THE SMOOTH WHITEHEAD SPECTRUM

OF A POINT AT ODD REGULAR PRIMES

John Rognes

Abstract. Let p be an odd regular prime, and assume that the Lichtenbaum– Quillen conjecture holds for K(ℤ[1/p]) at p. Then the p-primary homotopy type of the smooth Whitehead spectrum W h(∗) is described. A suspended copy of the cokernel-of-J spectrum splits off, and the torsion homotopy of the remainder equals the torsion homotopy of the fiber of the restricted S^{1}-transfer map t: ΣℂP^{∞} → S. The homotopy of W h(∗) is determined in a range of degrees, and the cohomology of W h(∗) is expressed as an A-module in all degrees, up to an extension. These results have geometric topological interpretations, in terms of spaces of concordances or diffeomorphisms of highly connected, high dimensional compact smooth manifolds.

Introduction

In this paper we study the smooth Whitehead spectrum W h(∗) of a point at an odd regular prime p, under the assumption that the Lichtenbaum–Quillen conjecture for K(ℤ[1/p]) holds at p. This is a reasonable assumption in view of recent work by Rost and Voevodsky. The results admit geometric topological interpretations in terms of the spaces of concordances (= pseudo-isotopies), h-cobordisms and diffeomorphisms of high-dimensional compact smooth manifolds that are as highly connected as their concordance stable range. Examples of such manifolds include discs and spheres.

Here is a summary of the paper.

In §1 we recall Waldhausen’s algebraic K-theory of spaces [Wa4], Quillen’s algebraic K-theory of rings [Qu1], the Lichtenbaum–Quillen conjecture in the strong formulation of Dwyer and Friedlander [DF1], and a theorem of Dundas [Du] about the relative properties of the cyclotomic trace map to the topological cyclic homology of Bökstedt, Hsiang and Madsen [BHM].

From §2 and onwards we assume that p is an odd regular prime and that the Lichtenbaum–Quillen conjecture holds for K(ℤ[1/p]) at p. In 2.1 and 2.2 we then call on Tate–Poitou duality for étale cohomology [Ta] to obtain a cofiber sequence

j ∨ Σ^{−2}ko ⟶ W h(∗) \underrightarrow{t̃rc} ΣℂP_{−1}^{∞} ⟶ Σj ∨ Σ^{−1}ko

of implicitly p-completed spectra. Here ℂP_{−1}^{∞} = T h(−γ^{1}) is a stunted complex projective spectrum with one cell in each even dimension ≥ −2, j is the connective

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary: 19D10, Secondary: 19F27, 55P42, 55Q52, 57R50, 57R80.

Key words and phrases. Algebraic K-theory, topological cyclic homology, Lichtenbaum–Quillen conjecture, transfer, h-cobordism, concordance, pseudoisotopy.

image-of-J spectrum at p, and ko is the connective real K-theory spectrum. In 2.5 we use this to obtain a splitting

W h(∗) ≃ Σc ∨ W h(∗)/Σc

of the suspended cokernel-of-J spectrum Σc off from W h(∗), and in 2.7 we obtain a cofiber sequence

Σ^{2}ko ⟶ W h(∗)/Σc ⟶^{τ} P_{0}Σ\overline{ℂP} _{−1}^{∞} ⟶ Σ^{3}ko ,

where π_{∗}(τ) identifies the p-torsion in the homotopy of W h(∗)/Σc with that of Σ\overline{ℂP} _{−1}^{∞}. The latter spectrum equals the homotopy fiber of the restricted S^{1}-transfer map

t: ΣℂP^{∞} → S .

Hence the homotopy of W h(∗) is as complicated as the (stable) homotopy of infinite complex projective space ℂP^{∞}, and the associated transfer map above.

In §3 we make a basic homotopical analysis, following Mosher [Mo] and Knapp [Kn], to compute π_{∗}Σ\overline{ℂP} _{−1}^{∞} and thus π_{∗}W h(∗) at p in degrees up to |β_{2}| − 2 = (2p + 1)q − 4. See 3.7 and 3.9. The first p-torsion to appear in π_{m}W h(∗) is ℤ/p for m = 4p − 2 when p ≥ 5, and ℤ/3{Σβ_{1}} for m = 11 when p = 3.

In §4 we make the corresponding mod p cohomological analysis and determine H^{∗}(W h(∗); 𝔽_{p}) as a module over the Steenrod algebra is all degrees, up to an extension. See 4.4 and 4.5. The extension is trivial for p = 3, and nontrivial for p ≥ 5. Taken together, this homotopical and cohomological information gives a detailed picture of the homotopy type W h(∗).

In §5 we recall the relation between the Whitehead spectrum W h(∗), the concordance space C(M) and the diffeomorphism group DIF F (M) of suitably highly connected and high dimensional compact smooth manifolds M. As a sample application we show in 5.3 that for p ≥ 5 and M a compact smooth k-connected n-manifold with k ≥ 4p − 2 and n ≥ 12p − 5, the first p-torsion in the homotopy of the smooth concordance space C(M) is π_{4p−4}C(M)_{(p)} ≅ ℤ/p. Specializing to M = D^{n} we conclude in 5.4 that π_{4p−4}DIF F (D^{n+1}) or π_{4p−4}DIF F (D^{n}) contains an element of order exactly p. Comparable results hold for p = 3.

A 2-primary analog of this study was presented in [Ro]. Related results on the homotopy fiber of the linearization map L: A(∗) → K(ℤ) were given in [KR].

§1. Algebraic K-theory and topological cyclic homology

Algebraic K-theory of spaces. Let A(X) be Waldhausen’s algebraic K-theory spectrum [Wa4, §2.1] of a space X. There is a natural cofiber sequence [Wa4, §3.3], [Wa5]

Σ^{∞}(X_{+}) ⟶^{η_{X}} A(X) ⟶^{π} W h(X) ,

where W h(X) = W h^{DIF F} (X) is the smooth Whitehead spectrum of X, and a natural trace map [Wa2] tr_{X}: A(X) → Σ^{∞}(X_{+}) which splits the above cofiber sequence up to homotopy. Let ι: W h(X) → A(X) be the corresponding homotopy section to π. When X = ∗ is a point, Σ^{∞}(∗_{+}) = S is the sphere spectrum, and the splitting simplifies to A(∗) ≃ S ∨ W h(∗).

Topological cyclic homology of spaces. Let p be a prime and let T C(X; p) be Bökstedt, Hsiang and Madsen’s topological cyclic homology [BHM, 5.12(i)] of the space X. There is a natural cofiber sequence [BHM, 5.17]

hofib(trf_{S^{1}}) ⟶^{ι} T C(X; p) ⟶^{β_{X}} Σ^{∞}(ΛX_{+})

after p-adic completion, where ΛX is the free loop space of X and

trf_{S^{1}} : Σ^{∞}(Σ(ES^{1} ×_{S^{1}} ΛX)_{+}) → Σ^{∞}(ΛX_{+})

is the dimension-shifting S^{1}-transfer map for the canonical S^{1}-bundle ES^{1} ×ΛX → ES^{1} ×_{S^{1}} ΛX; see e.g. [MS, §2]. When X = ∗ the S^{1}-transfer map simplifies to trf_{S^{1}} : Σ^{∞}ΣℂP_{+}^{∞} → S. Its homotopy fiber is ΣℂP_{−1}^{∞} [MS, §3], where the stunted complex projective spectrum ℂP_{−1}^{∞} = T h(−γ^{1} ↓ ℂP^{∞}) is defined as the Thom spectrum of minus the tautological line bundle over ℂP^{∞}. The map ι identifies ΣℂP_{−1}^{∞} with the homotopy fiber of β_{∗}: T C(∗; p) → S, after p-adic completion.

Integrally H_{∗}(ℂP_{−1}^{∞}) ≅ ℤ{b_{k} | k ≥ −1} and H^{∗}(ℂP_{−1}^{∞}) ≅ ℤ{y^{k} | k ≥ −1} with y^{k} dual to b_{k}, both in degree 2k. In mod p cohomology the Steenrod operations act by P^{i}(y^{k}) = \binom{k}{i}y^{k+(p−1)i} and β(y^{k}) = 0. In particular P^{i}(y^{−1}) = (−1)^{i}y^{−1+(p−1)i} ≠ 0 for all i ≥ 0.

The cyclotomic trace map for spaces. Let trc_{X}: A(X) → T C(X; p) be the natural cyclotomic trace map of Bökstedt, Hsiang and Madsen [BHM, 5.12(ii)]. It lifts the Waldhausen trace map, in the sense that tr_{X} ≃ ev ∘β_{X} ∘ trc_{X}, where ev: Σ^{∞}(ΛX_{+}) → Σ^{∞}(X_{+}) evaluates a free loop at a base point. Hence there is a map of (split) cofiber sequences of spectra:

\xymatrix{W h(X) \ar[r]^{ι} \ar[d]^{t̃rc} & A(X) \ar[r]^{^{tr^{X}}} \ar[d]^{_{trc_{X}}} & Σ^{∞}(X_{+}) \ar[d]^{=} \\ hofib(ev ∘β_{X}) \ar[r]^{ι} & T C(X; p) \ar[r]^{ev ∘β_{X}} & Σ^{∞}(X_{+})}

after p-adic completion. When X = ∗ the left hand square simplifies as follows:

Theorem 1.1 (Waldhausen, Bökstedt–Hsiang–Madsen). There is a homotopy cartesian square

\xymatrix{W h(∗) \ar[r]^{ι} \ar[d]^{t̃rc} & A(∗) \ar[d]^{_{trc_{∗}}} \\ ΣℂP_{−1}^{∞} \ar[r]^{ι} & T C(∗; p)}

after p-adic completion. Hence there is a p-complete equivalence hofib(t̃rc) ≃ hofib(trc_{∗}).

