THE FIBER OF THE LINEARIZATION MAP A(∗) → K(ℤ)

John R. Klein and John Rognes

Introduction

Waldhausen’s algebraic K-theory of spaces is a homotopy functor X ↦ A(X) from spaces to infinite loop spaces. It is related to diffeomorphisms of smooth manifolds through stable smooth pseudoisotopies, or concordances. Recall that when Y is a smooth manifold its smooth pseudoisotopy space P (Y) is the space of self-diffeomorphisms of Y × I relative to Y × 0 ∪ ∂Y × I. The stable smooth pseudoisotopy space is defined as P(Y) = hocolim_{N} P (Y × I^{N}), where the maps in the colimit are certain suspension maps. The smooth Whitehead space functor X ↦ Wh^{Diff}(X) is a homotopy functor defined in [33], satisfying Ω^{2} Wh^{Diff}(Y) ≃ P(Y) when Y is a smooth manifold.

Let Q(X_{+}) = hocolim_{N} Ω^{N}Σ^{N}(X_{+}) represent the unreduced stable homotopy of X. Then A(X) satisfies the following theorem, proved in [34]:

Theorem 0.1 (Waldhausen). A(X) ≃ Q(X_{+}) × Wh^{Diff}(X).

In the particular case X = ∗ this asserts that A(∗) ≃ Q(S^{0}) × Wh^{Diff}(∗), with

Ω^{2} Wh^{Diff}(∗) ≃ P(∗) ≃ hocolim_{N} Diff(D^{N+1} rel D^{N}).

Here D^{N} ⊂ S^{N} can be viewed as a hemispherical disc on the boundary of D^{N+1}. Furthermore there is a fibration sequence

Diff(D^{N+1} rel ∂D^{N+1}) ⟶ Diff(D^{N+1} rel D^{N}) ⟶ Diff(D^{N} rel ∂D^{N})

(when restricted to the path components in the image of the second map). These spaces clearly have geometric interest. On the level of homotopy groups we find

π_{i}A(∗) ≅ π_{i}Q(S^{0}) ⊕ π_{i−2}P(∗)
π_{i−2}P(∗) ≅ colim_{N} π_{i−2} Diff(D^{N+1} rel D^{N}).

By Igusa’s stability theorem [20], the colimit is actually achieved for some finite N ≫ i. Hence homotopy in P(∗) reappears as homotopy in Diff(D^{N} rel ∂D^{N}) for suitable (large) N.

Quillen’s higher algebraic K-theory of a ring R is represented by the K-theory space K(R) = K_{0}(R) × BGL(R)^{+}, where GL(R) = colim_{k} GL_{k}(R) and the superscript + on BGL(R) denotes Quillen’s plus construction, which abelianizes the fundamental group and leaves homology unchanged [4]. The higher algebraic Kgroups are K_{i}(R) = π_{i}K(R), and e.g. in the case when R = O_{E} is the ring of

Typeset by AMS-TEX

integers in a number field E they encode number theoretic information about R. The rational K-groups K_{i}(O_{E}) ⊗ ℚ were computed by Borel in [7]. In the case E = ℚ, R = ℤ, the result is

K_{i}(ℤ) ⊗ ℚ ≅ \{ \begin{matrix}ℚ & i = 0, or i ≡ 1 mod 4 and i ≥ 5, \\ 0 & otherwise. \end{matrix}

There is a linearization map L: A(∗) → K(ℤ) arising from thinking of A(∗) as the K-theory of the sphere spectrum viewed as a “ring up to homotopy” [32]. One definition of A(∗) which makes it clear how to define this map goes as follows. Let Hteq(⋁_{k} S^{n}) be the (topological) monoid of based self-homotopy equivalences of ⋁_{k} S^{n} = S^{n} ∨ ⋯ ∨ S^{n} (k summands). There is a map Hteq(⋁_{k} S^{n}) → GL_{k}(ℤ) taking a homotopy equivalence f: ⋁_{k} S^{n} → ⋁_{k} S^{n} to its induced isomorphism H̃_{n}(f): ℤ^{k} → ℤ^{k} on nth reduced homology. This map takes a point to its path component. There are stabilization maps increasing k and n, given respectively by wedge sum with additional sphere summands, and by suspension. Hence we may form hocolim_{n,k} B Hteq(⋁_{k} S^{n}), with fundamental group GL(ℤ), and define

A(∗) = ℤ × hocolim_{n,k} B Hteq(⋁_{k} S^{n})^{+}.

The linearization map L: A(∗) → K(ℤ) is then induced by naturality of the plus construction with respect to the map to path components. It is a rational equivalence, by Proposition 2.2 of [32]. Hence

π_{i}P(∗) ⊗ ℚ ≅ \{ \begin{matrix}ℚ & for i ≡ 3 mod 4, \\ 0 & otherwise. \end{matrix}

In fact π_{i} Diff(D^{N}, ∂D^{N}) ⊗ ℚ ≅ ℚ if i ≡ 3 mod 4 and N ≫ i is odd, and zero otherwise; compare [18].

The issue of torsion remains. It appears to be extremely difficult to explicitly determine the torsion groups in π_{i}A(∗) and K_{i}(ℤ). However, we may also pose the relative question of by how much these groups differ, and this is the subject of the present paper.

Let hofib(L) be the homotopy fiber of L, fitting into a fiber sequence

hofib(L) ⟶ A(∗) ⟶^{L} K(ℤ).

Then hofib(L) is rationally trivial, and since the homotopy groups of A(∗) and K(ℤ) are finitely generated [28, 16], its homotopy groups are finite in each degree. Thus no information is lost by localizing or completing hofib(L) at one prime p at a time. The first nonvanishing homotopy group of hofib(L) completed at p was determined by Waldhausen, e.g. in Corollary 3.7 of [33]. We recall the result below.

Let Q_{0}(S^{0}) and Q_{1}(S^{0}) be the base point and identity components of Q(S^{0}). We identify Q_{1}(S^{0}) with the monoid SG of stable orientation-preserving self-homotopy equivalences of spheres, cf. Definition 4.1. There is then an inclusion BSG → hocolim_{n,k} B Hteq(⋁_{k} S^{n}) corresponding to the inclusion of special 1 × 1-matrices BSL_{1}(R) → BGL(R), which induces a map BSG → A(∗). The self maps in SG induce the identity map on homology, so the composite

BSG ⟶ A(∗) ⟶^{L} K(ℤ)

is constant, and induces a map w : BSG → hofib(L).

