic, because ϕ and (ψ^{3} −1) commute and ∂_{3} ∘ (ψ^{3} − 1) ≃ ∗. Hence the left hand map is null and i_{3}^{#} is injective.

There is also an exact sequence

[Σko, ΣK(𝔽_{3})] ⟶^{η^{#}} [Σ^{2}ko, ΣK(𝔽_{3})] ⟶^{(Σ^{2}r∘β^{−1})^{#}} [ku, ΣK(𝔽_{3})] .

From 3.3 we see that [Σko, ΣK(𝔽_{3})] is zero, because postcomposition with (ψ^{3} − 1) acts injectively on the homotopy classes of maps ko → ku, see [MST]. Thus also (Σ^{2}r ∘ β^{−1})^{#} is injective, which completes the proof. □

Fix choices of maps j^{red} and j̄, as above.

Proposition 3.11. There is a cofiber sequence of spectra

K(𝔽_{3}) ⟶ hofib(j^{red}) ⟶ Σ^{2}ko ⟶^{∂} ΣK(𝔽_{3}) .

The connecting map ∂ is homotopic to the composite

Σ^{2}ko ⟶^{Σ^{2}c} Σ^{2}ku ⟶_{≃}^{β} bu ⟶^{∂_{3}} ΣK(𝔽_{3}) .

Proof. The map Σko → K(ℤ[\frac{1}{2}]) induces an isomorphism on 2-adic homotopy modulo torsion in dimensions ∗ ≡ 1 mod 8, and multiplication by 2 times a 2-adic unit in dimensions ∗ ≡ 5 mod 8. By 3.9 the same holds for the composite map from Σko to K(ℚ_{2}), and by 3.6 the same holds for the lift j^{red} : Σko → K^{red}(ℚ_{2}), as well as the composite map Σko → K^{red}(ℚ_{2}) → Σku. Any such map factors as a self-map ϕ of Σko followed by the suspended complexification map Σc: Σko → Σku. Since the suspended complexification map induces the identity in dimensions ∗ ≡ 1 mod 8, and multiplication by 2 in dimensions ∗ ≡ 5 mod 8, it follows that ϕ is a 2-adic homotopy equivalence. We obtain the following diagram of horizontal and vertical cofiber sequences:

\xymatrix{K(𝔽_{3}) \ar[r] \ar[d] & ∗ \ar[r] \ar[d] & ΣK(𝔽_{3}) \ar[d]^{f_{ℂ}} \\ hofib(j^{red}) \ar[r] \ar[d] & Σko \ar[r]^{j^{red}} \ar[d]^{ϕ}_{≃} & K^{red}(ℚ_{2}) \ar[d] \\ Σ^{2}ko \ar[r]^{Ση} & Σko \ar[r]^{Σc} & Σku .}

The connecting map ∂: Σ^{2}ko → ΣK(𝔽_{3}) is detected by its precomposition with Σ^{2}r ∘ β^{−1} : ku → Σ^{2}ko, because [Σko, ΣK(𝔽_{3})] = 0. By the diagram above, the composite ∂ ∘ Σ^{2}r ∘ β^{−1} is the desuspended connecting map Σ^{−1}∂_{1} = ∂_{3} ∘ (1 − ψ^{−1}) from 3.6. Thus ∂ = ∂_{3} ∘ β ∘ Σ^{2}c in the stable category, by the calculation

∂_{3} ∘ β ∘ Σ^{2}c ∘ Σ^{2}r ∘ β^{−1} = ∂_{3} ∘ (1 − ψ^{−1})

which uses c ∘ r = 1 + ψ^{−1}, and ψ^{k} ∘ β = β ∘ Σ^{2}(kψ^{k}). □

Proposition 3.12. There is a cofiber sequence of spectra

Σ^{3}ko ⟶ hofib(j^{red}) ⟶ ku ⟶^{∂} Σ^{4}ko .

The connecting map ∂ is homotopic to the composite

ku ⟶^{ψ^{3}−1} bu ⟶_{≃}^{β^{−1}} Σ^{2}ku ⟶^{Σ^{2}(Σ^{2}r∘β^{−1})} Σ^{4}ko

with the same notation as in (3.2). It is characterized by the following homotopy commutative diagram

\xymatrix{ku \ar[rrr]^{∂} \ar[d]^{cov} &  &  & Σ^{4}ko \ar[d]^{cov} \\ KU \ar[r]^{ψ^{3}−1} & KU & Σ^{4}KU \ar[l]_{β^{2}}^{≃} \ar[r]^{Σ^{4}r} & Σ^{4}KO .}

The maps labeled cov are k-connective covering maps, for suitable k.

Proof. We use the factorization of the connecting map in 3.11 to form the following diagram of horizontal and vertical cofiber sequences:

[figure: Σ^{3}ko; hofib(j^{red}); ku; β^{−1}∘(ψ^{3}−1); Σ^{3}ko; Σ^{2}η; Σ^{2}ko; Σ^{2}c; ∂; Σ^{2}ku; ∂_{3}∘β; ∗; ΣK(𝔽_{3}); ΣK(𝔽_{3}) .]

The right hand column is a variant of the sequence in 3.3. It follows that the connecting map ku → Σ^{4}ko for the top row is the composite of

ku ⟶^{ψ^{3}−1} bu ⟶_{≃}^{β^{−1}} Σ^{2}ku

and the connecting map for the middle row, i.e., the double suspension of the connecting map Σ^{2}r ∘ β^{−1} = ∂: ku → Σ^{2}ko of (3.2).

The covering maps induce an injection cov_{#} : [ku, Σ^{4}ko] → [ku, Σ^{4}KO] and a bijection cov^{#} : [KU, Σ^{4}KO] ≅ [ku, Σ^{4}KO], and the connecting map ∂ corresponds to Σ^{4}r ∘ β^{−2} ∘ (ψ^{3} − 1) in [KU, Σ^{4}KO]. Hence ∂ is characterized by the given diagram. □

The following theorem is the main result of this chapter.

Theorem 3.13. There is a cofiber sequence of spectra

Σ^{3}ko ⟶ hofib(trc) ⟶ Σ^{−2}ku ⟶^{δ} Σ^{4}ko .

The connecting map δ is characterized by the following homotopy commutative diagram

[figure: Σ^{−2}ku; cov; δ; Σ^{4}ko; cov; Σ^{−2}KU; KU; β; ≃; ψ^{3}−1; KU; Σ^{4}KU; β^{2}; ≃; Σ^{4}r; Σ^{4}KO .]

The maps labeled cov are suitable covering maps.

Proof. Consider the following diagram of horizontal and vertical cofiber sequences of spectra, obtained by combining 3.7, 3.8, 3.10 and 3.12:

[figure: Σ^{3}ko; hofib(j); ku; Σ^{3}ko; hofib(trc); X; ∗; Σ^{−2}Hℤ; Σ^{−2}Hℤ .]

Here X is the cofiber of the composite map Σ^{3}ko → hofib(j) → hofib(trc). It is classified as an extension of Σ^{−2}Hℤ by ku by an element in [Σ^{−2}Hℤ, Σku] ≅ ℤ_{2}, whose mod 2 reduction is detected by the composite k: Σ^{−2}Hℤ → Σku → ΣHℤ in [Σ^{−2}Hℤ, ΣHℤ] ≅ ℤ/2. (Here Σku → ΣHℤ is the map inducing an isomorphism on π_{1}.) This composite k is the k-invariant of X relating the homotopy groups in dimensions −2 and 0.

Since Σ^{3}ko is 2-connected, this lowest k-invariant is the same for X as for hofib(trc). By combining 1.11 with 1.21 we obtain a cofiber sequence

(3.14)  ℂP_{−1}^{∞} ⟶^{i} hofib(trc) ⟶^{j} Wh^{Diff}(∗)

whose connecting map is identified with t̃rc. Since Wh^{Diff}(∗) is connected, it follows that the lowest k-invariants for hofib(trc) and ℂP_{−1}^{∞} are equal. By 2.16 the latter is nonzero. Hence k is the essential map.