Algebraic K-theory of rings. Let K(R) be Quillen’s algebraic K-theory spectrum of a ring R [Qu1, §2]. When R is commutative, noetherian and 1/p ∈ R the étale K-theory spectrum K^{ét}(R) of Dwyer and Friedlander [DF1, §4] is defined, and comes equipped with a natural comparison map ϕ: K(R) → K^{ét}(R). By construction K^{ét}(R) is a p-adically complete K-local spectrum [Bo]. Let R be the ring of p-integers in a local or a global field of characteristic ≠ p. The Lichtenbaum–Quillen conjecture [Li1], [Li2], [Qu3] for K(R) at p, in the strong form due to Dwyer and Friedlander, then asserts:

Conjecture (Lichtenbaum–Quillen) 1.2. The comparison map ϕ induces a homotopy equivalence

P_{1}ϕ_{p}^{∧}: P_{1}K(R)_{p}^{∧} ⟶ P_{1}K^{ét}(R)

of 0-connected covers after p-adic completion.

Here P_{n}E denotes the (n−1)-connected cover of any spectrum E. In the cases of concern to us the p-completed map ϕ_{p}^{∧} will also induce an isomorphism in degree 0, so the covers P_{1} above can be replaced by P_{0}.

The conjecture above has been proven for p = 2 by Rognes and Weibel [RW, 0.6], based on Voevodsky’s proof [Vo1], [Vo2] of the Milnor conjecture. The oddprimary version of this conjecture would follow [SV] from results on the Bloch–Kato conjecture [BK] announced as “in preparation” by Rost and Voevodsky, but have not yet formally appeared.

Topological cyclic homology of rings. Let T C(R; p) be Bökstedt, Hsiang and Madsen’s topological cyclic homology of a (general) ring R. There is a natural cyclotomic trace map trc_{R}: K(R) → T C(R; p). When X is a based connected space with fundamental group π = π_{1}(X), and R = ℤ[π] is the group ring, there are natural linearization maps L: A(X) → K(R) [Wa1, §2] and L: T C(X; p) → T C(R; p) which commute with the cyclotomic trace maps. Moreover, by Dundas [Du] the square

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

is homotopy cartesian after p-adic completion. In the special case when X = ∗ and R = ℤ this simplifies to:

Theorem 1.3 (Dundas). There is a homotopy cartesian square

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

after p-adic completion. Hence there is a p-complete equivalence hofib(trc_{∗}) ≃ hofib(trc_{ℤ}).

The cyclotomic trace map for rings. When k is a perfect field of characteristic p > 0, W (k) its ring of Witt vectors, and R is an algebra of finite rank over W (k), then by Hesselholt and Madsen [HM, Thm. D] there is a cofiber sequence of spectra

K(R) ⟶^{trc_{R}} T C(R; p) ⟶ Σ^{−1}HW (R)_{F}

after p-adic completion. Here W (R)_{F} equals the coinvariants of the Frobenius action on the Witt ring of R, and Σ^{−1}HW (R)_{F} is the associated desuspended Eilenberg–Mac Lane spectrum. The Witt ring of k = 𝔽_{p} is the ring W (𝔽_{p}) = ℤ_{p} of p-adic integers, so the above applies to R = ℤ_{p}[π] for finite groups π. In particular, when X = ∗ and π = 1 there is a cofiber sequence

K(ℤ_{p}) ⟶^{trc_{ℤp}} T C(ℤ_{p}; p) ⟶ Σ^{−1}Hℤ_{p}

after p-adic completion. This uses that W (ℤ_{p})_{F} ≅ ℤ_{p}.

The completion map. Let κ: ℤ → ℤ_{p} and κ′: ℤ[1/p] → ℚ_{p} be the p-completion homomorphisms, where ℚ_{p} is the field of p-adic numbers. By naturality of trc_{R} with respect to κ there is a commutative square

\xymatrix{K(ℤ) \ar[r]^{κ} \ar[d]^{_{trc_{ℤ}}} & K(ℤ_{p}) \ar[d]^{_{trc_{ℤp}}} \\ T C(ℤ; p) \ar[r]^{≃} & T C(ℤ_{p}; p) .}

The lower map is a p-adic equivalence, since topological cyclic homology is insensitive to p-adic completion, cf. [HM, §6]. Hence there is a cofiber sequence of homotopy fibers

hofib(κ) ⟶ hofib(trc_{ℤ}) ⟶ Σ^{−2}Hℤ_{p} .

By the localization sequences in K-theory [Qu1, §5] there is a homotopy cartesian square

\xymatrix{K(ℤ) \ar[r] \ar[d]^{κ} & K(ℤ[1/p]) \ar[d]^{κ′} \\ K(ℤ_{p}) \ar[r] & K(ℚ_{p})}

so hofib(κ) ≃ hofib(κ′).

Topological K-theory and related spectra. Let ko and ku be the connective real and complex topological K-theory spectra, respectively. There is a complexification map c: ko → ku, and a cofiber sequence

Σko ⟶^{η} ko ⟶^{c} ku ⟶^{rβ^{−1}} Σ^{2}ko

related to real Bott periodicity, cf. [MQR, V.5.15]. Here η is multiplication by the stable Hopf map η: S^{1} → S^{0}, which is null-homotopic at odd primes, β: Σ^{2}ku → ku covers the Bott equivalence, and r: ku → ko is realification.

Suppose p is odd, and let q = 2p − 2. There are splittings ku_{(p)} ≃ ⋁_{i=0}^{p−2} Σ^{2i}ℓ and

(1.4)  ko_{(p)} ≃ ⋁_{i=0}^{(p−3)/2} Σ^{4i}ℓ ,

where ℓ is the connective p-local Adams summand of ku [Ad]. There is a cofiber sequence Σ^{q}ℓ → ℓ → Hℤ_{(p)} that identifies Σ^{q}ℓ with P_{q}ℓ. Let r be a topological generator of the p-adic units ℤ_{p}^{∗}, and let ψ^{r} be the Adams operation. The p-local image-of-J spectrum j is defined [MQR, V.5.16] by the cofiber sequence

j ⟶ ℓ ⟶^{ψ^{r}−1} Σ^{q}ℓ .

We now briefly write S for the p-local sphere spectrum. There is a unit map e: S → j representing (minus) the Adams e-invariant on homotopy [Qu4], and the p-local cokernel-of-J spectrum c is defined by the cofiber sequence

(1.5)  c ⟶^{f} S ⟶^{e} j .

Here e induces a split surjection on homotopy, so π_{∗}(f) is split injective. The map e identifies j with the connective cover P_{0}L_{K}S of the K-localization of S, localized at p [Bo, 4.3].

Lemma 1.6. Suppose that n ≤ 2q. If n ≠ q + 1 there are no essential spectrum maps Hℤ_{(p)} → Σ^{n}ℓ. If n = q + 1 the group of spectrum maps Hℤ_{(p)} → Σ^{q+1}ℓ is ℤ_{(p)}, generated by the connecting map ∂ of the cofiber sequence Σ^{q}ℓ → ℓ → Hℤ_{(p)}.

Lemma 1.7. There are no essential spectrum maps Σ^{n}ℓ → j for n ≥ 0 even. Hence there are no essential spectrum maps Σko_{(p)} → Σj.

The proofs are easy, using [Mis] for 1.6, and [MST, Cor. C] or [Mit, 2.4] for 1.7.

§2. Splittings at odd regular primes

The completion map in étale K-theory. When R = ℤ[1/p] and p is an odd regular prime there is a homotopy equivalence P_{0}K^{ét}(ℤ[1/p]) ≃ j ∨Σko after p-adic completion [DF2, 2.3]. Taking into account that ϕ is an equivalence in degree 0 and that K(ℤ[1/p]) has finite type [Qu2], the Lichtenbaum–Quillen conjecture for ℤ[1/p] at p amounts to the assertion that K(ℤ[1/p]) ≃ j ∨ Σko after p-localization. By the localization sequence in K-theory, this is equivalent to the assertion that K(ℤ) ≃ j ∨ Σ^{5}ko, after p-localization.

Hereafter we (often implicitly) complete all spectra at p.

When R = ℚ_{p} and p is an odd prime there is a p-adic equivalence P_{0}K^{ét}(ℚ_{p}) ≃ j∨Σj∨Σku. The Lichtenbaum–Quillen conjecture for ℚ_{p} at p asserts that K(ℚ_{p}) ≃ j ∨ Σj ∨ Σku [DM, 13.3], which again is equivalent to the assertion that K(ℤ_{p}) ≃ j ∨ Σj ∨ Σ^{3}ku, after p-adic completion. This is now a theorem, following from the calculation by Bökstedt and Madsen of T C(ℤ; p) [BM1, 9.17], [BM2].