Theorem 0.2 (Waldhausen). After completion at p the homotopy fiber hofib(L) is (2p−3)-connected, and the natural map w: BSG → hofib(L) induces an isomorphism

ℤ/p ≅ π_{2p−2}BSG ⟶ π_{2p−2} hofib(L).

In fact w is (4p − 3)-connected for odd p, by our Corollary 3.6 and Theorem 5.4 below. For a sketch why w is at least (4p − 6)-connected see [13].

We extend this result at odd primes p to a complete calculation of π_{∗} hofib(L) in the initial range of degrees where the (complex) j-map U → Q(S^{0}) induces isomorphisms on homotopy, i.e. for ∗ < deg(β_{1}) = 2p(p − 1) − 2. As a corollary we obtain torsion classes in π_{i−2}P(∗) coming from π_{i}A(∗) and mapping to zero in K_{i}(ℤ), which represent new families of diffeomorphisms of discs as described above. These families were not directly detectable by the linear K-theoretic invariants. See Theorem 0.3 and Corollary 0.4 below for precise statements.

Furthermore, in the range of degrees where the S^{1}-transfer map Q(Σ_{+}ℂP^{∞}) → Q(S^{0}) is surjective, i.e. for ∗ < deg(β_{p+1}) = 2p(p + 2)(p − 1) − 2, we reduce the analysis of π_{∗} hofib(L) essentially to the calculation of the stable homotopy groups of ℂP^{∞}. These results are summarized in Theorem 5.4, and we refer the reader to that theorem for the complete results.

Let p be an odd prime, and suppose all spaces and groups are completed at p.

Theorem 0.3. (i) When ∗ < deg(β_{1}) = 2(p − 1)p − 2, the homotopy fiber of the linearization map L: A(∗) → K(ℤ) has homotopy groups

π_{∗} hofib(L) ≅ \{ \begin{matrix}ℤ/p & if ∗ = 2n where m(p − 1) ≤ n < mp for some 1 ≤ m < p, \\ 0 & otherwise. \end{matrix}

(ii) The classes in degrees 2n with n ≡ 0 mod p − 1 are in the image of the natural map w: BSG → hofib(L), and map to zero in π_{2n}A(∗).

(iii) The remaining classes, in degrees 2n with m(p − 1) < n < mp for some 1 < m < p, inject onto direct summands in π_{2n}A(∗).

Thus π_{∗} hofib(L) ≅ ℤ/p precisely in the even degrees

2p − 2, 4p − 4, 4p − 2, 6p − 6, 6p − 4, 6p − 2, …, 2(p − 1)^{2}, …, 2(p − 1)p − 4

and is trivial otherwise, as long as ∗ < 2(p − 1)p − 2.

Corollary 0.4. (i) When n = m(p − 1) and 1 ≤ m < p there are classes of infinite order in K_{2n+1}(ℤ) mapping to classes of order p in π_{2n} hofib(L) under the connecting map ΩK(ℤ) → hofib(L).

(ii) There are torsion classes of order p in π_{2n−2}P(∗) when n satisfies m(p−1) < n < mp for some 1 < m < p, which come from π_{2n}A(∗) and map to zero in K_{2n}(ℤ). These classes are detected in π_{∗}T C(∗) (which will be reviewed momentarily) as the torsion classes of π_{∗}Q(ΣℂP^{∞}).

The theorem and corollary follow by specialization from Theorem 5.4, and are proved in Section 5. Again we emphasize that Theorem 5.4 below gives further information along these lines in the range of degrees from deg(β_{1}) to deg(β_{p+1}).

The method of proof uses the topological cyclic homology functor of Bökstedt, Hsiang and Madsen [12], and the associated cyclotomic trace map from K-theory. These are defined for suitable rings up to homotopy, namely those arising from “functors with smash product”, or FSPs, which were introduced in [11]. (See [12] for a published definition.) For such an FSP R one can define its K-theory K(R), its topological cyclic homology T C(R), and the cyclotomic trace map trc: K(R) → T C(R). These constructions are natural in R, and include the cases discussed above as special cases. Namely, the sphere spectrum is the ring up to homotopy associated with the identity FSP, with K-theory equal to A(∗) and topological cyclic homology denoted T C(∗). The integers are also the ring up to homotopy associated with an FSP, whose K-theory is K(ℤ) and whose topological cyclic homology is denoted T C(ℤ). The linearization map is induced by a map of FSPs, representing the path component map Q(S^{0}) → π_{0}Q(S^{0}) ≅ ℤ on underlying ring spaces. Hence there is a commutative square of infinite loop spaces and maps

(0.5)  [figure: A(∗); trc; L; T C(∗); L; K(ℤ); trc; T C(ℤ).]

We can access hofib(L: A(∗) → K(ℤ)) by means of the following recent theorem due to Dundas [15]:

Theorem 0.6 (Dundas). The square above is homotopy cartesian after completion at any prime p. Hence there is a homotopy equivalence

hofib(L: A(∗) → K(ℤ)) ⟶ hofib(L: T C(∗) → T C(ℤ))

after p-adic completion.

This allows us to reduce the problem posed to the study of T C(∗), T C(ℤ) and the linearization map between them. And, unlike the largely unknown spaces A(∗) and K(ℤ), the homotopy types of these spaces are known, by Theorem 5.17 of [12] in the case of T C(∗), and by Theorem 9.17 of Bökstedt and Madsen [13] in the case of T C(ℤ) at odd primes p. Hence only the identification of the linearization map in topological cyclic homology remains. This is achieved in a range of degrees by our Theorems 0.3 and 5.4.

In general Dundas’ theorem holds for the square diagram induced by a map R_{1} → R_{2} of FSPs inducing a nilpotent extension on π_{0}, i.e. when π_{0}R_{1} → π_{0}R_{2} is a surjective ring homomorphism with nilpotent kernel. This is a generalization to FSPs of a similar theorem due to McCarthy [25], which holds for maps of simplicial rings. Both of these theorems may be viewed as p-adic versions of a rational theorem due to Goodwillie [19], asserting that in the case of simplicial rings, if topological cyclic homology is replaced by negative cyclic homology, then the corresponding square is rationally homotopy cartesian.