It follows that X is classified by a map u · ∂ where ∂: Σ^{−2}Hℤ → Σku classifies Σ^{−2}ku and u is a 2-adic unit. We get a homotopy equivalence of cofiber sequences

[figure: X; ≃; Σ^{−2}Hℤ; u·∂; u; ≃; Σku; Σ^{−2}ku; Σ^{−2}Hℤ; ∂; Σku .]

Hence X ≃ Σ^{−2}ku, as claimed.

To characterize δ, we compare with the connecting map ∂ of 3.12. Precomposition with β: ku → Σ^{−2}ku, or its K-localization, induces the vertical maps in the following commutative diagram:

\xymatrix{[Σ^{−2}ku, Σ^{4}ko] \ar[r]^{cov_{#}} \ar[d]^{β^{#}} & [Σ^{−2}ku, Σ^{4}KO] \ar[d]^{β^{#}}_{≅} & [Σ^{−2}KU, Σ^{4}KO] \ar[d]^{β^{#}}_{≅} \ar[l]_{cov^{#}}^{≅} \\ [ku, Σ^{4}ko] \ar[r]^{cov_{#}} & [ku, Σ^{4}KO] & [KU, Σ^{4}KO] . \ar[l]_{cov^{#}}^{≅}}

Here the maps labeled cov_{#} are injective, and the maps labeled cov^{#} are bijective. The class δ in [Σ^{−2}ku, Σ^{4}ko] maps to ∂ under β^{#}, which in turn maps to Σ^{4}r ∘ β^{−2} ∘ (ψ^{3} − 1) in [KU, Σ^{4}KO] by 3.12. The right hand β^{#} is bijective, so this characterizes the image of δ in [Σ^{−2}KU, Σ^{4}KO] as Σ^{4}r ∘ β^{−2} ∘ (ψ^{3} − 1) ∘ β^{−1}. This characterizes δ up to homotopy, by the injectivity claims above. □

Remark 3.15. By [6A] or 1.8 this theorem also determines the homotopy fiber of the cyclotomic trace map trc_{X}: A(X) → T C(X) completed at 2 for any 1-connected space X, since the natural map

hofib(trc_{X}) ⟶^{≃} hofib(trc)

is a homotopy equivalence.

Let v_{2}(k) be the 2-adic valuation of k.

Corollary 3.16. In positive dimensions (n > 0) the homotopy groups of hofib(trc) are

π_{n}(hofib(trc)) ≅ \{ \begin{matrix}0 & for n ≡ 0, 1 mod 8, \\ ℤ & for n ≡ 2, 6 mod 8, \\ ℤ/16 & for n ≡ 3 mod 8, \\ ℤ/2 & for n ≡ 4, 5 mod 8, and \\ ℤ/2^{v_{2}(k)+4} & for n = 8k − 1. \end{matrix}

Also π_{n}(hofib(trc)) ≅ ℤ for n = −2 and n = 0. The remaining homotopy groups are zero.

Proof. This is a routine calculation, given the action of ψ^{3} − 1 and Σ^{4}r on homotopy. □

The spectrum cohomology of hofib(trc) is given in 4.4 below. The Adams E_{2}- term

(3.17)  E_{2}^{s,t} = Ext_{A}^{s,t}(H^{∗}(hofib(trc)), 𝔽_{2}) ⟹ π_{t−s}(hofib(trc))_{2}^{∧} .

is then easily deduced from the E_{2}-terms in the Adams spectral sequences for π_{∗}(ko)_{2}^{∧} and π_{∗}(ku)_{2}^{∧}. Furthermore only one pattern of differentials is compatible with 3.16: There is an infinite h_{0}-tower of nonzero d^{r}-differentials from column t − s = 8k for all k ≥ 1, with r = v_{2}(k) + 2, and no other differentials. The spectral sequence is displayed in Tables 3.18(a) and (b) below.

4. Cohomology of the smooth Whitehead spectrum

We now determine the mod two spectrum cohomology of the smooth Whitehead spectrum of a point, as a module over the Steenrod algebra.

Consider the following diagram:

(4.1)  [figure: ℂP_{−1}^{∞}; i; ℂP_{−1}^{∞}; ϵ; Σ^{3}ko; hofib(trc); j; Σ^{−2}ku; δ; Σ^{4}ko; Σ^{3}ko; Wh^{Diff}(∗); Σ hofib(ϵ); Σ^{4}ko .]

Table 3.18(a). The Adams E_{2}-term for hofib(trc)

The middle row is the cofiber sequence from Theorem 3.13, and the left column is (3.14). We let ϵ be the composite map ℂP_{−1}^{∞} → hofib(trf) → Σ^{−2}ku. Then the right column and bottom row are also cofiber sequences.

Proposition 4.2. The map ϵ induces the unique surjection of A-modules

Σ^{−2}A/A(Sq^{1}, Sq^{3}) ≅ H^{∗}(Σ^{−2}ku) ⟶^{ϵ^{∗}} H^{∗}(ℂP_{−1}^{∞}) ≅ Σ^{−2}A/C .

Hence

H^{∗}(Σ hofib(ϵ)) ≅ Σ^{−2}C/A(Sq^{1}, Sq^{3})

as an A-module.

Proof. We use that Σ^{4}ko and Wh^{Diff}(∗) are connective spectra. Hence Σ hofib(ϵ) is connective, and so ϵ induces an isomorphism in dimension −2. This determines ϵ^{∗} since H^{∗}(Σ^{−2}ku) is a cyclic A-module, and ϵ^{∗} is surjective because H^{∗}(ℂP_{−1}^{∞}) is a cyclic A-module. We identify H^{∗}(Σ hofib(ϵ)) with ker(ϵ^{∗}). □

Table 3.18(b). The Adams E_{2}-term for hofib(trc)

Proposition 4.3. The connecting map δ induces the zero homomorphism on cohomology.

Proof. In fact, the group of A-module homomorphisms

H^{∗}(Σ^{4}ko) ≅ Σ^{4}A/A(Sq^{1}, Sq^{2}) ⟶ Σ^{−2}A/A(Sq^{1}, Sq^{3}) ≅ H^{∗}(Σ^{−2}ku)

is zero. For A/A(Sq^{1}, Sq^{3}) is 𝔽_{2}{Sq^{6}, Sq^{4}Sq^{2}} in dimension 6, while Sq^{1} ∘ Sq^{6} ≠ 0 and Sq^{2} ∘ Sq^{4}Sq^{2} ≠ 0 in this A-module. □

Theorem 4.4. The mod two spectrum cohomology of hofib(trc) is the unique nontrivial extension of A-modules

Σ^{−2}A/A(Sq^{1}, Sq^{3}) ⟶ H^{∗}(hofib(trc)) ⟶ Σ^{3}A/A(Sq^{1}, Sq^{2}) .

Theorem 4.5. The mod two spectrum cohomology of Wh^{Diff}(∗) is the unique nontrivial extension of A-modules

Σ^{−2}C/A(Sq^{1}, Sq^{3}) ⟶ H^{∗}(Wh^{Diff}(∗)) ⟶ Σ^{3}A/A(Sq^{1}, Sq^{2}) .

The mod two spectrum cohomology of A(∗) is given by the splitting of A-modules

H^{∗}(A(∗)) ≅ H^{∗}(Wh^{Diff}(∗)) ⊕ 𝔽_{2} .

Here 𝔽_{2} = H^{∗}(S^{0}) denotes the trivial A-module concentrated in degree zero. We prove these two theorems together.