Proposition 2.1. Let p be an odd regular prime. There are p-adic equivalences P_{0}K^{ét}(ℤ[1/p]) ≃ j ∨ Σko and P_{0}K^{ét}(ℚ_{p}) ≃ j ∨ Σj ∨ Σku such that

κ′: P_{0}K^{ét}(ℤ[1/p]) ⟶ P_{0}K^{ét}(ℚ_{p})

is homotopic to the wedge sum of the identity id: j → j, the zero map ∗ → Σj, and the suspended complexification map Σc: Σko → Σku. Thus hofib(κ′) ≃ j ∨ Σ^{2}ko.

Proof. Taking the topological generator r to be a prime power, there is a reduction map red: P_{0}K^{ét}(ℚ_{p}) → K(𝔽_{r}) ≃ j after p-adic completion [DM, §13], such that the composite map

S ⟶^{η} K(ℤ[1/p]) ⟶^{ϕ} P_{0}K^{ét}(ℤ[1/p]) ⟶^{κ′} P_{0}K^{ét}(ℚ_{p}) ⟶^{red} j

is homotopic to e. Since K^{ét}(ℤ[1/p]) is K-local, ϕη also factors through e. These maps split off a common copy of j from P_{0}K^{ét}(ℤ[1/p]) and P_{0}K^{ét}(ℚ_{p}). There are no essential spectrum maps Σko → Σj by 1.7, so after p-adic completion κ′ is homotopic to a wedge sum of maps id: j → j, ∗ → Σj and a map κ′′: Σko → Σku. Any such κ′′ lifts over Σc: Σko → Σku, so it suffices to show that π_{2i−1}(κ′′) is a p-adic isomorphism for all odd i ≥ 1.

Equivalently we must show that κ′ induces an isomorphism on homotopy modulo torsion subgroups in degree 2i − 1 for all odd i > 1, or that

K_{2i−1}^{ét}(κ′; ℚ_{p}/ℤ_{p}): K_{2i−1}^{ét}(ℤ[1/p]; ℚ_{p}/ℤ_{p}) ⟶ K_{2i−1}^{ét}(ℚ_{p}; ℚ_{p}/ℤ_{p})

is injective. This equals the completion map

κ′: H_{ét}^{1}(ℤ[1/p]; ℚ_{p}/ℤ_{p}(i)) → H_{ét}^{1}(ℚ_{p}; ℚ_{p}/ℤ_{p}(i))

in étale cohomology, by the collapsing spectral sequence in [DF, 5.1]. By the 9-term exact sequence expressing Tate–Poitou duality [Ta, 3.1], [Mil2, I.4.10], its kernel is a quotient of A^{#} = H_{ét}^{2}(ℤ[1/p]; ℤ_{p}(1−i))^{#}, where A^{#} = Hom(A, ℚ/ℤ) denotes the Pontryagin dual of an abelian group A. But A = H_{ét}^{2}(ℤ[1/p]; ℤ_{p}(1−i)) is an abelian pro-p-group, with A/p ≅ H_{ét}^{2}(ℤ[1/p]; ℤ/p(1 − i)) contained as a direct summand in B = H_{ét}^{2}(ℤ[1/p, ζ_{p}]; ℤ/p), which is independent of i. Here R = ℤ[1/p, ζ_{p}] is the ring of p-integers in the p-th cyclotomic field ℚ(ζ_{p}). Kummer theory gives a short exact sequence

0 → Pic(R)/p ⟶ B ⟶ {p} Br(R) → 0

where Pic(R) and Br(R) are the Picard and Brauer groups of R, respectively. (See [Mil2, §IV] and [Ho].) Here Pic(R)/p = 0 because p is a regular prime, and {p} Br(R) = ker(p: Br(R) → Br(R)) = 0 because p is odd and (p) does not split in R [Mil1, p. 109], so B = 0. Thus A/p = 0 and it follows that A = 0, since A is an abelian pro-p-group. □

The fiber of the cyclotomic trace map. Hereafter we make the standing assumptions that:

(a) p is an odd regular prime, and

(b) the Lichtenbaum–Quillen conjecture 1.2 holds for K(ℤ[1/p]) at p.

Proposition 2.2. There is a homotopy equivalence hofib(trc_{ℤ}) ≃ j ∨ Σ^{−2}ko after p-adic completion.

Proof. By assumption κ′: K(ℤ[1/p]) → K(ℚ_{p}) agrees with κ′: P_{0}K^{ét}(ℤ[1/p]) → P_{0}K^{ét}(ℚ_{p}) after p-adic completion, so we have a cofiber sequence

j ∨ Σ^{2}ko ⟶ hofib(trc_{ℤ}) ⟶ Σ^{−2}Hℤ_{p} .

The connecting map Σ^{−2}Hℤ_{p} → Σj ∨ Σ^{3}ko is homotopic to a wedge sum of maps Σ^{−2}Hℤ_{p} → Σj and Σ^{−2}Hℤ_{p} → Σ^{4i−1}ℓ for 1 ≤ i ≤ (p − 1)/2. All such maps are null-homotopic by 1.6, with the exception of the map ∂′: Σ^{−2}Hℤ_{p} → Σ^{2p−3}ℓ corresponding to i = (p − 1)/2.

We claim that multiplication by v_{1} acts nontrivially from degree −2 to degree 2p − 4 in π_{∗}(hofib(trc_{ℤ}); ℤ/p), from which it follows that ∂′ is a p-adic unit times the connecting map ∂ in the cofiber sequence Σ^{q−2}ℓ → Σ^{−2}ℓ → Σ^{−2}Hℤ_{p}. This implies that

hofib(trc_{ℤ}) ≃ j ∨ Σ^{−2}ℓ ∨ ⋁_{i=1}^{(p−3)/2} Σ^{4i−2}ℓ ≃ j ∨ Σ^{−2}ko .

To prove the claim, consider the homotopy cartesian squares in 1.1 and 1.3. By the Atiyah–Hirzebruch spectral sequence

E_{s,t}^{2} = H_{s}(ℂP_{−1}^{∞}; π_{t}(S; ℤ/p)) ⟹ π_{s+t}(ℂP_{−1}^{∞}; ℤ/p)

we find π_{−2}(ℂP_{−1}^{∞}; ℤ/p) ≅ ℤ/p{b_{−1}} and π_{2p−4}(ℂP_{−1}^{∞}; ℤ/p) ≅ ℤ/p{v_{1}b_{−1}}. Hence multiplication by v_{1} acts nontrivially from π_{−1}(T C(∗; p); ℤ/p) ≅ ℤ/p{Σb_{−1}} to π_{2p−3}(T C(∗; p); ℤ/p) ≅ ℤ/p{α_{1}, Σv_{1}b_{−1}}, also modulo the image from the unit map η: S → T C(∗; p).

The map L: S → Hℤ is (2p − 3)-connected, hence so is L: T C(∗; p) → T C(ℤ; p) by [BM1, 10.9] and [Du]. Here π_{2p−3}(T C(ℤ; p); ℤ/p) ≅ ℤ/p{α_{1}} ⊕ ℤ/p since P_{0}T C(ℤ; p) ≃ K(ℤ_{p}) ≃ j ∨ Σj ∨ Σ^{3}ku. So the surjection π_{2p−3}(L; ℤ/p) is in fact a bijection, and multiplication by v_{1} acts nontrivially from π_{−1}(T C(ℤ; p); ℤ/p) to π_{2p−3}(T C(ℤ; p); ℤ/p), also modulo the image from the unit map η: S → T C(ℤ; p).

By the assumed p-adic equivalence K(ℤ) ≃ j ∨ Σ^{5}ko, this image equals the image from the cyclotomic trace map trc_{ℤ}: K(ℤ) → T C(ℤ; p). Hence we can pass to cofibers, and conclude that multiplication by v_{1} acts nontrivially from π_{−2}(hofib(trc_{ℤ}); ℤ/p) to π_{2p−4}(hofib(trc_{ℤ}); ℤ/p), as claimed. □

We let d be the homotopy cofiber map of t̃rc. Combining 1.1, 1.3 and 2.2 we have:

Corollary 2.3. There is a diagram of horizontal cofiber sequences:

\xymatrix{j ∨ Σ^{−2}ko \ar[r] \ar[d]^{=} & W h(∗) \ar[r]^{t̃rc} \ar[d]^{ι} & ΣℂP_{−1}^{∞} \ar[r]^{d} \ar[d]^{ι} & Σj ∨ Σ^{−1}ko \ar[d]^{=} \\ j ∨ Σ^{−2}ko \ar[r] \ar[d]^{=} & A(∗) \ar[r]^{_{trc_{∗}}} \ar[d]^{L} & T C(∗; p) \ar[r] \ar[d]^{L} & Σj ∨ Σ^{−1}ko \ar[d]^{=} \\ j ∨ Σ^{−2}ko \ar[r] & K(ℤ) \ar[r]^{_{trc_{ℤ}}} & T C(ℤ; p) \ar[r] & Σj ∨ Σ^{−1}ko .}