The paper is organized as follows. Section 1 fixes notation and recalls the image of J and cokernel of J spaces. Section 2 reviews the calculations of T C(∗) and T C(ℤ). Proposition 2.10 gives a first product splitting of hofib(L). The difficult piece is related to the homotopy fiber of the S^{1}-transfer map Q(Σ_{+}ℂP^{∞}) → Q(S^{0}). Section 3 reviews Segal’s splitting of the stable homotopy of ℂP^{∞} as BU and a space F with finite homotopy groups in each degree. Corollary 3.6 gives the homotopy groups of F in a beginning range of degrees. Section 4 collects previous geometric results on the j-maps and the S^{1}-transfer, summarized in diagram (4.3). This is applied in Corollary 4.16 to give, essentially, a splitting of U off from the homotopy fiber of the S^{1}-transfer map, and a description of the remainder term. In Section 5 we assemble the available information to prove our main Theorem 5.4, from which Theorem 0.3 and Corollary 0.4 above follow. A final Section 6 contains some speculative remarks on the space level structure of hofib(L).

Acknowledgements. The authors would like to thank Lars Hesselholt and Ib Madsen for independently suggesting this problem. We also thank Karl-Heinz Knapp for illuminating discussions about the S^{1}-transfer and the J-homomorphism.

1. Stable homotopy of spheres and the image of J

We begin by presenting our conventions.

Let p be an odd prime. All groups, spaces and spectra will hereafter be implicitly completed at p. After this section we will only work with infinite loop spaces, thought of as connective spectra, and hence we will write Q(S^{0}) for (the underlying space of) the sphere spectrum, etc. In particular T C(R) denotes the underlying space of the topological cyclic homology spectrum of an FSP R. By the infinite loop space cofiber of a map f: X → Y of infinite loop spaces we mean the infinite loop space Z fitting into a fiber sequence X ⟶^{f} Y → Z, so that ΩZ ≃ hofib(f). Similarly, by the infinite loop space smash product X ∧ Y we mean the underlying space of the spectrum level smash product of the spectra representing X and Y.

Let k be a topological generator for the p-adic units ℤ_{p}^{×}, i.e. a generator of (ℤ/p^{2})^{×} ≅ ℤ/(p − 1)p. Then the image of J space ImJ, is defined by the fiber sequence

(1.1)  ImJ ⟶ ℤ × BU ⟶^{ψ^{k}−1} BU.

If k is chosen as a prime power then ImJ ≃ K(𝔽_{k}) by [27]. The Adams operation ψ^{k} acts as multiplication by k^{i} in degree 2i, so

(1.2)  π_{∗} ImJ ≅ \{ \begin{matrix}ℤ & for ∗ = 0, \\ ℤ/(k^{i} − 1) & for ∗ = 2i − 1 > 0, \\ 0 & otherwise. \end{matrix}

Here the p-adic valuations are

(1.3)  v_{p}(k^{i} − 1) = \{ \begin{matrix}1 + v_{p}(i) & for i ≡ 0 mod p − 1, \\ 0 & otherwise. \end{matrix}

The K-theory unit Q(S^{0}) → ℤ×BU lifts through an infinite loop map e: Q(S^{0}) → ImJ, which we call the Adams e-invariant. For any element x ∈ π_{∗}Q(S^{0}) the image e(x) ∈ π_{∗} ImJ has the same order as the complex Adams e-invariant e_{ℂ}(x) ∈ ℚ/ℤ, by [29]. We can identify ImJ with the connective cover L_{K}Q(S^{0})[0, ∞) of the Bousfield K-localization of the sphere spectrum [9], and then e corresponds to the connective cover of the localization map

(1.4)  e: Q(S^{0}) → L_{K}Q(S^{0})[0, ∞) ≃ ImJ.

Let the cokernel of J space CokJ, be defined by the fiber sequence

(1.5)  CokJ ⟶ Q(S^{0}) ⟶^{e} ImJ.

By the solution to the Adams conjecture [1], e admits a section α: ImJ → Q(S^{0}) up to homotopy, so

(1.6)  π_{∗}Q(S^{0}) ≅ π_{∗} ImJ ⊕ π_{∗} CokJ.

There is a j-map U → Q(S^{0}) taking a unitary isometry A: ℂ^{n} ⟶^{≅} ℂ^{n} to the induced map of one-point compactifications j(A): S^{2n} ⟶^{≃} S^{2n}, inducing the Jhomomorphism

π_{∗}(j): π_{∗}U ⟶ π_{∗}Q(S^{0}).

For p odd Im(π_{∗}(j)) = π_{∗} ImJ ⊂ π_{∗}Q(S^{0}) under the splitting above; hence the terminology.

Let J_{⊕} and J_{⊗} be the zero- and one-components of ImJ, with respectively the additive and multiplicative infinite loop space structures. Then there are infinite loop space fiber sequences

(1.7)  \begin{matrix}C_{⊕} ⟶ Q_{0}(S^{0}) & ⟶^{e_{⊕}} J_{⊕} \\ C_{⊗} ⟶ SG & ⟶^{e_{⊗}} J_{⊗}. \end{matrix}

Here C_{⊕} and C_{⊗} are both homotopy equivalent to CokJ as spaces.

The first elements in π_{∗} CokJ are generated by the β-family. These are classes β_{i} ∈ π_{∗} CokJ ⊂ π_{∗}Q(S^{0}) for i ≥ 1, with deg(β_{1}) = 2(p − 1)p − 2 and deg(β_{i}) = deg(β_{1}) + 2(i − 1)(p^{2} − 1). The class β_{1} is the first nontrivial class in π_{∗} CokJ, so e: Q(S^{0}) → ImJ induces an isomorphism on homotopy groups for ∗ < deg(β_{1}). We shall also have reason to consider β_{p+1} in degree deg(β_{p+1}) = 2(p − 1)p(p + 2) − 2, since by a theorem of Knapp (see Theorem 4.14 and the remarks following that theorem) the S^{1}-transfer map Q(Σ_{+}ℂP^{∞}) → Q_{0}(S^{0}) induces a surjection on homotopy groups up to this degree.

2. Review of Topological Cyclic Homology

The topological cyclic homology of the identity FSP, or conceptually of the sphere spectrum, was determined in [12]. Likewise the topological cyclic homology of the integers, completed at an odd prime p, was determined in [13]. We now review these results.

Theorem 2.1 (Bökstedt–Hsiang–Madsen). There is a homotopy cartesian square of infinite loop spaces and maps

[figure: T C(∗); Q(Σ_{+}ℂP^{∞}); _{trf_{S^{1}}}; Q(S^{0}); ≃∗; Q_{0}(S^{0})]

after p-adic completion.

Here Σ_{+}X = Σ(X_{+}) is the unreduced suspension of X, and the notation ≃ ∗ denotes a null-homotopic map.