Proof of 4.4 and 4.5. We apply mod 2 spectrum cohomology to (4.1). By 4.2 the map ϵ induces a surjection in each dimension, so Σ^{−2}ku → Σ hofib(ϵ) induces an injection in each dimension. By 4.3 the map δ induces the zero homomorphism in each dimension, and combining these facts we see that Σ hofib(ϵ) → Σ^{4}ko also induces the zero homomorphism in cohomology. Thus the long exact sequences in cohomology associated to the middle and lower horizontal cofiber sequences in (4.1) break up into short exact sequences. These express H^{∗}(hofib(trc)) and H^{∗}(Wh^{Diff}(∗)) as extensions of A-modules, as claimed.

(4.6)  [figure: Σ^{−2}A/C; Σ^{−2}A/C; Σ^{3}A/A(Sq^{1}, Sq^{2}); H^{∗}(hofib(trc)); i^{∗}; Σ^{−2}A/A(Sq^{1}, Sq^{3}); ϵ^{∗}; Σ^{3}A/A(Sq^{1}, Sq^{2}); H^{∗}(Wh^{Diff}(∗)); j^{∗}; Σ^{−2}C/A(Sq^{1}, Sq^{3})]

It remains to characterize the extensions, which are represented by elements of Ext_{A}^{1}. Recall that H^{∗}(ko) = A/A(Sq^{1}, Sq^{2}) = A//A_{1} where A_{1} ⊂ A is the sub-Hopf algebra generated by Sq^{1} and Sq^{2}. Hence there are change-of-rings isomorphisms

Ext_{A}^{1}(Σ^{3}A//A_{1}, Σ^{−2}A/A(Sq^{1}, Sq^{3})) ≅ Ext_{A_{1}}^{1}(Σ^{3}𝔽_{2}, Σ^{−2}A/A(Sq^{1}, Sq^{3}))

and

Ext_{A}^{1}(Σ^{3}A//A_{1}, Σ^{−2}C/A(Sq^{1}, Sq^{3})) ≅ Ext_{A_{1}}^{1}(Σ^{3}𝔽_{2}, Σ^{−2}C/A(Sq^{1}, Sq^{3})) .

An A_{1}-module extension of Σ^{−2}A/A(Sq^{1}, Sq^{3}) by Σ^{3}𝔽_{2} is determined by the values of Sq^{1} and Sq^{2} on the nonzero element of Σ^{3}𝔽_{2}, and these are connected by the Adem relation Sq^{1}Sq^{2}Sq^{1} = Sq^{2}Sq^{2}.

By inspection of Σ^{−2}A/A(Sq^{1}, Sq^{3}) and Σ^{−2}C/A(Sq^{1}, Sq^{3}) as A_{1}-modules, there are precisely two such A_{1}-module extensions in both cases; one trivial (split) and one nontrivial (not split). Furthermore the map of extensions induced by (4.1) induces an isomorphism

ℤ/2 ≅ Ext_{A_{1}}^{1}(Σ^{3}𝔽_{2}, Σ^{−2}C/A(Sq^{1}, Sq^{3})) ⟶^{≅}

Ext_{A_{1}}^{1}(Σ^{3}𝔽_{2}, Σ^{−2}A/A(Sq^{1}, Sq^{3})) ≅ ℤ/2 .

Thus to prove that each extension is the unique nontrivial extension of its kind, it suffices to show that H^{∗}(Wh^{Diff}(∗)) does not split as the sum of Σ^{3}A/A(Sq^{1}, Sq^{2}) and Σ^{−2}C/A(Sq^{1}, Sq^{3}).

Now Σ^{−2}C/A(Sq^{1}, Sq^{3}) is 3-connected, and by [BW, 1.3] the bottom homotopy group of Wh^{Diff}(∗) is π_{3}(Wh^{Diff}(∗)) ≅ ℤ/2. Hence there is a nontrivial Sq^{1} acting on the nonzero class in H^{3}(Wh^{Diff}(∗)), which tells us that Σ^{3}A/A(Sq^{1}, Sq^{2}) does not split off from H^{∗}(Wh^{Diff}(∗)).

This proves that both extensions are nontrivial, and completes the proofs. □

Remark 4.7. By (4.6), we see that the lifted cyclotomic trace map

t̃rc: Wh^{Diff}(∗) ⟶ T̃ C(∗) ≃ ΣℂP_{−1}^{∞}

induces the zero homomorphism on mod 2 spectrum cohomology. The map is nevertheless very useful.

Question 4.8. The map ϵ lives in the group

[ℂP_{−1}^{∞}, Σ^{−2}ku] ≅ [ℂP_{+}^{∞}, ku] = KU^{0}(ℂP^{∞}) ≅ ℤ[[γ^{1}]]

where the first isomorphism is the Thom isomorphism in complex topological Ktheory for the virtual complex bundle −γ^{1} over ℂP_{+}^{∞}. To which power series in γ^{1} does ϵ correspond ?

Proposition 4.9. The linearization map L: T C(∗) → T C(ℤ) and the suspended map Σϵ: ΣℂP_{−1}^{∞} → Σ^{−1}ku induce the same homomorphisms up to 2-adic units, on homotopy groups modulo torsion in dimensions ∗ ≡ 3 mod 4.

Proof. The suspended map Σϵ is the composite

ΣℂP_{−1}^{∞} ⟶ T C(∗) ⟶^{L} T C(ℤ) ⟶ Σ hofib(trc) ⟶ Σ^{−1}ku .

The first map splits T C(∗), the second is the linearization map, the third is the connecting map in the cofiber sequence generated by trc_{ℤ}, and the fourth suspends a map that appears in 3.13. The first map induces an isomorphism on homotopy groups modulo torsion in all positive dimensions, since the other summand Σ^{∞}S^{0} has finite homotopy groups in positive dimensions. The third and fourth maps also induce an isomorphism on homotopy groups modulo torsion in dimensions ∗ ≡ 3 mod 4, by the calculation of trc_{ℤ} in [R5, 9.1], and the description of δ in 3.13 □

5. Two-primary homotopy of Wh^{Diff}(∗)

Let ι_{−2} be the generator in dimension −2 of H^{∗}(Σ^{−2}ku) ≅ Σ^{−2}A/A(Sq^{1}, Sq^{3}), and let ι_{3} be the generator in dimension 3 of H^{∗}(Σ^{3}ko) ≅ Σ^{3}A/A(Sq^{1}, Sq^{2}). By 4.2 the map ϵ: ℂP_{−1}^{∞} → Σ^{−2}ku of (4.1) induces a surjection on cohomology, and we regard

ker(ϵ^{∗}) = Σ^{−2}C/A(Sq^{1}, Sq^{3}) ⊂ Σ^{−2}A/A(Sq^{1}, Sq^{3}) = H^{∗}(Σ^{−2}ku)

as a submodule of H^{∗}(Σ^{−2}ku). It is thus spanned by suitable monomials Sq^{I}ι_{−2} taken modulo A(Sq^{1}, Sq^{3})ι_{−2}. By inspection ker(ϵ^{∗}) is 3-connected. The bottom cofiber sequence in (4.1) induces the nontrivial extension

0 → ker(ϵ^{∗}) ⟶ H^{∗}(Wh^{Diff}(∗)) ⟶ H^{∗}(Σ^{3}ko) → 0 .

dim x  Sq^{1}(x)  Sq^{2}(x)  Sq^{4}(x)  Sq^{8}(x)
3 ι_{3}  Sq^{4}Sq^{2}ι_{−2}  Sq^{7}ι_{−2}  Sq^{4}ι_{3}  Sq^{8}ι_{3}
4 Sq^{4}Sq^{2}ι_{−2}  0  Sq^{6}Sq^{2}ι_{−2}  0  Sq^{8}Sq^{4}Sq^{2}ι_{−2}
5 Sq^{7}ι_{−2}  0  Sq^{9}ι_{−2}  Sq^{11}ι_{−2}  Sq^{13}Sq^{2}ι_{−2}
≡ Sq^{7}Sq^{2}ι_{−2}  +Sq^{11}Sq^{4}ι_{−2}
6 Sq^{6}Sq^{2}ι_{−2} Sq^{9}ι_{−2} 0 Sq^{10}Sq^{2}ι_{−2} Sq^{10}Sq^{4}Sq^{2}ι_{−2}
≡ Sq^{7}Sq^{2}ι_{−2}
7 Sq^{9}ι_{−2} 0 0 Sq^{11}Sq^{2}ι_{−2}