The restricted S^{1}-transfer map. There is a stable splitting in_{1} ∨ in_{2} : S^{1} ∨ ΣℂP^{∞} ≃ ΣℂP_{+}^{∞}. Let the restricted S^{1}-transfer map t = trf_{S^{1}} ∘ in_{2} : ΣℂP^{∞} → S be the restriction of trf_{S^{1}} to the second summand [Mu, §2]. The restriction to the first summand is the stable Hopf map η = trf_{S^{1}} ∘ in_{1} : S^{1} → S^{0}, which is null-homotopic at odd primes. Hence the inclusion in_{1} lifts to a map Σb_{0} : S^{1} → hofib(trf_{S^{1}}) = ΣℂP_{−1}^{∞}, with Hurewicz image Σb_{0} ∈ H_{1}(ΣℂP_{−1}^{∞}). Dually the projection pr_{1} : ΣℂP_{+}^{∞} → S^{1} yields a map Σy^{0} : ΣℂP_{−1}^{∞} → S^{1} with dual Hurewicz image Σy^{0} ∈ H^{1}(ΣℂP_{−1}^{∞}). We obtain a diagram of horizontal and split vertical cofiber sequences:

(2.4)  \xymatrix{S^{1} \ar[d]^{Σb_{0}} \ar[r]^{=} & S^{1} \ar[d]^{_{in_{1}}} \\ ΣℂP_{−1}^{∞} \ar[d] \ar[r] \ar[u]^{Σy^{0}} & ΣℂP_{+}^{∞} \ar[d]^{_{pr_{2}}} \ar[r]^{_{trf_{S^{1}}}} \ar[u]^{_{pr_{1}}} & S \ar[d]^{=} \\ hofib(t) \ar[r] \ar[u] & ΣℂP^{∞} \ar[r]^{t} \ar[u]^{_{in_{2}}} & S .}

Writing \overline{ℂP} _{−1}^{∞} for the homotopy cofiber of b_{0} : S → ℂP_{−1}^{∞}, we have hofib(t) ≃ Σ\overline{ℂP} _{−1}^{∞}. Then H_{∗}(Σ\overline{ℂP} _{−1}^{∞}) = ℤ\{Σb_{k} | k ≥ −1, k ≠ 0\} and H^{∗}(Σ\overline{ℂP} _{−1}^{∞}) ≅ ℤ\{Σy^{k} | k ≥ −1, k ≠ 0}. It has been shown by Knapp [Kn] that π_{∗}(t): π_{∗}(ΣℂP^{∞}) → π_{∗}(S) is surjective for 0 < ∗ < |β_{p+1}| = p(p + 2)q − 2, so the homotopy of Σ\overline{ℂP} _{−1}^{∞} is as well understood in this range as that of ΣℂP^{∞}.

The suspended cokernel-of-J spectrum. We can split off the suspension of the cofiber sequence (1.5) defining the cokernel-of-J from the top cofiber sequence in 2.3.

Proposition 2.5. There is a diagram of horizontal and split vertical cofiber sequences:

\xymatrix{j \ar[r] \ar[d]^{_{in_{1}}} & Σc \ar[r]^{Σf} \ar[d]^{g} & S^{1} \ar[r]^{Σe} \ar[d]^{Σb_{0}} & Σj \ar[d]^{_{in_{1}}} \\ j ∨ Σ^{−2}ko \ar[r] \ar[d]^{_{pr_{2}}} \ar[u]^{_{pr_{1}}} & W h(∗) \ar[r]^{t̃rc} \ar[d] \ar[u] & ΣℂP_{−1}^{∞} \ar[r]^{d} \ar[d] \ar[u]^{Σy^{0}} & Σj ∨ Σ^{−1}ko \ar[d]^{_{pr_{2}}} \ar[u]^{_{pr_{1}}} \\ Σ^{−2}ko \ar[r] \ar[u]^{_{in_{2}}} & W h(∗)/Σc \ar[r] \ar[u] & Σ\overline{ℂP} _{−1}^{∞} \ar[r] \ar[u] & Σ^{−1}ko . \ar[u]^{_{in_{2}}}}

In particular there is a splitting

W h(∗) ≃ Σc ∨ (W h(∗)/Σc)

where W h(∗)/Σc is defined as the homotopy cofiber of g.

Proof. The composite d ∘ Σb_{0} represents the generator of π_{1}(Σj ∨ Σ^{−1}ko), hence factors as in_{1} ∘Σe: S^{1} → Σj → Σj ∨ Σ^{−1}ko. We define g: Σc → W h(∗) as the induced map of homotopy fibers. It is well-defined up to homotopy since π_{2}(Σj ∨ Σ^{−1}ko) = 0. This explains the downward cofiber sequences of the diagram.

To split g we must show that pr_{1} ∘ d factors as Σe ∘ Σy^{0}, or equivalently that the composite

Σ\overline{ℂP} _{−1}^{∞} → ΣℂP_{−1}^{∞} ⟶^{d} Σj ∨ Σ^{−1}ko ⟶^{pr_{1}} Σj

is null-homotopic. But this map lies in a zero group, because in the Atiyah– Hirzebruch spectral sequence

E_{s,t}^{2} = H^{−s}(Σ\overline{ℂP} _{−1}^{∞}; π_{t}(Σj)) ⟹ [Σ\overline{ℂP} _{−1}^{∞}, Σj]_{s+t}

all the groups E_{s,t}^{2} with s + t = 0 are zero. □

Remark 2.6. Let G/O be the homotopy fiber of the map of spaces BO → BG, and let Cok J = Ω^{∞}c be the cokernel-of-J space. There is a (Sullivan) fiber sequence Cok J → G/O → BSO [MM, §5C]. Waldhausen [Wa3, 3.4] constructed a space level map hw: G/O → ΩΩ^{∞}W h(∗), using manifold models for A(∗). Hence there is a geometrically defined composite map Cok J → G/O → ΩΩ^{∞}W h(∗). Presumably this is homotopic to the infinite loop map Ω^{∞}Σ^{−1}g.

A cofiber sequence. We can analyze a variant of the lower cofiber sequence in 2.5 by passing to connective covers. There is a map of homotopy cartesian squares from

\xymatrix{Σc \ar[r]^{Σf} \ar[d]^{∗} & S^{1} \ar[d]^{in_{2} ∘Σe} & to & W h(∗) \ar[r]^{t̃rc} \ar[d]^{Lι} & P_{0}ΣℂP_{−1}^{∞} \ar[d]^{Lι} \\ j \ar[r]^{_{in_{1}}} & j ∨ Σj &  & K(ℤ) \ar[r]^{_{trc_{ℤ}}} & P_{0}T C(ℤ; p)}

induced by g, Σb_{0}, in_{1} and in_{1} ∨ in_{2} in the upper left, upper right, lower left and lower right corners, respectively. In the lower rows we are using the splittings K(ℤ) ≃ j ∨ Σ^{5}ko and P_{0}T C(ℤ; p) ≃ K(ℤ_{p}) ≃ j ∨ Σj ∨ Σ^{3}ku derived from 2.1. Let τ: W h(∗)/Σc → P_{0}Σ\overline{ℂP} _{−1}^{∞}, ℓ: W h(∗)/Σc → Σ^{5}ko and ℓ: P_{0}Σ\overline{ℂP} _{−1}^{∞} → Σ^{3}ku be the cofiber maps induced by t̃rc: W h(∗) → P_{0}ΣℂP_{−1}^{∞}, Lι: W h(∗) → K(ℤ) and Lι: P_{0}ΣℂP_{−1}^{∞} → P_{0}T C(ℤ; p), respectively.

Theorem 2.7. There is a diagram of horizontal and vertical cofiber sequences:

\xymatrix{ & hofib(ℓ) \ar[r]^{=} \ar[d] & hofib(ℓ) \ar[d] \\ Σ^{2}ko \ar[r] \ar[d]^{=} & W h(∗)/Σc \ar[r]^{τ} \ar[d]^{ℓ} & P_{0}Σ\overline{ℂP} _{−1}^{∞} \ar[r] \ar[d]^{ℓ} & Σ^{3}ko \ar[d]^{=} \\ Σ^{2}ko \ar[r]^{∗} & Σ^{5}ko \ar[r]^{c} & Σ^{3}ku \ar[r]^{rβ^{−1}} & Σ^{3}ko .}

The map τ: W h(∗)/Σc → P_{0}Σ\overline{ℂP} _{−1}^{∞} induces a split injection on homotopy groups in all degrees, and each map ℓ is (2p − 3)-connected. Thus

π_{∗}(τ): tors π_{∗}(W h(∗)/Σc) ≅ tors π_{∗}(Σ\overline{ℂP} _{−1}^{∞}) .

Here tors A denotes the torsion subgroup of an abelian group A.

Proof. It follows from 2.1 and localization in algebraic K-theory that the map Σ^{5}ko → Σ^{3}ku induced by trc_{ℤ}: K(ℤ) → P_{0}T C(ℤ; p) ≃ K(ℤ_{p}) is the lift of Σc: Σko → Σku to the 1-connected covers. This identifies the central homotopy cartesian square in the diagram.

By comparing the vertical homotopy fibers in the last three homotopy cartesian squares we obtain a cofiber sequence c ∨ Σc → hofib(L) → hofib(ℓ), as in [KR, 3.6]. Hence each map ℓ is (2p − 3)-connected because L is. There is a (4p − 3)-connected space level map from SU to Ω^{∞}Σ\overline{ℂP} _{−1}^{∞}, as in [KR, (17)].