Remark 2.2. The image of the S^{1}-transfer map trf_{S^{1}} is connected and thus contained in the zero-component Q_{0}(S^{0}) of Q(S^{0}). In Bökstedt, Hsiang and Madsen’s non-connective spectrum-level formulation of the theorem the lower right hand space is Q(S^{0}), and π_{0}Q(S^{0}) ≅ ℤ maps isomorphically by a connecting map to the single nontrivial negative homotopy group T C_{−1}(∗) ≅ ℤ.

Hence T C(∗) ≃ Q(S^{0}) × hofib(trf_{S^{1}}). There is a cofiber sequence

ΣℂP^{∞} ⟶ Σ_{+}ℂP^{∞} → Σ_{+}∗ = S^{1}

split by a choice of a point in ℂP^{∞}, e.g. the point ℂP^{0}. Hence we have a product splitting Q(Σ_{+}ℂP^{∞}) ≃ Q(ΣℂP^{∞}) × Q(S^{1}).

Definition 2.3. Let t: Q(ΣℂP^{∞}) → Q_{0}(S^{0}) be the restricted transfer map given as the composite

Q(ΣℂP^{∞}) ⟶ Q(Σ_{+}ℂP^{∞}) ⟶^{trf_{S^{1}}} Q_{0}(S^{0}).

Then trf_{S^{1}} splits as the sum of t and the Hopf map η: Q(S^{1}) → Q_{0}(S^{0}) in terms of the splitting above. (To see this, use a version of diagram (4.3) below restricted to ℂP^{0} ⊂ ℂP^{∞} to see that trf_{S^{1}} restricted over S^{1} ⊂ Q(S^{1}) ⊂ Q(Σ_{+}ℂP^{∞}) is homotopic to the complex j-map S^{1} ≅ U(1) → Ω^{2}S^{2} → Q(S^{0}). The latter is well known to represent η ∈ π_{1}Q(S^{0}).)

The Hopf map is null-homotopic when completed at an odd prime, and so we obtain a further splitting hofib(trf_{S^{1}}) ≃ hofib(t) × Q(S^{1}). Hence

(2.4)  T C(∗) ≃ Q(S^{0}) × Q(S^{1}) × hofib(t).

At odd primes p, T C(ℤ) is shown in [13] to be the connective cover of its Klocalization (modulo a slight correction in degrees zero and one). Thus we have a factorization of the inclusion of the first two terms of the splitting above, through the connective covers of their K-localizations. Hence we have a commutative diagram

(2.5)  [figure: Q(S^{0}) × Q(S^{1}); e×Be; T C(∗); L; ImJ × B ImJ; T C(ℤ).]

The mod p calculations in [13] suffice to show that the infinite loop space cofiber of the bottom map is SU, and that the resulting fiber sequence is split, as explained in [30]. This gives the following theorem:

Theorem 2.6 (Bökstedt–Madsen). There is a product splitting of infinite loop spaces

T C(ℤ) ≃ ImJ × B ImJ × SU

for each odd prime p.

To complete the picture, we review from [10] that there is a natural map

Φ: K(ℤ) → JK(ℤ) ≃ ImJ × BBO

at odd primes p. Bökstedt has noted that according to the Lichtenbaum–Quillen conjecture for K(ℤ) [17, 26] the homotopy fiber of Φ has the homotopy groups of Ω hofib(ρ^{k}). Here ρ^{k}: BSpin → BSpin_{⊗} is the Bott cannibalistic class [2, 24], which is an equivalence for primes p which are regular in the sense of number theory. Writing this extension as a twisted product ( ˜×) and the conjectured equivalence as ∼, we can rewrite the homotopy cartesian square (0.5) as follows:

(2.7) [figure: Q(S^{0}) × Wh^{Diff}(∗) ≃ A(∗); trc; L; T C(∗) ≃ Q(S^{0}) × Q(S^{1}) × hofib(t); L; ImJ × BBO ˜× Ω hofib(ρ^{k}) ∼ K(ℤ); trc; T C(ℤ) ≃ ImJ × B ImJ × SU]

We now turn to analyzing the right hand vertical map.

Definition 2.8. Let ℓ: hofib(t) → SU be the infinite loop map induced on infinite loop space cofibers by e × Be and L in (2.5) above.

Then we have a diagram of infinite loop space fiber sequences:

(2.9)  [figure: Ω ImJ × ImJ; Ω hofib(ℓ); a; CokJ × B CokJ; hofib(L); hofib(ℓ); Ω hofib(t); ≃∗; Q(S^{0}) × Q(S^{1}); e×Be; T C(∗); L; hofib(t); ℓ; ImJ × B ImJ; T C(ℤ); SU]

Proposition 2.10. The map a above is null-homotopic as a map of spaces. Hence there is a space level splitting

Ω hofib(L) ≃ Ω CokJ × CokJ × Ω hofib(ℓ)

and

π_{∗} hofib(L) ≅ π_{∗} CokJ ⊕ π_{∗}B CokJ ⊕ π_{∗} hofib(ℓ).

Presumably this splitting can be improved to an infinite loop space splitting by means of the multiplicative sequence in (1.7).

Proof. The map a factors over Ω ImJ×ImJ because the composite to Q(S^{0})×Q(S^{1}) factors through the null-homotopic map Ω hofib(t) → Q(S^{0}) × Q(S^{1}). By the solution to the Adams conjecture there is a space level homotopy section α: ImJ → Q(S^{0}) to the map e in the fiber sequence below:

CokJ ⟶ Q(S^{0}) ⟶^{e} ImJ ⟶^{≃∗} B CokJ.

Hence the right hand connecting map is null-homotopic as a map of spaces. Thus a factors through a product of null-homotopic maps, and must be null-homotopic on the space level. The lemma follows. □

It remains to identify t: Q(ΣℂP^{∞}) → Q_{0}(S^{0}) and ℓ: hofib(t) → SU on the homotopy group level. In the next section we commence this project by recalling a splitting of the source of t.

3. The Segal splitting

Consider the natural map

(3.1)  ℂP_{+}^{∞} = BU(1)_{+} ⟶ ∐_{n≥0} BU(n) ⟶ ℤ × BU

induced by viewing line bundles as virtual vector bundles. There is an infinite loop space extension of (3.1) using the additive infinite loop space structure on ℤ × BU, which we denote as

(3.2)  ϵ: Q(ℂP_{+}^{∞}) ⟶ ℤ × BU.

Let ϵ′ be the infinite loop map (unique up to homotopy) making the diagram below homotopy commute:

[figure: Q(ℂP^{∞}); ϵ′; BU; Q(ℂP_{+}^{∞}); ϵ; ℤ × BU]

Definition 3.3. Let F by the infinite loop space defined as the homotopy fiber of the map ϵ′: Q(ℂP^{∞}) → BU.