≡ Sq^{7}Sq^{2}ι_{−2}

Sq^{4}ι_{3}  0  Sq^{6}ι_{3}  Sq^{13}ι_{−2}
8 Sq^{8}Sq^{2}ι_{−2}  0  Sq^{10}Sq^{2}ι_{−2}  Sq^{12}Sq^{2}ι_{−2}
9 Sq^{11}ι_{−2}  0  Sq^{13}ι_{−2}  Sq^{13}Sq^{2}ι_{−2}
Sq^{6}ι_{3}  Sq^{7}ι_{3}  0  Sq^{13}Sq^{2}ι_{−2}
+Sq^{11}Sq^{4}ι_{−2}
+Sq^{10}ι_{3}
10 Sq^{10}Sq^{2}ι_{−2}  Sq^{11}Sq^{2}ι_{−2} 0  0
Sq^{8}Sq^{4}ι_{−2}  0  Sq^{10}Sq^{4}ι_{−2}  Sq^{12}Sq^{4}ι_{−2}
Sq^{7}ι_{3}  0  0  Sq^{11}ι_{3}
11 Sq^{13}ι_{−2}  0  0
Sq^{11}Sq^{2}ι_{−2}  0  Sq^{13}Sq^{2}ι_{−2}
Sq^{8}ι_{3}  Sq^{8}Sq^{4}Sq^{2}ι_{−2} Sq^{10}ι_{3}
12 Sq^{12}Sq^{2}ι_{−2}  Sq^{13}Sq^{2}ι_{−2} Sq^{14}Sq^{2}ι_{−2}
Sq^{10}Sq^{4}ι_{−2}  Sq^{11}Sq^{4}ι_{−2} 0
Sq^{8}Sq^{4}Sq^{2}ι_{−2} 0  Sq^{10}Sq^{4}Sq^{2}ι_{−2}
13 Sq^{15}ι_{−2}  0
Sq^{13}Sq^{2}ι_{−2}  0
Sq^{11}Sq^{4}ι_{−2}  0
Sq^{10}ι_{3}  Sq^{11}ι_{3}
14 Sq^{14}Sq^{2}ι_{−2}

Sq^{12}Sq^{4}ι_{−2} Sq^{10}Sq^{4}Sq^{2}ι_{−2} Sq^{11}ι_{3}

Table 5.1. H^{∗}(Wh^{Diff}(∗)) in dimensions ≤ 14.

We let ι_{3} ∈ H^{3}(Wh^{Diff}(∗)) denote the unique lift of ι_{3} ∈ H^{3}(Σ^{3}ko). With these notations we list a basis for H^{∗}(Wh^{Diff}(∗)) in dimensions ∗ ≤ 14 in Table 5.1, together with generators for the A-module structure.

We now consider the Adams spectral sequences associated with the spectra in the cofiber sequence of spectra

(5.2)  ℂP_{−1}^{∞} ⟶^{i} hofib(trc) ⟶^{j} Wh^{Diff}(∗)

appearing vertically in (4.1). They are

(5.3)  _{c}E_{2}^{s,t} = Ext_{A}^{s,t}(H^{∗}(ℂP_{−1}^{∞}), 𝔽_{2}) ⟹ π_{t−s}(ℂP_{−1}^{∞})_{2}^{∧}
(5.4)  _{f}E_{2}^{s,t} = Ext_{A}^{s,t}(H^{∗}(hofib(trc)), 𝔽_{2}) ⟹ π_{t−s}(hofib(trc))_{2}^{∧}
(5.5)  _{w}E_{2}^{s,t} = Ext_{A}^{s,t}(H^{∗}(Wh^{Diff}(∗)), 𝔽_{2}) ⟹ π_{t−s}(Wh^{Diff}(∗))_{2}^{∧} .

The prefix ‘c’ refers to the truncated complex projective space, ‘f’ refers to the homotopy fiber of the cyclotomic trace map, and ‘w’ refers to the Whitehead spectrum. The spectral sequence _{c}E_{∗} was already studied in (2.17), 2.18 and 2.21, while the spectral sequence _{f}E_{∗} appeared in (3.17) and 3.18. The spectral sequence _{w}E_{∗} is displayed below, in Tables 5.7(a) and (b).

The diagram (5.2) induces a short exact sequence of A-modules in cohomology, by (4.6), and thus a long exact sequence of Ext-groups

(5.6)  ⋯ → _{c}E_{2}^{s,t} ⟶^{i_{∗}} _{f}E_{2}^{s,t} ⟶^{j_{∗}} _{w}E_{2}^{s,t} ⟶^{∂} _{c}E_{2}^{s+1,t} → ….

By the geometric boundary theorem [Ra, 2.3.4], the connecting map ∂ is induced by the spectrum map t̃rc: Wh^{Diff}(∗) → ΣℂP_{−1}^{∞} extending (5.2), and so each map in the long exact sequence is part of a map of spectral sequences. Furthermore these maps are compatible with the maps in the long exact sequence in 2-completed homotopy induced by (5.2).

The E_{2}-term of the Adams spectral sequence (5.5) for Wh^{Diff}(∗) is displayed in dimensions t − s ≤ 21 in Tables 5.7(a) and (b). This was obtained by hand from a minimal resolution of H^{∗}(Wh^{Diff}(∗)) in internal degrees t ≤ 14, using Table 5.1, and using Bruner’s Ext-calculator program [Br] in higher dimensions. The notation in these tables is that the maps in (5.6) take a class denoted ‘•’ in one spectral sequence to a class denoted ‘∘’ in the following spectral sequence, i.e., • ↦ ∘. As before, the classes hit by or supporting differentials are shaded gray. In bidegree (2, 14) a d_{2}-differential hits the sum of the two given generators, which are therefore both half-shaded.

Proposition 5.8. The map i: ℂP_{−1}^{∞} → hofib(trc) induces a map

i_{∗}: _{c}E_{2}^{s,t} ⟶ _{f}E_{2}^{s,t}

of Adams E_{2}-terms, which is surjective in dimensions t − s ≤ 2, t − s = 4 and t − s ≡ 5, 6 mod 8. In positive dimensions t − s ≡ 3 mod 8 its image equals the three h_{0}-divisible classes. In other dimensions the map is zero.

Proof. Note that _{f}E_{2} has dimension 0 or 1 in each bidegree. In the range of bidegrees displayed in Tables 2.18, 3.18 and 5.7, the claim follows by a dimension count using exactness in (5.6). Since hofib(trc) agrees with its Bousfield K-localization in dimensions ∗ ≥ 1 by 3.13, the result is propagated to higher dimensions by applying suitable periodicity operators. □

Proposition 5.9. In the Adams spectral sequence _{w}E_{∗} the nonzero differentials landing in homotopical dimension ≤ 21 are

(i) d_{2}^{s,s+8} ≠ 0 for s ≥ 0.

(ii) d_{2}^{s,s+11} ≠ 0 for s ≥ 1, with image divisible by h_{0}^{s+2}.

Table 5.7(a). The Adams E_{2}-term for Wh^{Diff}(∗)

(iii) d_{2}^{s,s+13} ≠ 0 for 0 ≤ s ≤ 3. The image of d_{2}^{0,13} contains h_{0} · x + h_{1} · y for nonzero classes x, y. The image of d_{2}^{s,13+s} for 1 ≤ s ≤ 3 is divisible by h_{0}^{s+1}.

(iv) d_{2}^{1,15} ≠ 0 has image divisible by h_{1}^{2}.