Bϕ: SU ⟶ Ω^{∞}Σ\overline{ℂP} _{−1}^{∞} ⟶^{Ω^{∞}ℓ} SU .

Its composite with Ω^{∞}ℓ to Ω^{∞}Σ^{3}ku = SU loops to an H-map ϕ: BU → BU. Any such H-map is a series of Adams operations ψ^{k}, as in [MST, 2.3], so π_{∗}(ϕ; ℤ/p) only depends on ∗ mod q in positive degrees. Since ℓ is (2p−3)-connected it follows that ϕ is (2p−4)-connected, so π_{∗}(ϕ; ℤ/p) is an isomorphism for 0 < ∗ < q, and so π_{∗}(ϕ) is an isomorphism for all ∗ ≢ 0 mod q. Hence π_{∗}(ℓ) is (split) surjective whenever ∗ ≢ 1 mod q, cf. [KR, 6.3(i)].

Finally rβ^{−1} is split surjective as a spectrum map, and π_{∗}(Σ^{3}ko) is zero for ∗ ≡ 1 mod q, so rβ^{−1}ℓ: P_{0}Σ\overline{ℂP} _{−1}^{∞} → Σ^{3}ko induces a split surjection on homotopy in all degrees. □

Remark 2.8. We still do not know the behavior of ℓ: W h(∗)/Σc → Σ^{5}ko in degrees ∗ ≡ 1 mod q. It induces the same homomorphism on homotopy as ℓ: P_{0}Σ\overline{ℂP} _{−1}^{∞} → Σ^{3}ku, since π_{∗}(τ) and π_{∗}(c) are isomorphisms in these degrees.

The suspended quaternionic projective spectrum. After p-adic completion ℂP_{−1}^{∞} splits as a wedge sum of (p − 1) eigenspectra ℂP_{−1}^{∞}[a] for −1 ≤ a ≤ p − 3, much like the Adams splitting in [Ad]. Here H^{∗}(ℂP_{−1}^{∞}[a]) ≅ ℤ_{p}{Σy^{k} | k ≥ −1, k ≡ a mod p − 1}, and similarly with mod p coefficients.

Let ℍP^{∞} be the infinite quaternionic projective spectrum. The “quaternionification” map q: ℂP_{−1}^{∞} → ℍP_{+}^{∞} ≃ S ∨ ℍP^{∞} admits a (stable p-adic) section s: ℍP_{+}^{∞} → ℂP_{−1}^{∞}. (It can be obtained by Thomifying the Becker–Gottlieb transfer map Σ^{∞}(BS_{+}^{3}) → Σ^{∞}(BS_{+}^{1}) associated to the sphere bundle S^{2} → BS^{3} → BS^{1}, with respect to minus the tautological quaternionic line bundle over BS^{3} = ℍP^{∞}, and collapsing the bottom −4-cell.) This section s identifies S ∨ ℍP^{∞} with the wedge sum of the even ℂP_{−1}^{∞}[a] for a = 2i with 0 ≤ i ≤ (p − 3)/2.

Splitting off S, suspending once and passing to connected covers, we obtain maps s′: ΣℍP^{∞} → P_{0}Σ\overline{ℂP} _{−1}^{∞} and q′: P_{0}Σ\overline{ℂP} _{−1}^{∞} → ΣℍP^{∞} whose composite is a p-adic equivalence.

Proposition 2.9. The map s′: ΣℍP^{∞} → P_{0}Σ\overline{ℂP} _{−1}^{∞} admits a lift s̃: ΣℍP^{∞} → W h(∗)/Σc over τ, which is unique up to homotopy, and whose composite with q′ ∘ τ: W h(∗)/Σc → ΣℍP^{∞} is a p-adic equivalence.

Proof. The composite map rβ^{−1}ℓ ∘ s′: ΣℍP^{∞} → Σ^{3}ko lies in a zero group, by the Atiyah–Hirzebruch spectral sequence

E_{s,t}^{2} = H^{−s}(ΣℍP^{∞}; π_{t}Σ^{3}ko) ⟹ [ΣℍP^{∞}, Σ^{3}ko]_{s+t} .

Hence s′ admits a lift s̃, as claimed. In fact the lift is unique up to homotopy, since also [ΣℍP^{∞}, Σ^{3}ko]_{1} = 0. □

A second cofiber sequence. We define W h(∗)/(Σc, ΣℍP^{∞}) ≃ hofib(q′τ) as the homotopy cofiber of s̃, and write

(2.10)  \frac{P_{0}Σ\overline{ℂP} _{−1}^{∞}}{ΣℍP^{∞}} ≃ P_{0}ΣℂP_{−1}^{∞}[−1] ∨ ⋁_{i=1}^{(p−3)/2} ΣℂP_{−1}^{∞}[2i − 1]

for the suspended homotopy cofiber of s′. Then:

Theorem 2.11. There is a splitting

W h(∗) ≃ Σc ∨ ΣℍP^{∞} ∨ \frac{W h(∗)}{(Σc, ΣℍP^{∞})}

and a cofiber sequence

Σ^{2}ko ⟶ \frac{W h(∗)}{(Σc, ΣℍP^{∞})} ⟶^{τ} \frac{P_{0}Σ\overline{ℂP} _{−1}^{∞}}{ΣℍP^{∞}} ⟶^{δ} Σ^{3}ko .

The map τ induces a split injection on homotopy groups in all degrees, and the map δ induces an injection on mod p cohomology in degrees ≤ 2p − 3. Thus

π_{∗}(W h(∗)) ≅ π_{∗}(Σc) ⊕ π_{∗}(ΣℍP^{∞}) ⊕ tors π_{∗}(\frac{Σ\overline{ℂP} _{−1}^{∞}}{ΣℍP^{∞}} ) .

Proof. The cofiber sequence arises by splitting off ΣℍP^{∞} from the middle horizontal cofiber sequence in 2.7. The assertion about τ follows by retraction from the corresponding statement in 2.7. The map δ is the composite of the maps

\frac{P_{0}Σ\overline{ℂP} _{−1}^{∞}}{ΣℍP^{∞}} ⟶^{in} P_{0}Σ\overline{ℂP} _{−1}^{∞} ⟶^{ℓ} Σ^{3}ku ⟶^{rβ^{−1}} Σ^{3}ko .

On mod p cohomology (rβ^{−1})^{∗} is split injective and ℓ^{∗} is injective in degrees ≤ 2p−3 by 2.7. The kernel of in^{∗} is ΣH^{∗}(ℍP^{∞}; 𝔽_{p}), which is concentrated in degrees ≡ 1 mod 4. But in degrees ≤ 2p − 3 all of H^{∗}(Σ^{3}ko; 𝔽_{p}) is in degrees ≡ 3 mod 4, so also the composite δ^{∗} is injective in this range of degrees. □

Remark 2.12. Note that the upper cofiber sequence in 2.3 maps as in 2.5 to the middle horizontal cofiber sequence in 2.7, which in turn maps to the cofiber sequence in 2.11. In 4.4 we will see that δ is (4p − 2)-connected.

§3. Homotopical analysis

Homotopy of the fiber of the restricted S^{1}-transfer map. To make the p-primary homotopy groups of W h(∗) explicit we refer to 2.7 and compute the ptorsion in the homotopy of \overline{ℂP} _{−1}^{∞} in an initial range of degrees. This is related to ℂP^{∞} by the cofiber sequence

(3.1)  Σ\overline{ℂP} _{−1}^{∞} ⟶ ΣℂP^{∞} ⟶^{t} S

extracted from (2.4). We also use the cofiber sequence

c ∧ ℂP^{∞} ⟶^{f∧1} ℂP^{∞} ⟶^{e∧1} j ∧ ℂP^{∞}

obtained by smashing (1.5) with ℂP^{∞}. There are Atiyah-Hirzebruch spectral sequences:

(3.2)  E_{s,t}^{2} = H_{s}(ℂP^{∞}; π_{t}(j)) ⟹ j_{s+t}(ℂP^{∞})
(3.3)  E_{s,t}^{2} = H_{s}(ℂP^{∞}; π_{t}(S)) ⟹ π_{s+t}(ℂP^{∞})
(3.4)  E_{s,t}^{2} = H_{s}(\overline{ℂP} _{−1}^{∞}; π_{t}(S)) ⟹ π_{s+t}(\overline{ℂP} _{−1}^{∞}) .

We will now account for the abutment of (3.2) in all degrees, and for (3.3) and (3.4) in total degrees ∗ < |β_{2}b_{1}| = (2p + 1)q and ∗ < |β_{2}b_{−1}| = (2p + 1)q − 4, respectively.

Let v_{p}(n) be the p-adic valuation of a natural number n. In degrees ∗ < |β_{2}| = (2p + 1)q − 2 the p-torsion in π_{∗}(S) = π_{∗}^{S} is generated by the image-of-J classes ᾱ_{i} ∈ π_{qi−1}^{S} of order p^{1+v_{p}(i)} for i ≥ 1, and the cokernel-of-J classes [Ra, 1.1.14]

β_{1} ∈ π_{pq−2}^{S}, α_{1}β_{1} ∈ π_{(p+1)q−3}^{S}, β_{1}^{2} ∈ π_{2pq−4}^{S} and α_{1}β_{1}^{2} ∈ π_{(2p+1)q−5}^{S} ,

each of order p.