Theorem 3.4 (Segal). The map ϵ′: Q(ℂP^{∞}) → BU is a rational equivalence and a retraction up to homotopy. Hence there is a space level splitting

Q(ℂP^{∞}) ≃ F × BU

using the H-space structure on Q(ℂP^{∞}). In each degree π_{∗}F is thus identified with the torsion subgroup of π_{∗}Q(ℂP^{∞}), which is finite.

Segal gave a proof using representation theory in [31], and Becker gave a proof using the Becker–Gottlieb transfer in [5]. Theorem 4.2 below, due to Crabb and Knapp, deloops this result once, while also giving an explicit right homotopy inverse.

The natural map e: Q(S^{0}) → ImJ induces a map Q(ℂP^{∞}) → ImJ ∧ ℂP^{∞} (infinite loop space smash product) which induces an isomorphism up to degree deg(β_{1}) + 2 = 2(p − 1)p on homotopy groups. The order of the homotopy groups of the target are known, e.g. by Theorem 1 of [21].

Theorem 3.5 (Knapp). The group π_{∗}(ImJ ∧ ℂP^{∞}) = (ImJ)_{∗}(ℂP^{∞}) is infinite cyclic in positive even degrees, and finite in odd degrees. Up to a p-adic unit its order is

#(ImJ)_{2n−1}(ℂP^{∞}) = \frac{1}{n!} ∏_{i=1}^{n−1}(k^{i} − 1),

which has p-adic valuation

v_{p}(#(ImJ)_{2n−1}(ℂP^{∞})) = − ∑_{j=1}^{[log_{p}(n)]}[\frac{n}{p^{j}}] + ∑_{j=1}^{[ \frac{n−1}{p−1} ]}(1 + v_{p}(j)).

Here k denotes a topological generator of the p-adic units, as in Section 1. We sketch the argument, to give some indication of where the torsion classes appearing in Theorem 0.3 and Corollary 0.4 come from.

Sketch of proof. Let ku_{∗}(X) and bu_{∗}(X) denote the connective and connected reduced K-homology groups of X. These are the homology theories represented by ℤ × BU and BU. The cofiber sequence ℂP^{n−1} → ℂP^{n} → S^{2n} and the operation ψ^{k} − 1: ℤ × BU → BU gives rise to the map of short exact sequences

[figure: 0; ku_{2n}(ℂP^{n−1}); ψ^{k}−1; ku_{2n}(ℂP^{n}); ψ^{k}−1; ku_{2n}(S^{2n}); ψ^{k}−1; 0; 0; bu_{2n}(ℂP^{n−1}); bu_{2n}(ℂP^{n}); bu_{2n}(S^{2n}); 0.]

Abstractly the upper row is ℤ^{n−1} → ℤ^{n} → ℤ, and the lower row is ℤ^{n−1} → ℤ^{n−1} → 0. The vertical kernels and cokernels give the following exact sequence of ImJ-homology groups

0 ⟶ (ImJ)_{2n}(ℂP^{n−1}) ⟶ (ImJ)_{2n}(ℂP^{n}) ⟶ (ImJ)_{2n}(S^{2n}) ⟶

⟶ (ImJ)_{2n−1}(ℂP^{n−1}) ⟶ (ImJ)_{2n−1}(ℂP^{n}) ⟶ (ImJ)_{2n−1}(S^{2n}) ⟶ 0.

Let x = H −1 ∈ K̃^{0}(ℂP^{∞}) where H is the Hopf line bundle, and let u ∈ K^{−2}(∗) be the Bott periodicity element. Then K^{0}(ℂP^{∞}) ≅ ℤ[[x]] with ψ^{k}(x) = (x + 1)^{k} − 1 and ψ^{k}(u) = ku. Let b_{i} ∈ K_{0}(ℂP^{∞}) be dual to x^{i}, so ⟨x^{i}, b_{j}⟩ = δ_{ij} (Kronecker delta). Then K̃^{2n}(ℂP^{n−1}) ≅ ℤ^{n−1} with basis {u^{−n}x, …, u^{−n}x^{n−1}}, and dually ku_{2n}(ℂP^{n−1}) ≅ bu_{2n}(ℂP^{n−1}) ≅ ℤ^{n−1} with basis {u^{n}b_{1}, …, u^{n}b_{n−1}}. In Kcohomology (ψ^{k} − 1)(u^{−n}x^{i}) = (k^{i−n} − 1)u^{−n}x^{i} plus terms involving higher powers of x. Thus ψ^{k} − 1 is represented by a lower triangular matrix, with diagonal entries k^{i−n} −1 for i = 1, …, n−1. Dually, ψ^{k} −1 in homology is represented by the transposed matrix, with determinant k^{−\binom{n}{2}} ∏_{i=1}^{n−1}(k^{i} − 1). Hence (ImJ)_{2n}(ℂP^{n−1}) = 0 and (ImJ)_{2n−1}(ℂP^{n−1}) has order ∏_{i=1}^{n−1}(k^{i} − 1) up to a p-adic unit.

Clearly (ImJ)_{2n−1}(S^{2n}) = 0 and (ImJ)_{2n}(S^{2n}) ≅ ℤ, so by a diagram chase (ImJ)_{2n}(ℂP^{n}) ≅ (ImJ)_{2n}(ℂP^{∞}) ≅ ℤ. We claim that the pinch map ℂP^{n} → S^{2n} induces multiplication by n! on (ImJ)_{2n}. The fundamental class of ℂP^{1} ⊂ ℂP^{∞} represents the generator [ℂP^{1}] ∈ (ImJ)_{2}(ℂP^{∞}), and maps to ub_{1} in ku_{2}(ℂP^{∞}). Using the Pontrjagin product induced by the H-space structure on ℂP^{∞}, we can form

[ℂP^{1}]^{n} ∈ (ImJ)_{2n}(ℂP ∞) ≅ (ImJ)_{2n}(ℂP^{n}),

mapping to (ub_{1})^{n} ∈ ku_{2n}(ℂP^{∞}). By a calculation (involving Stirling numbers) which we omit, (ub_{1})^{n} = u^{n}b_{1} + ⋯ + n!u^{n}b_{n} in the given basis. Thus [ℂP^{1}]^{n} is not divisible, since the u^{n}b_{1}-coefficient is a unit, and must be a generator of (ImJ)_{2n}(ℂP^{n}). The pinch map takes (ub_{1})^{n} to n!u^{n}b_{n}, which is n! times the generator of (ImJ)_{2n}(S^{2n}). The claim follows.