(v) d_{2}^{0,16} ≠ 0 has image divisible by h_{0}. (vi) d_{3}^{s,s+16} ≠ 0 for s ≥ 1 is zero on the h_{0}-torsion classes. (vii) d_{2}^{s,s+18} ≠ 0 for s = 0, 1. (viii) d_{3}^{2,20} ≠ 0 is zero on the h_{1}-divisible classes. (ix) d_{2}^{s,s+19} ≠ 0 for s = 3, 4 is zero on the h_{1}-divisible classes, and takes h_{0}-

torsion values.

Table 5.7(b). The Adams E_{2}-term for Wh^{Diff}(∗)

(x) d_{2}^{s,s+19} ≠ 0 for s ≥ 5, with image divisible by h_{0}^{s+2}.

(xi) d_{2}^{3,24} ≠ 0, d_{2}^{4,25} ≠ 0, d_{2}^{5,26} ≠ 0, d_{2}^{6,27} ≠ 0, d_{2}^{7,28} ≠ 0, d_{2}^{0,22} ≠ 0 and d_{2}^{5,27} ≠ 0 all have rank 1.

Proof. The differentials in _{f}E_{∗} given in Tables 3.18(a) and (b) induce differentials in_{w}E_{∗} by naturality with respect to the spectral sequence map j_{∗} in (5.6). Likewise the differentials in _{c}E_{∗} given in 2.21 lift by the connecting map ∂ in (5.6) to detect differentials in _{w}E_{∗}. Taking the h_{0}-multiplications in _{w}E_{2} into account, this gives rise to all the differentials listed above.

It remains to check that there are no further differentials in _{w}E_{∗}. Any such would have to map from classes ‘•’ detected by ∂ to classes ‘∘’ in the image of j_{∗}. For bidegree reasons the only possible targets are the h_{1}-divisible classes ‘∘’ in bidegree (s, t) = (4k, 12k + 3) with k ≥ 1. These classes represent the image of π_{8k+3}(hofib(trc)) in π_{8k+3}(Wh^{Diff}(∗)). Now the generator of π_{8k+3}(hofib(trc)) ≅ ℤ/16 maps to the order 2 class η^{2}μ_{8k+1} in K_{8k+3}(ℤ) ≅ ℤ/16, which generates the kernel of the cyclotomic trace map trc_{ℤ} to π_{8k+3}(T C(ℤ)) ≅ ℤ ⊕ ℤ/8 by [R5, 9.1]. Hence by the diagram in 1.11, the image of π_{8k+3}(hofib(trc)) in π_{8k+3}(Wh^{Diff}(∗)) is nontrivial, and so the cited class in bidegree (4k, 12k + 3) must survive to the E_{∞}-term. Hence it is not hit by a differential. □

Theorem 5.10. The 2-primary homotopy groups of Wh^{Diff}(∗) in dimensions ∗ ≤ 21 are as follows:

π_{n}(Wh^{Diff}(∗)) = 0  for n ≤ 2,
π_{3}(Wh^{Diff}(∗)) = ℤ/2
π_{4}(Wh^{Diff}(∗)) = 0
π_{5}(Wh^{Diff}(∗)) = ℤ
π_{6}(Wh^{Diff}(∗)) = 0
π_{7}(Wh^{Diff}(∗)) = ℤ/2
π_{8}(Wh^{Diff}(∗)) = 0
π_{9}(Wh^{Diff}(∗)) = ℤ/2 ⊕ ℤ
π_{10}(Wh^{Diff}(∗)) = (ℤ/2)^{2} ⊕ ℤ/8
π_{11}(Wh^{Diff}(∗)) = ℤ/2
π_{12}(Wh^{Diff}(∗)) = ℤ/4
π_{13}(Wh^{Diff}(∗)) = ℤ
π_{14}(Wh^{Diff}(∗)) = ℤ/4
π_{15}(Wh^{Diff}(∗)) = (ℤ/2)^{2}
π_{16}(Wh^{Diff}(∗)) = ℤ/2 ⊕ ℤ/8
π_{17}(Wh^{Diff}(∗)) = (ℤ/2)^{2} ⊕ ℤ
π_{18}(Wh^{Diff}(∗)) = (ℤ/2)^{3} ⊕ ℤ/32
π_{19}(Wh^{Diff}(∗)) = ℤ/2 ⋊ ℤ/2 ⋊ ℤ/8 ⋊ ℤ/2
π_{20}(Wh^{Diff}(∗)) = #2^{7}
π_{21}(Wh^{Diff}(∗)) = #2^{4} ⊕ ℤ .

In the long exact sequence in homotopy induced by the cofiber sequence

ℂP_{−1}^{∞} ⟶^{i} hofib(trc) ⟶^{j} Wh^{Diff}(∗)

the image of j_{∗} is ℤ/2 in dimensions n ≡ 3 mod 8 and zero otherwise, for n ≤ 21.

Proof. This follows by inspection of the E_{∞}-term of the Adams spectral sequence for Wh^{Diff}(∗), and the long exact sequence

⋯ → π_{n}(ℂP_{−1}^{∞}) ⟶^{i_{∗}} π_{n}(hofib(trc)) ⟶^{j_{∗}} π_{n}(Wh^{Diff}(∗)) ⟶^{t̃rc_{∗}} π_{n−1}(ℂP_{−1}^{∞}) → ….

The long exact sequence shows that π_{18}(Wh^{Diff}(∗)) ≅ π_{17}(ℂP_{−1}^{∞}), which was found in 2.13. Next π_{19}(Wh^{Diff}(∗)) is an extension of the torsion in π_{18}(ℂP_{−1}^{∞}) ≅ ℤ/2 ⋊ ℤ/8 ⋊ ℤ/2 ⊕ ℤ by ℤ/2. Also π_{20}(Wh^{Diff}(∗)) is the kernel of a homomorphism from π_{19}(ℂP_{−1}^{∞}) ≅ ℤ/2 ⊕ ℤ/8 ⊕ ℤ/64 with image ℤ/8. This is some group of order 2^{7}, denoted #2^{7} in the statement of the theorem. Lastly π_{21}(Wh^{Diff}(∗)) is the sum of a group of order 2^{4} and an infinite cyclic group, as can be read off from the E_{∞}-term of _{w}E_{∗}. □

Regarding multiplicative structure, we have the following addendum.

Lemma 5.11. The homomorphism η_{#} : π_{n}(Wh^{Diff}(∗)) → π_{n+1}(Wh^{Diff}(∗)) has image (ℤ/2)^{2} for n = 9, image ℤ/2 for n = 10 and is zero for all other n ≤ 14.

The homomorphism ν_{#} : π_{n}(Wh^{Diff}(∗)) → π_{n+3}(Wh^{Diff}(∗)) has image ℤ/2 for n = 7 and is zero for all other n ≤ 14.

The homomorphism σ_{#} : π_{n}(Wh^{Diff}(∗)) → π_{n+7}(Wh^{Diff}(∗)) has image ℤ/2 for n = 5 and is zero for all other n ≤ 11.

Proof. The nonzero multiplications listed are all detected by nontrivial h_{1}, h_{2} or h_{3}-multiplications in the Adams spectral sequence (5.5) for Wh^{Diff}(∗). To see that there are no other nonzero multiplications in this range one can use Adams filtration arguments in this spectral sequence, combined with naturality with respect to the map t̃rc: Wh^{Diff}(∗) → ΣℂP_{−1}^{∞}. For example, π_{14}(Wh^{Diff}(∗)) is detected in π_{13}(ℂP_{−1}^{∞}), but the image of π_{7}(Wh^{Diff}(∗)) in π_{6}(ℂP_{−1}^{∞}) is divisible by ν and σν = 0, so σ_{#} = 0 for n = 7. □

6. Cohomology of K(ℤ) and the linearization map

We continue to implicitly complete all spectra at 2. Bökstedt’s spectrum JK(ℤ) is the homotopy fiber of the composite

ko ⟶^{ψ^{3}−1} bspin ⟶^{c} bsu .