Theorem 3.5. Above the horizontal axis and in total degrees ∗ < |β_{2}| − 2, the Atiyah–Hirzebruch E_{s,t}^{∞}-term for π_{∗}\overline{ℂP} _{−1}^{∞} agrees with that for j_{∗}(ℂP^{∞}), plus the ℤ/p-module generated by β_{1}b_{m}, α_{1}β_{1}b_{mp}, β_{1}^{2}b_{m} (and α_{1}β_{1}^{2}b_{mp}) for 1 ≤ m ≤ p − 3, minus the ℤ/p-module generated by α_{1}b_{mp} for m ≥ p − 2.

We give the proof in a couple of steps.

Connective J-theory of complex projective space. On the horizontal axis the E^{2}-terms of (3.2) and (3.3) have the form E_{∗,0}^{2} = H_{∗}(ℂP^{∞}) = ℤ{b_{n} | n ≥ 1}, which has the structure of a divided power algebra on b_{1}. By Toda [To] or Mosher [Mo, 2.1], the corresponding part of the E^{∞}-term of (3.3) consists of the polynomial algebra on b_{1}, i.e.,

(3.6)  E_{2n,0}^{∞} = ℤ{n! b_{n}} ⊆ E_{2n,0}^{2} = ℤ{b_{n}}

for all n ≥ 1. Hence the order of the images of the differentials d_{2n,0}^{r} landing in total degree 2n − 1 all multiply to n!.

It is known by [Mo, 4.7(a)] that these differentials from the horizontal axis land in the image-of-J, i.e., have the form θb_{k} with θ a multiple of some ᾱ_{i}. Hence (3.6) also gives the E^{∞}-term of (3.2) on the horizontal axis. Since the Atiyah–Hirzebruch spectral sequence for j_{∗}(ℂP^{∞}) only has classes in (even, odd) bidegrees above the horizontal axis, there can be no further differentials in (3.2). In even total degrees it follows that j_{2n}(ℂP^{∞}) ≅ ℤ{n! b_{n}} for n ≥ 1.

In odd total degrees, the E^{2}-term of (3.2) contains the classes p^{e}ᾱ_{i}b_{k} in bidegree (s, t) = (2k, qi − 1), for 0 ≤ e ≤ v_{p}(i). It follows that the p-valuation of the order of the groups E_{s,t}^{2} in total degree s + t = 2n − 1 equals ∑_{e≥0} [(n − 1)/p^{e}(p − 1)], so the p-valuation of the order of the finite group j_{2n−1}(ℂP^{∞}) is

∑_{e≥0} [\frac{n − 1}{p^{e}(p − 1)}] − ∑_{e≥0} [\frac{n}{p^{e} · p}] .

Here the second sum equals v_{p}(n!). Compare [KR, 4.3] due to Knapp. For n ≤ p^{2}(p − 1) the terms with e ≥ 2 vanish.

Stable homotopy of complex projective space. We now return to (3.3) where the E^{2}-term contains additional classes from H_{∗}(ℂP^{∞}; π_{∗}(c)). The primary operation P^{1} detects α_{1}, and P^{1}(y^{k}) = ky^{k+p−1} in mod p cohomology, so there are differentials d^{q}(θb_{k+p−1}) = kα_{1}θb_{k} for all θ ∈ π_{∗}(S). In the case θ = 1 these differentials were already accounted for by the differentials leading to (3.6), but for t < |β_{2}| there are also differentials

d^{q}(β_{1}b_{k+p−1}) = α_{1}β_{1}b_{k}  and  d^{q}(β_{1}^{2}b_{k+p−1}) = α_{1}β_{1}^{2}b_{k}

up to unit multiples, for k ≢ 0 mod p, k ≥ 1. This leaves the classes α_{1}b_{mp} (already in j_{∗}(ℂP^{∞})), α_{1}β_{1}b_{mp} and α_{1}β_{1}^{2}b_{mp} for m ≥ 1 in odd total degrees, and the classes β_{1}b_{1}, …, β_{1}b_{p−2}, β_{1}b_{mp−1} for m ≥ 1, β_{1}^{2}b_{1}, …, β_{1}^{2}b_{p−2} and β_{1}^{2}b_{mp−1} for m ≥ 1 in even total degrees.

The (well-known) p-fold Toda bracket β_{1} = ⟨α_{1}, …, α_{1}⟩ implies differentials

d^{(p−1)q}(θα_{1}b_{k+(p−1)^{2}}) = θβ_{1}b_{k}

when k +(p−1)^{2} = mp, up to unit multiples. So the classes α_{1}b_{mp} (from j_{∗}(ℂP^{∞})) and α_{1}β_{1}b_{mp} for m ≥ p − 1 support d^{(p−1)q}-differentials, which kill the classes β_{1}b_{mp−1} and β_{1}^{2}b_{mp−1} for m ≥ 1. For bidegree reasons this accounts for all differentials in (3.3) in total degrees ∗ < |β_{2}b_{1}|.

To pass from ℂP^{∞} to \overline{ℂP} _{−1}^{∞} we must take into account the differentials in (3.4) that cross the vertical axis, which amounts to the restricted S^{1}-transfer map t as in (3.1). The image-of-J in its target π_{∗}(S) is hit by classes on the horizontal axis of (3.3), by [Mu, 4.3] or Crabb and Knapp, cf. [KR, 5.8]. The cokernel-of-J classes are hit by the differentials

d^{q}(β_{1}b_{p−2}) = α_{1}β_{1}b_{−1} ,  d^{(p−1)q}(α_{1}b_{(p−2)p}) = β_{1}b_{−1} ,
d^{q}(β_{1}^{2}b_{p−2}) = α_{1}β_{1}^{2}b_{−1} ,  d^{(p−1)q}(α_{1}β_{1}b_{(p−2)p}) = β_{1}^{2}b_{−1}

in (3.4). Looking over the bookkeeping concludes the proof of Theorem 3.5.

Torsion homotopy of the smooth Whitehead spectrum.

Theorem 3.7. (a) The torsion homotopy of W h(∗) decomposes as

tors π_{∗}(W h(∗)) ≅ π_{∗}(Σc) ⊕ tors π_{∗}(Σ\overline{ℂP} _{−1}^{∞})

in all degrees.

(b) In degrees ∗ < |β_{2}| + 1 = (2p + 1)q − 1

π_{∗}(Σc) ≅ ℤ/p{Σβ_{1}, Σα_{1}β_{1}, Σβ_{1}^{2}, Σα_{1}β_{1}^{2}}

with generators in degrees pq −1, (p+1)q −2, 2pq −3 and (2p+1)q −4, respectively.

(c) In even degrees ∗ < |β_{2}| − 1 = (2p + 1)q − 3 the p-valuation of the order of tors π_{2n}(Σ\overline{ℂP} _{−1}^{∞}) equals

([\frac{n − 1}{p − 1}] + [\frac{n − 1}{p(p − 1)}]) − ([\frac{n}{p}] + [\frac{n}{p^{2}} ]) ,

plus 1 when n = p^{2} − 2 + mp for 1 ≤ m ≤ p − 3, minus 1 when n = p − 1 + mp for m ≥ p − 2.

(d) In odd degrees ∗ < |β_{2}| − 1 = (2p + 1)q − 3 the p-valuation of the order of tors π_{2n+1}(Σ\overline{ℂP} _{−1}^{∞}) equals 1 when n = p^{2} − p − 1 + m or n = 2p^{2} − 2p − 2 + m for 1 ≤ m ≤ p − 3, and is 0 otherwise.

Example 3.8. (a) When p = 3, the 3-torsion in π_{∗}W h(∗) has order 3 in degrees 11, 16, 18, 20, 21 and 22, order 3^{2} in degree 24, order 3^{3} in degree 14, and is trivial in the remaining degrees ∗ < 25.

(b) When p = 5, the 5-torsion in π_{∗}W h(∗) has order 5 in degrees 18, 26, 28, 34, 36, 39, 41, 43, 48, 50, 52, 54, 58, 60, 62, 64, 68, 70, 72, 77, 78, 79, 80 and 81, order 5^{2} in degrees 42, 44, 56, 74 and 76, order 5^{3} in degrees 46, 66 and 82, order 5^{4} in degree 84, and is trivial in the remaining degrees ∗ < 85.

In roughly half this range we can give the following simpler statement.

Corollary 3.9. (a) For p ≥ 5, the low-degree p-torsion in π_{∗}W h(∗) is ℤ/p in degrees ∗ = 2n for m(p − 1) < n < mp and 1 < m < p, except in degree 2p^{2} − 2p − 2 (corresponding to n = mp − 1 and m = p − 1). The next p-torsion is ℤ/p{Σβ_{1}} in degree 2p^{2} − 2p − 1, and a group of order p^{2} in degree 2p^{2} − 2p + 2.

(b) For p = 3 the bottom 3-torsion in π_{∗}W h(∗) is ℤ/3{Σβ_{1}} in degree 11, followed by ℤ/3{Σα_{1}β_{1}} ⊕ ℤ/9 in degree 14.

The asserted group structure of π_{14}W h(∗)_{(3)} can be obtained from 4.5(a) below and the mod 3 Adams spectral sequence.

§4. Cohomological analysis

We can determine the mod p cohomology of W h(∗) as a module over the Steenrod algebra A, up to an extension, in all degrees. To do this, we apply cohomology to the splitting and cofiber sequence in 2.11.