Hence (ImJ)_{2n−1}(ℂP^{n}) ≅ (ImJ)_{2n−1}(ℂP^{∞}) has order 1/n! · ∏_{i=1}^{n−1}(k^{i} − 1) up to a p-adic unit, as claimed. □

Corollary 3.6. In degrees ∗ < deg(β_{1}) + 2 = 2(p − 1)p the torsion group π_{∗}F is cyclic of order p when ∗ satisfies ∗ = 2n − 1 with m(p − 1) < n < mp for some 1 < m < p, and trivial otherwise.

Proof. When n ≤ (p − 1)p we have (n − 1)/(p − 1) < p and log_{p}(n) < 2, so π_{2n−1}Q(ℂP^{∞}) ≅ π_{2n−1}F has finite order with p-adic valuation [(n − 1)/(p − 1)] − [n/p]. This difference equals 1 if n/p < m ≤ (n − 1)/(p − 1) for some integer m, and 0 otherwise, in this range. The corollary is a simple reformulation of this fact. □

Remark 3.7. Using mod p methods, a closed formula for the number of cyclic summands in (ImJ)_{∗}(ℂP^{∞}) can be determined. This is, however, not sufficient to determine the structure of these groups in higher degrees. And beginning in degree deg(β_{1})+2 the cokernel of J enters into the torsion in π_{∗}Q(ℂP^{∞}) and thus in π_{∗}F. Thus it is rather complicated to describe π_{∗}F in any higher degrees. We will not proceed any further in this direction.

4. The S^{1}-transfer and j-maps

We now use classical results on the S^{1}-transfer and the J-homomorphism to split off a copy of SU from hofib(t), at least on the level of homotopy groups, in a range. This amounts to fibering a delooped Segal splitting Q(ΣℂP^{∞}) ≃ BF × SU over the restricted transfer map t.

Definitions 4.1. Let SG be the monoid of stable unbased orientation-preserving self-homotopy equivalences of spheres, under composition. The natural inclusion Q_{1}(S^{0}) → SG is a homotopy equivalence.

Let SG(S^{1}) be the monoid of stable orientation preserving S^{1}-equivariant selfhomotopy equivalences of free S^{1}-spheres, under composition. SG(S^{1}) is called the Becker–Schultz space, and it is proved in [6] that there is a homotopy equivalence SG(S^{1}) ≃ Q(Σ_{+}ℂP^{∞}).

An explicit such homotopy equivalence is constructed in Chapters 2, 3 and 6 of Crabb’s book [14], as a difference class of relative Euler classes. This is a homotopy equivalence ζ_{S^{1}} : SG(S^{1}) → Q(Σ_{+}ℂP^{∞}). There is a similar homotopy equivalence in the non-equivariant case ζ: SG → Q_{0}(S^{0}), which is homotopic to the component shift map

SG ≃ Q_{1}(S^{0}) ⟶^{[−1]} Q_{0}(S^{0})

using the additive H-space structure on Q(S^{0}), by Lemma 3.11 of loc. cit.

There is a forgetful monoid map i^{∗}: SG(S^{1}) → SG induced by restricting the S^{1}-actions to the trivial group. It is compatible under ζ_{S^{1}} and ζ with the S^{1}- transfer trf_{S^{1}}, almost by definition. For these claims, see p. 18 and p. 70 of loc. cit.

A unitary map A: ℂ^{n} → ℂ^{n} restricts to an S^{1}-homotopy equivalence of the S^{1}- free unit spheres S^{2n−1} ⊂ ℂ^{n}, which defines the S^{1}-equivariant j-map j_{S^{1}} : U → SG(S^{1}). This is an H-map from the additive H-structure on U to the multiplicative H-structure on SG(S^{1}). Composing with the forgetful map we obtain the complex j-map j: U → SG with i^{∗} ∘ j_{S^{1}} = j.

The local obstruction θ: U → Q(Σ_{+}ℂP^{∞}) is defined in a similar manner as ζ_{S^{1}}, and satisfies θ = ζ_{S^{1}} ∘ j_{S^{1}} (see p. 70 of loc. cit.). Next there is a rotation map R: Σ_{+}ℂP^{∞} → U taking a pair (L, z) with L ∈ ℂP^{∞} and z ∈ S^{1} to the unitary transformation rotating by z along L and fixing the orthogonal complement L^{⊥}. The composite θ ∘ R is homotopic to the stabilization map ι: Σ_{+}ℂP^{∞} → Q(Σ_{+}ℂP^{∞}) by a theorem of James (see Proposition 2.7 of loc. cit.).

Let Δ: U → J_{⊕} be the connecting map in the fiber sequence (1.7) defining J_{⊕}. A solution to the complex Adams conjecture gives a map α: J_{⊕} → SG such that α ∘ Δ ≃ j. Let the restrictions of the Adams e-invariant e: Q(S^{0}) → ImJ to the zero– and one-components be e_{⊕}: Q_{0}(S^{0}) → J_{⊕} and e_{⊗}: SG ≃ Q_{1}(S^{0}) → J_{⊗}. Then e_{⊕}∘[−1] ≃ [−1]∘e_{⊗} since e is a ring map. The cannibalistic class ρ^{k}: J_{⊕} → J_{⊗} is the composite ρ^{k} = e_{⊗} ∘ α, and is an infinite loop map homotopy equivalence (see e.g. Chapter V.3 of [24]).

Finally recall the extension ϵ: Q(Σ_{+}ℂP^{∞}) → U of R over ι, using the additive infinite loop space structure on U. Thus ϵ ∘ ι = R. This ϵ deloops that of (3.2), as is seen by choosing an explicit Bott periodicity equivalence ΩU ≃ ℤ × BU as in [8].

The following result appears as Proposition 6.10 of [14].

Theorem 4.2 (Crabb–Knapp). The composite ϵ ∘ θ: U → U is homotopic to the identity map.

The proof is an application of the splitting principle in K-theory. First ϵ ∘ θ is shown to be an H-map. Thus it suffices to check the result on line bundles, which amounts to noting that the restrictions over R agree.

We have now produced the diagram below. The maps Δ, j_{S^{1}}, j, ρ^{k}, i^{∗}, e_{⊗}, trf_{S^{1}}, ϵ and e_{⊕} are defined on the infinite loop space level, while the remaining maps are only defined on the space level. By the discussion above the diagram homotopy commutes, with one exception: The bottom right hand square does not commute.^{1}

The horizontal composite ϵ ∘ θ: U → U is the identity, while the composite e_{⊕} ∘ ζ ∘ α: J_{⊕} → J_{⊕} is a homotopy equivalence [−1] ∘ ρ^{k} different from the identity. Thus the composite maps Δ ∘ ϵ and e_{⊕} ∘ trf_{S^{1}} are not homotopic, because the restrictions over ι are respectively Δ ∘ R and [−1] ∘ ρ^{k} ∘ Δ ∘ R, which are different.