It is also homotopy equivalent to the algebraic K-theory spectrum K(ℤ), by [RW], [We]. Hence there is a diagram of horizontal and vertical cofiber sequences of spectra:

(6.1)  [figure: bso; η; bso; t; ∗; Σbso; η; spin; ζ; c; j; i; ko; ψ^{3}−1; bspin; c; su; K(ℤ); ko; bsu; Σbso; Σbso; ∗; Σ^{2}bso .]

The right hand column is a connected covering of (3.2), and the second row defines the connective real image of J spectrum j. We let t = ζ ∘ η be the composite of the Bott map η: bso → spin and the connecting map ζ: spin → j.

Miller and Priddy [MP] define spectra g/o_{⊕} and ibo as the pullbacks in the following diagram:

(6.2)  \xymatrix{g/o_{⊕} \ar[r] \ar[d] & ibo \ar[r] \ar[d] & S^{0} \ar[d]^{e} \\ bso \ar[r]^{η} & spin \ar[r]^{ζ} & j .}

(More precisely, they define the underlying infinite loop spaces G/O_{⊕} = Ω^{∞}g/o_{⊕} and IBO = Ω^{∞}ibo.) Here e: S^{0} → j is the map representing the real Adams e-invariant. Its fiber c is the cokernel of J spectrum, which is K-acyclic. Thus the unit map i: S^{0} → K(ℤ) factors, uniquely up to homotopy, as e composed with i: j → K(ℤ). By (6.1) the cofiber of the lower composite map ζη in (6.2) is K(ℤ). Hence there is a cofiber sequence

(6.3)  g/o_{⊕} ⟶ S^{0} ⟶^{i} K(ℤ)

of 2-complete spectra. Thus there is a fiber sequence G/O_{⊕} → QS^{0} → K(ℤ) of underlying infinite loop spaces, and we might write G/O_{⊕} = IK(ℤ) as the ‘ideal’ in QS^{0} = Ω^{∞}S^{0} defining K(ℤ) (at the prime 2).

We compute the mod 2 spectrum cohomology H^{∗}(K(ℤ)) by means of the cofiber sequence su → K(ℤ) → ko, where su ≃ Σ^{3}ku. In view of (6.3) this also determines H^{∗}(g/o_{⊕}). Miller and Priddy conjecture in [MP] that G/O_{⊕} ≃ G/O as infinite loop spaces. If confirmed, this would also lead to a calculation of the spectrum cohomology H^{∗}(g/o). It is known that G/O ≃ G/O_{⊕} as 2-complete spaces, and that H_{∗}(G/O; 𝔽_{2}) ≅ H_{∗}(G/O_{⊕}; 𝔽_{2}) as Hopf algebras over the Steenrod– and Dyer– Lashof algebras, by unpublished calculations of J. Tornehave.

Theorem 6.4. The mod 2 spectrum cohomology of K(ℤ) is the unique nontrivial extension of A-modules

A/A(Sq^{1}, Sq^{2}) ⟶ H^{∗}(K(ℤ)) ⟶ Σ^{3}A/A(Sq^{1}, Sq^{3}) .

The A-module H^{∗}(Σg/o_{⊕}) is the connected cover of H^{∗}(K(ℤ)), i.e., the kernel of the augmentation H^{∗}(K(ℤ)) → 𝔽_{2}.

Proof. We use the cofiber sequence K(ℤ) → ko → bsu where the right hand map is the composite of ψ^{3} − 1: ko → bspin and c: bspin → bsu. The induced map

Σ^{4}A/A(Sq^{1}, Sq^{3}) ≅ H^{∗}(bsu) ⟶ H^{∗}(ko) ≅ A/A(Sq^{1}, Sq^{2})

is the zero homomorphism. For the complexification map c induces multiplication by 2 on π_{4}, and thus the zero map on H^{4}. Thus the long exact sequence in spectrum cohomology decomposes as the A-module extension above. The second claim follows from the cofiber sequence S^{0} → K(ℤ) → Σg/o_{⊕}.

It remains to characterize the extension. There are precisely two such A-module extensions, since

Ext_{A}^{1}(Σ^{3}A/A(Sq^{1}, Sq^{3}), A/A(Sq^{1}, Sq^{2})) ≅ Ext_{E_{1}}^{1}(Σ^{3}𝔽_{2}, A/A(Sq^{1}, Sq^{2})) ≅ 𝔽_{2} .

Here E_{1} ⊂ A is the exterior algebra generated by Sq^{1} and Q_{1} = Sq^{3} + Sq^{2}Sq^{1}. We know that H^{∗}(K(ℤ)) is a nontrivial extension, because H_{3}^{spec}(K(ℤ); ℤ_{2}) ≅ π_{2}(g/o_{⊕}) ≅ ℤ/2 implies that there is a nonzero Sq^{1} from dimension 3 to dimension 4 in H^{∗}(K(ℤ)). □

We list a monomial basis for H^{∗}(K(ℤ)) in dimensions ≤ 14 in Table 6.5. It differs from H^{∗}(Σg/o_{⊕}) only in dimension 0. The notation is that ι_{0} ∈ H^{0}(K(ℤ)) pulls back from the generator of H^{0}(ko), while ι_{3} ∈ H^{3}(K(ℤ)) is the unique lift of the generator in dimension 3 of H^{∗}(su) ≅ Σ^{3}A/A(Sq^{1}, Sq^{3}). We have chosen Sq^{9}(ι_{3}) = Sq^{1}Sq^{8}(ι_{3}) as the lift in H^{∗}(K(ℤ)) of Sq^{9}ι_{3} = Sq^{7}Sq^{2}ι_{3} in H^{∗}(su).

The linearization map L: A(∗) → K(ℤ) from [Wa1] and 1.8 is compatible with the unit maps from S^{0}. When combined with the pullback diagram (6.2) defining g/o_{⊕} it yields the following spectrum level diagram with horizontal cofiber sequences:

(6.6)  [figure: S^{0}; i; A(∗); L; Wh^{Diff}(∗); L̄; S^{0}; i; e; K(ℤ); Σg/o_{⊕}; j; i; K(ℤ); Σbso .]

Proposition 6.7. The reduced linearization map L̄: Wh^{Diff}(∗) → Σg/o_{⊕} is a rational equivalence, but induces the zero homomorphism between the bottom homotopy groups π_{3}(Wh^{Diff}(∗)) ≅ ℤ/2 and π_{3}(Σg/o_{⊕}) ≅ ℤ/2. The induced map on mod 2 spectrum cohomology

L̄^{∗}: H^{∗}(Σg/o_{⊕}) ⟶ H^{∗}(Wh^{Diff}(∗))

is zero in all dimensions.

Proof. The linearization map L: A(∗) → K(ℤ) is a rational equivalence between spectra of finite type, by [Wa1, 2.2], so its 2-adic completion is also a rational equivalence. Comparison with (6.6) shows that also L̄ is a rational equivalence.

The homomorphism π_{3}(L̄) is induced from the homomorphism

π_{3}(L): π_{3}(A(∗)) ≅ ℤ/24 ⊕ ℤ/2 ⟶ K_{3}(ℤ) ≅ ℤ/48

by passage to the quotient with respect to subgroups π_{3}^{S} ≅ ℤ/24 on both sides. Algebraically, the only possibility is that π_{3}(L̄) = 0.

In cohomology we have the following map of extensions of A-modules:

[figure: H^{∗}(Σg/o_{⊕}); L̄^{∗}; H^{∗}(K(ℤ)); i^{∗}; L^{∗}; 𝔽_{2}; H^{∗}(Wh^{Diff}(∗)); H^{∗}(A(∗)); i^{∗}; 𝔽_{2} .]