Some cohomology modules. Let us briefly write H^{∗}(X) = H^{∗}(X; 𝔽_{p}) for the mod p cohomology of a spectrum X, where p is an odd prime. It is naturally a left module over the mod p Steenrod algebra A [St]. Let A_{n} be the subalgebra of A generated by the Bockstein operation β and the Steenrod powers P^{1}, …, P^{p^{n−1}} and let E_{n} be the exterior subalgebra generated by the Milnor primitives β, Q_{1}, …, Q_{n}, where Q_{0} = β and Q_{n+1} = [P^{p^{n}}, Q_{n−1}]. For an augmented subalgebra B ⊂ A we write I(B) = ker(ϵ: B → 𝔽_{p}) for the augmentation ideal, and let A//B = A ⊗_{B} 𝔽_{p} = A/A · I(B).

Proposition 4.1. (a) H^{∗}(Hℤ) ≅ A//E_{0} = A/A(β) and H^{∗}(ℓ) ≅ A//E_{1} = A/A(β, Q_{1}).

(b) The cofiber sequence Σ^{pq−1}ℓ → j → ℓ induces a nontrivial extension

A//A_{1} → H^{∗}(j) → Σ^{pq−1}A//A_{1}

of A-modules. As an A-module H^{∗}(j) is generated by two classes 1 and b in degree 0 and pq − 1, respectively, with β(b) = P^{p}(1).

(c) The cofiber sequence S ⟶^{e} j → Σc induces an identification H^{∗}(Σc) ≅ ker(e^{∗}: H^{∗}(j) → 𝔽_{p}). There is a nontrivial extension

I(A)/A(β, P^{1}) → H^{∗}(Σc) → Σ^{pq−1}A//A_{1}

of A-modules.

Proof. For (a), see [AP, 2.1]. For (c), clearly the given cofiber sequence identifies H^{∗}(Σc) with the positive degree part of H^{∗}(j). The long exact sequence in cohomology associated to the cofiber sequence given in (b) is:

Σ^{q}A//E_{1} ⟶^{(ψ^{r}−1)^{∗}} A//E_{1} ⟶ H^{∗}(j) ⟶ Σ^{q−1}A//E_{1} ⟶^{(ψ^{r}−1)^{∗}} Σ^{−1}A//E_{1} .

The map e: S → j is (pq − 2)-connected [Ra, 1.1.14], so e^{∗}: H^{∗}(j) → H^{∗}(S) = 𝔽_{p} is an isomorphism for ∗ ≤ pq − 2. Thus P^{1} ∈ A//E_{1} is in the image of (ψ^{r} − 1)^{∗}, and so (ψ^{r} − 1)^{∗} is induced up over A_{1} ⊂ A by

Σ^{q}A_{1}//E_{1} ⟶^{P^{1}} A_{1}//E_{1} ,

which has kernel Σ^{pq}𝔽_{p} generated by Σ^{q}P^{p−1} and cokernel 𝔽_{p} generated by 1. Hence there is an extension A//A_{1} → H^{∗}(j) → Σ^{pq−1}A//A_{1}. Note that the bottom classes in A//A_{1} are 1 and P^{p} in degrees 0 and pq, respectively. Let b ∈ H^{pq−1}(j) be the class mapped to Σ^{pq−1}(1) in Σ^{pq−1}A//A_{1}. By the Hurewicz theorem for Σc it is dual to the Hurewicz image of the bottom class Σβ_{1} ∈ π_{pq−1}(Σc). Since β_{1} ∈ π_{pq−2}(c) ⊂ π_{pq−2}(S) has order p there is a nontrivial Bockstein β(b) in H^{∗}(Σc), and thus also in H^{∗}(j). The only possible value in degree pq is P^{p}(1). Part (c) now follows easily from (b). □

Proposition 4.2. (a) H^{∗}(ΣℍP^{∞}) ≅ 𝔽_{p}{Σy^{k} | k ≥ 2 even}.

(b) H^{∗}(ΣℂP_{−1}^{∞}[−1]) ≅ Σ^{−1}A/C. Here C ⊂ A is the annihilator ideal of Σy^{−1}, which is spanned over 𝔽_{p} by all admissible monomials in A except 1 and the P^{i} for i ≥ 1.

(c) The cofiber sequence P_{0}ΣℂP_{−1}^{∞}[−1] → ΣℂP_{−1}^{∞}[−1] → Σ^{−1}Hℤ induces an identification H^{∗}(P_{0}ΣℂP_{−1}^{∞}[−1]) ≅ Σ^{−2}C/A(β).

(d) For 1 ≤ i ≤ (p−3)/2 there are isomorphisms H^{∗}(ΣℂP_{−1}^{∞}[2i−1]) ≅ 𝔽_{p}{Σy^{k} | k = 2i − 1 + m(p − 1), m ≥ 0}.

Proof. Any admissible monomial P^{I} with I = (i_{1}, …, i_{n}) and n ≥ 2 acts trivially on Σy^{−1} because z = P^{i_{n}}(Σy^{−1}) is in the image from H^{∗}(ΣℂP^{∞}), which is an unstable A-module, and then P^{i_{n−1}}(z) = 0 by instability. □

Cohomology of the smooth Whitehead spectrum.

Proposition 4.3. The A-module homomorphism

δ^{∗}: H^{∗}(Σ^{3}ko) → H^{∗}(P_{0}Σ\overline{ℂP} _{−1}^{∞}/ΣℍP^{∞})

splits as the direct sum of the injection

Σ^{q−1}A//E_{1} ⟶ Σ^{−2}C/A(β)

taking Σ^{q−1}(1) to Σ^{−2}Q_{1}, and the homomorphisms

δ_{i}^{∗}: Σ^{4i−1}A//E_{1} ⟶ H^{∗}(ΣℂP_{−1}^{∞}[2i − 1])
≅ 𝔽_{p}{Σy^{k} | k = 2i − 1 + m(p − 1), m ≥ 0}

taking Σ^{4i−1}(1) to Σy^{2i−1} for 1 ≤ i ≤ (p − 3)/2.

Proof. By (1.4) and 4.1(a) the source of δ^{∗} splits as the direct sum of the cyclic A-modules Σ^{4i−1}A//E_{1} for 1 ≤ i ≤ (p − 1)/2. Here 4i − 1 = q − 1 for i = (p − 1)/2. Hence δ^{∗} is determined as an A-module homomorphism by its value on the generators Σ^{4i−1}(1). These are all in degrees ≤ q − 1 = 2p − 3, and δ^{∗} is injective in this range by 2.11. By (2.10), 4.2(c) and (d) the target of δ^{∗} splits as the direct sum of 𝔽_{p}{Σy^{k} | k ≡ 2i − 1 + m(p − 1), m ≥ 0} for 1 ≤ i ≤ (p − 3)/2 and Σ^{−2}C/A(β). The bottom class of the latter is Σ^{−2}Q_{1}, in degree q − 1. Hence the target of δ^{∗} has rank 1 in each degree 4i − 1 for 1 ≤ i ≤ (p − 1)/2, and so (up to a unit which we suppress) δ^{∗} maps Σ^{4i−1}(1) to Σy^{2i−1} for 1 ≤ i ≤ (p − 3)/2 and Σ^{q−1}(1) to Σ^{−2}Q_{1}.

The homomorphism Σ^{q−1}A//E_{1} → Σ^{−2}C/A(β) is injective, as its continuation into Σ^{−2}A//E_{0} is induced up over E_{1} ⊂ A from the injection Σ^{q−1}𝔽_{p} → Σ^{−2}E_{1}//E_{0} taking Σ^{q−1}(1) to Σ^{−2}Q_{1}. □

Theorem 4.4. There is a splitting

H^{∗}(W h(∗)) ≅ H^{∗}(Σc) ⊕ H^{∗}(ΣℍP^{∞}) ⊕ H^{∗}(\frac{W h(∗)}{(Σc, ΣℍP^{∞})})

and an extension of A-modules

0 → cok(δ^{∗}) ⟶ H^{∗}(\frac{W h(∗)}{(Σc, ΣℍP^{∞})}) ⟶ Σ^{−1} ker(δ^{∗}) → 0

where

cok(δ^{∗}) ≅ Σ^{−2}C/A(β, Q_{1}) ⊕ ⨁_{i=1}^{(p−3)/2} H^{∗}(ΣℂP_{−1}^{∞}[a])/A(Σy^{a})

and

Σ^{−1} ker(δ^{∗}) ≅ ⨁_{i=1}^{(p−3)/2} Σ^{2a}C_{a}/A(β, Q_{1}) .

In both sums we briefly write a = 2i − 1, so a is odd with 1 ≤ a ≤ p − 4. Here H^{∗}(ΣℂP_{−1}^{∞}[a]) = 𝔽_{p}{Σy^{k} | k ≡ a mod p − 1, k ≥ a}, A(Σy^{a}) ⊂ H^{∗}(ΣℂP_{−1}^{∞}[a]) is the submodule generated by Σy^{a}, and C_{a} ⊂ A is the annihilator ideal of Σy^{a} ∈ H^{∗}(ΣℂP_{−1}^{∞}[a]).