Remark 4.4. One might wish to use this diagram to split Δ: U → J_{⊕} off from trf_{S^{1}} : Q(Σ_{+}ℂP^{∞}) → Q_{0}(S^{0}). The fact that the right hand square only commutes

1The authors thank Karl-Heinz Knapp for pointing out to us that this square fails to commute.

(4.3)  [figure: U; θ; j_{S^{1}}; j; Δ; Σ_{+}ℂP^{∞}; R; ι; R; SG(S^{1}); ζ_{S^{1}}; ≃; i^{∗}; Q(Σ_{+}ℂP^{∞}); ϵ; _{trf_{S^{1}}}; U; Δ; J_{⊕}; α; ρ^{k}; ≃; SG; ζ; ≃; e_{⊗}; Q_{0}(S^{0}); e_{⊕}; J_{⊕}; J_{⊗}; [−1]; ≃]

up to an automorphism shows that this might not be possible. Instead we will settle for a splitting on the level of homotopy groups.

We will need the following maps related to θ, ϵ and Δ.

Definition 4.5. Let θ′: SU → Q(ΣℂP^{∞}) be the composite

SU ⟶ U ⟶^{θ} Q(Σ_{+}ℂP^{∞}) ⟶ Q(ΣℂP^{∞}),

let ϵ′: Q(ΣℂP^{∞}) → SU be the unique lift of Q(ΣℂP^{∞}) ⟶ Q(Σ_{+}ℂP^{∞}) ⟶^{ϵ} U over the universal covering SU → U, and let Δ′: SU → J_{⊕} be the composite

SU ⟶ U ⟶^{Δ} J_{⊕}.

Then we have a diagram as below:

(4.6)  [figure: SU; θ′; Q(ΣℂP^{∞}); ϵ′; SU; U; θ; Q(Σ_{+}ℂP^{∞}); ϵ; U; Q(S^{1})]

The following lemmas are clear.

Lemma 4.7. The difference of ϵ′ ∘ θ′ and id_{SU} factors as SU → Q(S^{1}) → SU, hence is zero on homotopy groups. Thus π_{∗}(ϵ′ ∘ θ′) is the identity homomorphism, and ϵ′ ∘ θ′ is a homotopy equivalence. □

In fact ϵ′ ∘ θ′ is homotopic to the identity, as self maps of SU are detected on rational homotopy. (See e.g. V.2.9 of [24].)

Lemma 4.8. The difference of SU → U composed with trf_{S^{1}} ∘ θ: U → Q_{0}(S^{0}), and t ∘ θ′: SU → Q_{0}(S^{0}) factors through the Hopf map

η: Q(S^{1}) ⟶ Q(Σ_{+}ℂP^{∞}) ⟶^{trf_{S^{1}}} Q_{0}(S^{0}),

hence is null-homotopic when completed at an odd prime p. □

Lemma 4.9. There is a fiber sequence

SU ⟶^{Ω(ψ^{k}−1)} SU ⟶^{Δ′} J_{⊕} ⟶ BSU ⟶^{ψ^{k}−1} BSU

at odd primes p. □

Proposition 4.10. There is a homotopy commutative square

[figure: SU; θ′; Δ′; Q(ΣℂP^{∞}); t; J_{⊕}; ζα; Q_{0}(S^{0})]

where Δ′ and t are infinite loop maps, while θ′ and ζα are not H-maps. Both maps θ′ and ζα are split injective up to homotopy.

Proof. We simply combine diagrams (4.3) and (4.6). The map θ′ does not lift θ, but the composites to Q_{0}(S^{0}) are homotopic nonetheless, by Lemma 4.8. And θ′ is split injective up to homotopy by Lemma 4.7. The left homotopy inverse to ζα is visible in diagram (4.3). □

The diagram in (4.10) gives rise to the following lattice of fiber sequences

(4.11)  [figure: Ω hofib(t); b; F′; SU; Ω(ψ^{k}−1); hofib(t); Q(ℂP^{∞}); c; Ωt; F; ≃∗; SU; θ′; Δ′; Q(ΣℂP^{∞}); t; ΩQ_{0}(S^{0}); d; Ω CokJ; ≃∗; J_{⊕}; ζα; Q_{0}(S^{0}).]

For ϵ′ ∘ θ′ is a homotopy equivalence, so the homotopy fiber of θ′ is homotopy equivalent to the loop space of the homotopy fiber of ϵ′: Q(ΣℂP^{∞}) → SU, which is BF by Definition 3.3. Similarly the homotopy fiber of ζα is homotopy equivalent to the loop space of the homotopy fiber of e_{⊕}: Q_{0}(S^{0}) → J_{⊕}, which is CokJ by 1.5. In particular the maps c and d admit right homotopy inverses. However, we do not know that these homotopy sections can be chosen compatibly. Hence we cannot directly conclude that b admits a right homotopy inverse.

Definition 4.12. A given choice of commuting homotopy in Proposition 4.10 induces a map of horizontal homotopy fibers F → Ω CokJ in the diagram above. Let F′ be defined as the homotopy fiber of this map.

Clearly F′ has finite homotopy groups in each degree, like F. It is not clear whether F′ can be defined so as to admit a delooping.

Proposition 4.13. The map labeled b in (4.11) induces a split surjection on the level of homotopy groups, at least in degrees ∗ < deg(β_{p+1}) − 2.

For the proof, we use the following theorem.

Theorem 4.14 (Knapp). At odd primes p the (restricted) S^{1}-transfer map

t: Q(ΣℂP^{∞}) → Q_{0}(S^{0})

induces a surjection on homotopy in all degrees up to deg(β_{p+1}).

A complete proof of this result has yet to be published. Diagram (4.3) shows that the image of π_{∗}(t) contains the image of the J-homomorphism, and Theorem 6 of [21] asserts that the image also contains the β-elements β_{1}, …, β_{p}, but not β_{p+1}. Hence all products of α- and β-elements in the range ∗ < deg(β_{p+1}) are hit by π_{∗}(t). In particular the first γ-element is of this form, and is thus hit. The remaining elements in π_{∗}Q(S^{0}) up to degree deg(β_{p+1}) are expressible as Toda brackets in α’s and β’s, and unpublished calculations of Knapp show that also all these classes are hit by π_{∗}(t). The theorem follows from this assertion. We are told by Knapp that the remaining calculations will appear shortly as [22].