The lower extension is split, as in 4.5. Here H^{∗}(K(ℤ)) is generated as an A-module by classes ι_{0} and ι_{3}, as in 6.4 and 6.5. The class ι_{0} maps to the split summand 𝔽_{2} of H^{∗}(A(∗)), hence the submodule it generates maps to zero in positive degrees. Likewise ι_{3} maps to zero by the π_{3}-calculation above and the Hurewicz theorem. Thus L^{∗} is zero in positive degrees, and L̄^{∗} is zero in all degrees. □

dim x  Sq^{1}(x) Sq^{2}(x)  Sq^{4}(x)  Sq^{8}(x)
0 ι_{0}  0  0  Sq^{4}ι_{0}  Sq^{8}ι_{0}

1 2

3 ι_{3}  Sq^{4}ι_{0}  Sq^{2}ι_{3}  Sq^{4}ι_{3}  Sq^{8}ι_{3}
4 Sq^{4}ι_{0}  0  Sq^{6}ι_{0}  0  Sq^{8}Sq^{4}ι_{0}
5 Sq^{2}ι_{3}  0  Sq^{7}ι_{0}  Sq^{4}Sq^{2}ι_{3} Sq^{8}Sq^{2}ι_{3}
6 Sq^{6}ι_{0}  Sq^{7}ι_{0}  0  Sq^{10}ι_{0}  Sq^{10}Sq^{4}ι_{0}
7 Sq^{7}ι_{0}  0  0  Sq^{11}ι_{0}
Sq^{4}ι_{3}  0  Sq^{6}ι_{3}  Sq^{6}Sq^{2}ι_{3}
8 Sq^{8}ι_{0}  0  Sq^{10}ι_{0}  Sq^{12}ι_{0}
9 Sq^{6}ι_{3}  Sq^{7}ι_{3}  0  Sq^{10}ι_{3}
+Sq^{8}Sq^{2}ι_{3}
Sq^{4}Sq^{2}ι_{3}  0  Sq^{6}Sq^{2}ι_{3} Sq^{13}ι_{0}
10 Sq^{10}ι_{0}  Sq^{11}ι_{0} 0  0
Sq^{7}ι_{3}  0  Sq^{8}Sq^{4}ι_{0} Sq^{11}ι_{3}
+Sq^{9}ι_{3}
11 Sq^{11}ι_{0}  0  Sq^{13}ι_{0}
Sq^{8}ι_{3}  Sq^{9}ι_{3}  Sq^{10}ι_{3}
Sq^{6}Sq^{2}ι_{3}  Sq^{8}Sq^{4}ι_{0} 0
+Sq^{9}ι_{3}
12 Sq^{12}ι_{0}  Sq^{13}ι_{0} Sq^{14}ι_{0}
Sq^{8}Sq^{4}ι_{0}  0  Sq^{10}Sq^{4}ι_{0}
Sq^{9}ι_{3}  0  Sq^{10}Sq^{4}ι_{0}

≡ Sq^{7}Sq^{2}ι_{3}

13 Sq^{13}ι_{0}  0
Sq^{10}ι_{3}  Sq^{11}ι_{3}
Sq^{8}Sq^{2}ι_{3}  0
14 Sq^{14}ι_{0}

Sq^{10}Sq^{4}ι_{0} Sq^{11}ι_{3}

Table 6.5. H^{∗}(K(ℤ)) in dimensions ≤ 14.

Corollary 6.8. There is a long exact sequence in mod 2 spectrum cohomology

⋯ → H^{∗}(T C(ℤ)) ⟶^{L^{∗}⊕trc_{ℤ}^{∗}} H^{∗}(T C(∗)) ⊕ H^{∗}(K(ℤ)) ⟶^{trc_{∗}^{∗} −L^{∗}} H^{∗}(A(∗)) ⟶^{∂} ….

Here L: A(∗) → K(ℤ) and trc_{∗}: A(∗) → T C(∗) induce zero maps in positive dimensions, ∂ induces an injective map in positive dimensions, and L: T C(∗) → T C(ℤ) and trc_{ℤ}: K(ℤ) → T C(ℤ) both induce surjections in all dimensions.

Proof. The sequence arises by applying mod 2 spectrum cohomology to the homotopy cartesian square in 1.8 for X = ∗. The assertions for L: A(∗) → K(ℤ) and trc_{∗} follow from 4.7 and 6.7. The rest follows by exactness. In fact L^{∗} ⊕ trc_{ℤ}^{∗} will be surjective in positive degrees, which is stronger than the stated conclusion. □

Remark 6.9. The ‘rigid tube’ map from [Wa3, §3] provides a space level map of horizontal fiber sequences

\xymatrix{G/O \ar[r] \ar[d]^{hw} & BSO \ar[r]^{j} \ar[d]^{s} & BSG \ar[d]^{w} \\ ΩWh^{Diff}(∗) \ar[r] & QS^{0} \ar[r]^{i} & A(∗) .}

We call the left vertical map hw the Hatcher–Waldhausen map. It was proved in [R1] that this gives a diagram of infinite loop maps if one uses a multiplicative infinite loop space structure on each of the spaces in the lower row. However, these are generally different from the additive infinite loop space structures we have been considering in this paper. Let ΩWh_{⊗}^{Diff}(∗) denote the spectrum with underlying infinite loop space given as the homotopy fiber of the unit map i: SG = Q(S^{0})_{1} → A(∗)_{1} with the multiplicative infinite loop space structures.

It can be read off from Tables 5.1, 5.7 and 6.5 that the (space level) Hatcher– Waldhausen map hw: G/O → ΩWh^{Diff}(∗) does not admit a four-fold delooping, when the target is given the additive infinite loop space structure. For by [Wa3], π_{2}(hw): ℤ/2 ≅ ℤ/2 is an isomorphism, and a k-invariant argument (see 7.5 below) shows that π_{4}(hw): ℤ → ℤ is a 2-adic equivalence. If hw were to admit a four-fold delooping then σ · hw(x) = hw(σ · x) for any x ∈ π_{4}(G/O). But π_{11}(G/O) = 0, while the minimal resolution leading to Table 5.7 shows that there is a nonzero h_{3}-multiplication from the class representing the generator of π_{4}(ΩWh^{Diff}(∗)) to the class representing the element of order 2 in π_{11}(ΩWh^{Diff}(∗)). See also 5.11. This contradicts the existence of the four-fold delooping. Note that we did not specify a choice of four-fold delooping of G/O in this argument, so it applies to both Ω^{∞}(Σ^{4}g/o) and Ω^{∞}(Σ^{4}g/o_{⊕}), in case they are different.

The spectrum map g/o → ΩWh_{⊗}^{Diff}(∗) constructed geometrically in [R1] thus shows that the spectra Wh^{Diff}(∗) and Wh_{⊗}^{Diff}(∗) cannot be homotopy equivalent.

7. A spectrum map from Wh^{Diff}(∗) to Σg/o_{⊕}

Observe by inspection of Tables 5.1 and 6.5 that H^{∗}(Wh^{Diff}(∗)) and H^{∗}(Σg/o_{⊕}) are abstractly isomorphic as A-modules in dimensions ∗ ≤ 9. In this chapter we construct a spectrum map

M: Wh^{Diff}(∗) ⟶ Σg/o_{⊕}

inducing an isomorphism in these dimensions. As before, all spectra are implicitly 2-completed in this chapter.

Lemma 7.1. There is a spectrum map m: hofib(trc) → K(ℤ) making the following diagram of horizontal cofiber sequences commute:

\xymatrix{hofib(trc) \ar[r] \ar[d]^{m} & Σ^{−2}ku \ar[r]^{δ} \ar[d]^{rβ^{−1}} & Σ^{4}ko \ar[d]^{β^{2}Σ^{4}c} \\ K(ℤ) \ar[r] & ko \ar[r]^{c(ψ^{3}−1)} & bsu .}

Proof. The maps in the right hand square are characterized (up to homotopy) by their K-localizations, and after K-localization we can compute

β^{2}Σ^{4}c ∘ L_{K}δ = β^{2} ∘ Σ^{4}c ∘ Σ^{4}r ∘ β^{−2} ∘ (ψ^{3} − 1) ∘ β^{−1} = c(ψ^{3} − 1) ∘ rβ^{−1} .