Proof. The splitting and extension follow by applying cohomology to 2.11. The cohomologies of Σc and ΣℍP^{∞} are given in 4.1(c) and 4.2(a), respectively. The descriptions of ker(δ^{∗}) and cok(δ^{∗}) are immediate from 4.3. □

Example 4.5. (a) When p = 3 there is a splitting

H^{∗}(W h(∗)) ≅ H^{∗}(Σc) ⊕ H^{∗}(ΣℍP^{∞}) ⊕ Σ^{−2}C/A(β, Q_{1}) .

(b) When p = 5 there is an extension

0 → Σ^{−2}C/A(β, Q_{1}) ⊕ H^{∗}(ΣℂP_{−1}^{∞}[1])/A(Σy)

⟶ H^{∗}(\frac{W h(∗)}{(Σc, ΣℍP^{∞})}) ⟶ Σ^{2}C_{1}/A(β, Q_{1}) → 0

where

H^{∗}(ΣℂP_{−1}^{∞}[1])/A(Σy) ≅ 𝔽_{p}{Σy^{k} | k ≡ 1 mod p − 1, k ≥ 1; k ≠ p^{e}, e ≥ 0}

and C_{1} ⊂ A is spanned over 𝔽_{p} by all admissible monomials in A except 1 and the P^{I} for I = (p^{e}, p^{e−1}, …, p, 1) with e ≥ 0.

Remark 4.6. (a) The A-module Σ^{−2}C/A(β, Q_{1}) can be shown to split off from H^{∗}(W h(∗)/(Σc, ΣℍP^{∞})) by considering the lower cofiber sequence in 2.5.

(b) For p ≥ 5 the extension of Σ^{2}C_{1}/A(β, Q_{1}) by H^{∗}(ΣℂP_{−1}^{∞}[1])/A(Σy) is not split. By 3.10 the bottom p-torsion homotopy of W h(∗) is ℤ/p in degree 4p − 2, which implies that there is a nontrivial mod p Bockstein relating the bottom classes Σ^{2}P^{2} and Σy^{2p−1} of these two A-modules, respectively.

§5. Applications to automorphism spaces

We now recall the relation between Whitehead spectra, smooth concordance spaces and diffeomorphism groups, to allow us to formulate a geometric interpretation of our calculations.

Spaces of concordances and h-cobordisms. Let M be a compact smooth nmanifold, possibly with corners, and let I = [0, 1] be the unit interval. To study the automorphism space DIF F (M) of self-diffeomorphisms of M relative to the boundary ∂M, one is led to study the concordance space

C(M) = DIF F (M × I, M × 1)

of smooth concordances on M, also known as the pseudo-isotopy space of M [Ig]. This equals the space of self-diffeomorphisms ψ of the cylinder M × I relative to the part ∂M × I ∪ M × 0 of the boundary. Both DIF F (M) and C(M) can be viewed as topological or simplicial groups, and there is a fiber sequence

(5.1)  DIF F (M × I) ⟶ C(M) ⟶^{r} DIF F (M)

where r restricts a concordance ψ to the upper end M × 1 of the cylinder.

Let J = [0, ∞). The smooth h-cobordism space H(M) of M [Wa3, §1] is the space of smooth codimension zero submanifolds W ⊂ M × J that are h-cobordisms with M = M × 0 at one end, relative to the trivial h-cobordism ∂M × I. There is a fibration over H(M) with C(M) as fiber and the contractible space of collars on M × 0 in M × J as total space. Hence H(M) is a non-connective delooping of C(M), i.e., C(M) ≃ ΩH(M). The homotopy types of the diffeomorphism group DIF F (M), the concordance space C(M) and the h-cobordism space H(M) are of intrinsic interest in geometric topology.

There are stabilization maps σ: C(M) → C(I ×M) and σ: H(M) → H(I ×M). By Igusa’s stability theorem [Ig], the former map is at least k-connected when n ≥ max{2k + 7, 3k + 4}. Then this is also a lower bound for the connectivity of the canonical map

Σ: C(M) ⟶ 𝒞(M) = hocolim_{ℓ} C(I^{ℓ} × M)

to the mapping telescope of the stabilization map σ repeated infinitely often. We call 𝒞(M) the stable concordance space of M, and call the connectivity of Σ: C(M) → 𝒞(M) the concordance stable range of M. Likewise there is a stable h-cobordism space ℋ(M) = hocolim_{ℓ} H(I^{ℓ} × M), and 𝒞(M) ≃ Ωℋ(M). The connectivity of the map H(M) → ℋ(M) is one more than the concordance stable range of M.

The stable parametrized h-cobordism theorem. Waldhausen proved in [Wa6] that when X = M is a compact smooth manifold there is a homotopy equivalence

(5.2)  ℋ(M) ≃ ΩΩ^{∞}W h(M) ,

i.e., that the Whitehead space Ω^{∞}W h(M) of M is a delooping of the stable hcobordism space ℋ(M) of M. This stable parametrized h-cobordism theorem is the fundamental result linking algebraic K-theory of spaces to concordance theory. At the level of π_{0} it recovers the (stable) h- and s-cobordism theorems of Smale, Barden, Mazur and Stallings. Waldhausen’s theorem includes in particular the assertion that the stable h-cobordism space ℋ(M) and the stable concordance space 𝒞(M) are infinite loop spaces.

The functor X ↦ A(X) preserves connectivity of mappings, in the sense that if X → Y is a k-connected map with k ≥ 2 then A(X) → A(Y) is also k-connected [Wa1, 1.1], [BM1, 10.9]. It follows that W h(M), ℋ(M) and 𝒞(M) take k-connected maps to k-, (k − 1)- and (k − 2)-connected maps, respectively, for k ≥ 2.

Let π = π_{1}(M) be the fundamental group of X = M. The classifying map M → Bπ for the universal covering of M is k-connected for some k ≥ 2, so also A(M) → A(Bπ) is k-connected. Let R = ℤ[π]. Then the linearization map L: A(Bπ) → K(R) is a rational equivalence by [Wa1, 2.2]. Hence rational information about K(R) gives rational information about A(M) up to degree k, and about 𝒞(M) up to degree k − 2, which in turn agrees with C(M) in the concordance stable range. For example, Farrell and Hsiang [FH] show that π_{m}C(D^{n}) ⊗ ℚ has rank 1 in all degrees m ≡ 3 mod 4, and rank 0 in other degrees, for n sufficiently large with respect to m.

For π a finite group, A(X) and W h(X) are of finite type by theorems of Dwyer [Dw] and Betley [Be], so the integral homotopy type is determined by the rational homotopy type and the p-adic homotopy type for all primes p. Therefore our results on the p-adic homotopy type of W h(∗) have following application:

Theorem 5.3. (a) Suppose p ≥ 5 and let M be a (4p − 2)-connected compact smooth manifold whose concordance stable range exceeds (4p − 4), e.g., an nmanifold with n ≥ 12p − 5. Then the first p-torsion in the homotopy of the smooth concordance space C(M), and in the homotopy of the smooth h-cobordism space H(M), is

π_{4p−4}C(M)_{(p)} ≅ π_{4p−3}H(M)_{(p)} ≅ ℤ/p .

(b) Suppose p = 3 and let M be an 11-connected compact smooth manifold whose concordance stable range exceeds 9, e.g., an n-manifold with n ≥ 34. Then the first 3-torsion in the homotopy of the smooth concordance space C(M), and in the homotopy of the smooth h-cobordism space H(M), is

π_{9}C(M)_{(3)} ≅ π_{10}H(M)_{(3)} ≅ ℤ/3 .

Proof. The first p-torsion in π_{∗}W h(∗) is ℤ/p in degree ∗ = 4p − 2 for p ≥ 5, and ℤ/3{Σβ_{1}} in degree ∗ = 11 for p = 3, and π_{∗}W h(∗) is finite in all of these degrees. When M is (4p−2)-connected, resp. 11-connected, the map π_{∗}W h(M) → π_{∗}W h(∗) is an isomorphism in this degree. And π_{∗−2}𝒞(M) ≅ π_{∗−1}ℋ(M) ≅ π_{∗}W h(M). So if the concordance stable range is at least (4p − 3), resp. 10, also π_{∗−2}C(M) ≅ π_{∗−2}𝒞(M) and π_{∗−1}H(M) ≅ π_{∗−1}ℋ(M) in this degree. □

Similar statements may of course be given for when the subsequent torsion groups in π_{∗}W h(∗) agree with π_{∗−2}C(M) and π_{∗−1}H(M), under stronger connectivity and dimension hypotheses.

Theorem 5.4. (a) Suppose p ≥ 5 and let M = D^{n} with n ≥ 12p − 5. Then π_{4p−4}DIF F (D^{n+1}) or π_{4p−4}DIF F (D^{n}) contains an element of order p.

(b) Suppose p = 3 and let M = D^{n} with n ≥ 34. Then π_{9}DIF F (D^{n+1}) or π_{9}DIF F (D^{n}) contains an element of order 3.

Proof. Consider the exact sequence in homotopy induced by (5.1), with D^{n} × I ≅ D^{n+1}. A ℤ/p in π_{m}C(D^{n}) either comes from π_{m}DIF F (D^{n+1}), which known to be finite in these cases, or maps to π_{m}DIF F (D^{n}). □

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Department of Mathematics, University of Oslo, Norway

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