Hence the right hand square homotopy commutes. We let m be the induced map of horizontal homotopy fibers. □

Lemma 7.2. There is a spectrum map M: Wh^{Diff}(∗) → Σg/o_{⊕} making the following diagram of horizontal cofiber sequences commute:

\xymatrix{ℂP_{−1}^{∞} \ar[r]^{i} \ar[d] & hofib(trc) \ar[r]^{j} \ar[d]^{m} & Wh^{Diff}(∗) \ar[d]^{M} \\ S^{0} \ar[r]^{i} & K(ℤ) \ar[r] & Σg/o_{⊕} .}

Proof. We must show that the composite map

ℂP_{−1}^{∞} ⟶^{i} hofib(trc) ⟶^{m} K(ℤ) ⟶ Σg/o_{⊕}

is null homotopic. Consider the following diagram of horizontal and vertical cofiber sequences:

[figure: S^{0}; i; S^{0}; i; K(ℤ); ko; c(ψ^{3}−1); bsu; Σg/o_{⊕}; ko/S^{0}; bsu .]

We have [ℂP_{−1}^{∞}, su] = 0 by an application of the Atiyah–Hirzebruch spectral sequence, so we can identify [ℂP_{−1}^{∞}, K(ℤ)] with the kernel of

c(ψ^{3} − 1)_{#} : [ℂP_{−1}^{∞}, ko] ⟶ [ℂP_{−1}^{∞}, bsu] .

By another calculation with the Atiyah–Hirzebruch spectral sequence using [Ad] and [AW], this kernel is isomorphic to ℤ_{2}, and is generated by the composite map

ℂP_{−1}^{∞} ⟶ ℂP_{+}^{∞} ⟶ S^{0} ⟶^{i} K(ℤ) .

The left hand map pinches the bottom cell to a point; the middle map retracts ℂP^{∞} to a point. The composite maps to zero in [ℂP_{−1}^{∞}, Σg/o_{⊕}], so m extends to a map M as claimed. □

Lemma 7.3. The map M: Wh^{Diff}(∗) → Σg/o_{⊕} induces an isomorphism on π_{3}.

Proof. Consider the maps of long exact sequences of homotopy groups induced by the diagrams in 7.1 and 7.2. The isomorphism π_{4}(β^{2}Σ^{4}c): ℤ ≅ ℤ passes to quotient isomorphisms π_{3}(m): ℤ/16 ≅ ℤ/16 and π_{3}(M): ℤ/2 ≅ ℤ/2. □

Theorem 7.4. There is a spectrum map

M: Wh^{Diff}(∗) → Σg/o_{⊕}

inducing an isomorphism on mod 2 spectrum cohomology in dimensions ∗ ≤ 9. So M is precisely 9-connected, and induces a map of spaces

ΩM: ΩWh^{Diff}(∗) ⟶ G/O_{⊕} ≃ G/O

such that π_{∗}(ΩM) is an isomorphism for ∗ ≤ 8.

Proof. The A-module homomorphism M^{∗}: H^{∗}(Σg/o_{⊕}) → H^{∗}(Wh^{Diff}(∗)) is an isomorphism in degree 3 by 7.3. We can then compute M^{∗} in dimensions ∗ ≤ 14 from Tables 5.1 and 6.5, finding that H^{∗}(hofib(M)) is 9-connected, has rank 1 in each dimension 10 ≤ ∗ ≤ 13, and has rank ≥ 1 in dimension 14. Thus ΩM is 8-connected, and the surjection π_{8}(ΩM) is in fact an isomorphism, since both its source and target are isomorphic to ℤ ⊕ ℤ/2. □

Theorem 7.5. The Hatcher–Waldhausen map hw: G/O → ΩWh^{Diff}(∗) induces an isomorphism on 2-primary homotopy in dimensions ∗ ≤ 8, and an injection on 2-primary homotopy in dimensions ∗ ≤ 13. Its 2-completion is thus precisely 8-connected.

Proof. Let P^{n}X denote the nth Postnikov section of a (simple) space X. The map

P^{2}(hw): P^{2}G/O → P^{2}ΩWh^{Diff}(∗)

is a homotopy equivalence by 7.3. The k-invariants of G/O and ΩWh^{Diff}(∗) all lift to spectrum cohomology, since these are infinite loop spaces, and are abstractly isomorphic for n ≤ 8 by 7.4. They can be partly read off from the minimal resolution for H^{∗}(Wh^{Diff}(∗)) that was used to generate Table 5.7, yielding the following facts: Let β_{1} : K(ℤ/2, n) → K(ℤ, n + 1) be the mod 2 Bockstein map, and let i_{1} : K(ℤ, n) → K(ℤ/2, n) be the mod 2 reduction map. Then i_{1}β_{1} = Sq^{1}. For m ≥ n let p_{n}^{m}: P^{m}X → P^{n}X be a projection in the Postnikov system. Then

k^{5} : K(ℤ/2, 2) ≃ P^{2}ΩWh^{Diff}(∗) ⟶ K(ℤ, 5)

is β_{1}Sq^{2}, while

k^{7} : P^{4}ΩWh^{Diff}(∗) ⟶ K(ℤ/2, 7)

factors as Sq^{5}p_{2}^{4}. The last k-invariant we consider is

k^{9} = k_{1}^{9} × k_{2}^{9} : P^{6}ΩWh^{Diff}(∗) ⟶ K(ℤ/2 ⊕ ℤ, 9) ≃ K(ℤ/2, 9) × K(ℤ, 9) .

Its projection k_{2}^{9} to K(ℤ, 9) factors over p_{4}^{6}, and the composite

K(ℤ, 4) ⟶ P^{4}ΩWh^{Diff}(∗) ⟶^{k̄_{2}^{9}} K(ℤ, 9)

is β_{1}Sq^{4}i_{1}. Here k_{2}^{9} = k̄_{2}^{9} ∘ p_{4}^{6}.

Considering the maps of Postnikov sections P^{n}(hw): P^{n}G/O → P^{n}ΩWh^{Diff}(∗) and comparing the k-invariants for G/O and ΩWh^{Diff}(∗), it follows that also P^{4}(hw) and P^{6}(hw) are homotopy equivalences, and that P^{8}(hw) induces an isomorphism on π_{8} modulo the torsion subgroups. Hence π_{∗}(hw) is an isomorphism for ∗ ≤ 7. In particular the image of ν^{2} ∈ π_{6}(SG) ≅ π_{6}^{S} in π_{6}(G/O) maps to the generator of π_{6}(ΩWh^{Diff}(∗)).

The 2-torsion in π_{8}(G/O) is the image of ν̄ ∈ π_{8}(SG) ≅ π_{8}^{S}, satisfying η·ν̄ = ν·ν^{2}. The image of ν̄ in π_{8}(ΩWh^{Diff}(∗)) is nonzero, because η · hw(ν̄) = ν · hw(ν^{2}) is nonzero, as can be seen from Table 5.7(a) or detected by ΩM. Hence π_{8}(hw) is also an isomorphism on the torsion in dimension 8. So hw is 8-connected, but cannot be 9-connected because π_{9}(G/O) = (ℤ/2)^{2} cannot surject to π_{9}(ΩWh^{Diff}(∗)) = (ℤ/2)^{2} ⊕ ℤ/8.

The nonzero multiplications by η in π_{∗}(ΩWh^{Diff}(∗)) given in 5.11 then imply that π_{n}(hw) is injective for 9 ≤ n ≤ 11 and n = 13. Finally π_{12}(hw) is injective since π_{12}(G/O) = ℤ and hw is a rational equivalence [B1]. □

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Department of Mathematics, University of Oslo, Norway

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