Corollary 4.15. The long exact sequence in homotopy for the fibration F′ → F → Ω CokJ splits into short exact sequences

0 ⟶ π_{∗−1}F′ ⟶ π_{∗}BF ⟶ π_{∗} CokJ ⟶ 0

for ∗ < deg(β_{p+1}). □

Proof of Proposition 4.13. By a diagram chase the composite map

f: F′ ⟶ F ⟶ Q(ℂP^{∞}) ⟶^{Ωt} ΩQ_{0}(S^{0})

has a lift over Ω(ζα). Here F → Q(ℂP^{∞}) is a homotopy section to c. Thus f is null-homotopic precisely if the further composite with Ωe_{⊕}: ΩQ_{0}(S^{0}) → ΩJ_{⊕} is null. We do not know if this is true, but at least we claim that the map Ωe_{⊕} ∘ f induces the zero homomorphism on homotopy.

To see this, use the fiber sequence

CokJ ∧ ℂP^{∞} ⟶ Q(ℂP^{∞}) ⟶^{e∧id} ImJ ∧ ℂP^{∞}.

Here ∧ denotes the infinite loop space smash product, given by forming a smash product of spectra and taking the resulting underlying space. The source CokJ ∧ ℂP^{∞} is connective and K-acyclic, and ΩJ_{⊕} is the connective cover of its Klocalization, so there are no essential maps CokJ ∧ ℂP^{∞} → ΩJ_{⊕}. Hence Ωe_{⊕} ∘ f factors through a map ImJ ∧ ℂP^{∞} → ΩJ_{⊕}.

Now π_{∗}ΩJ_{⊕} is concentrated in even degrees. As we saw in the proof of Theorem 3.5, the even homotopy of ImJ ∧ ℂP^{∞} is torsion free. Thus F′ → F → ImJ ∧ ℂP^{∞} induces the zero map on homotopy in even degrees, since π_{∗}F′ is torsion, and so our claim that π_{∗}(f) is zero in all degrees follows.

Finally π_{∗}Ω hofib(t) ⟶ π_{∗}Q(ℂP^{∞}) is injective in our range of degrees, by Knapp’s theorem. Thus π_{∗}F′ → π_{∗}F → π_{∗}Q(ℂP^{∞}) factors uniquely through a homomorphism π_{∗}F′ → π_{∗}Ω hofib(t), which is a homotopy section to b, compatible with the homotopy section to c. □

Corollary 4.16. For p odd there is a short exact sequence

0 ⟶ π_{∗}SU ⟶ π_{∗} hofib(t) ⟶ π_{∗−1}F′ ⟶ 0

which is split at least for ∗ < deg(β_{p+1}) − 1. □

5. Concluding arguments

Definition 5.1. Let ℓ′: SU → SU be the space level map given as the composite in the diagram below:

[figure: F′; SU; ℓ′; hofib(t); ℓ; SU]

We think of Ωℓ′: BU → BU as an additive reduced K-theory operation. Hence, by [23], it can be written as a power series in Adams operations ψ^{i} for various i.

Lemma 5.2. In degrees ∗ < deg(β_{p+1}) − 1 there is a short exact sequence

0 ⟶ π_{∗} hofib(ℓ′) ⟶ π_{∗} hofib(ℓ) ⟶ π_{∗−1}F′ ⟶ 0.

The subgroup π_{∗} hofib(ℓ′) vanishes in odd degrees.

Proof. Using 4.16, the long exact sequence in homotopy associated to the fiber sequence defining hofib(ℓ) splits into exact sequences

0 ⟶ π_{2n+1} hofib(ℓ) ⟶ π_{2n+1}SU ⊕ π_{2n}F′ ⟶^{ℓ_{∗}} π_{2n+1}SU ⟶

⟶ π_{2n} hofib(ℓ) ⟶ π_{2n−1}F′ ⟶ 0.

Here the second map ℓ_{∗} is a rational equivalence, and is zero on the torsion group π_{2n}F′. We can clearly identify its cokernel with π_{2n} hofib(ℓ′). □

Proposition 5.3. (i) ℓ′: SU → SU induces a p-adic equivalence on homotopy in degrees ∗ = 2n + 1 with n ≢ 0 mod p − 1.

(ii) In degrees ∗ < deg(β_{1}) = 2(p − 1)p − 2 its homotopy fiber has homotopy groups

π_{∗} hofib(ℓ′) ≅ \{ \begin{matrix}ℤ/p & if ∗ ≡ 0 mod 2p − 2, \\ 0 & otherwise. \end{matrix}

(iii) In the same range of degrees π_{2n+1} hofib(ℓ) = 0, and

π_{2n} hofib(ℓ) ≅ π_{2n−1}F′ ⊕ π_{2n} hofib(ℓ′).

Proof. Recall the map w: BSG → hofib(L) discussed in Theorem 0.2, which induces an isomorphisms on homotopy groups through degree 2p − 2. By Proposition 2.10 there are isomorphisms π_{∗} hofib(L) ≅ π_{∗} hofib(ℓ) for ∗ < deg(β_{1}).

Composition with the Adams map A: Σ^{2p−2}S^{0}/p → S^{0}/p induces the v_{1}-action on the mod p homotopy of infinite loop spaces. Thus such groups are graded ℤ/p[v_{1}]-modules. Now ℓ: hofib(t) → SU is an infinite loop map, and thus induces a ℤ/p[v_{1}]-module homomorphism on mod p homotopy. The mod p homotopy of the target is a free ℤ/p[v_{1}]-module on p − 1 generators in degrees 3, 5, …, 2p − 1, corresponding to the Adams splitting of SU [3]. Since the fiber of ℓ is (2p − 3)- connected and has bottom homotopy group ℤ/p in degree 2p − 2, we find that π_{∗}(ℓ; ℤ/p) maps onto the generators in degrees 3, 5, …, 2p − 3 and is trivial in degree 2p − 1. By Corollaries 3.6 and 4.15 the space F′ is (4p − 4)-connected, so π_{∗}SU → π_{∗} hofib(t) is an isomorphism well beyond the range ∗ ≤ 2p − 1. Hence

π_{∗}(ℓ′; ℤ/p): π_{∗}(SU; ℤ/p) ⟶ π_{∗}(SU; ℤ/p)

maps the mod p homotopy of the first p − 2 Adams summands surjectively (and thus isomorphically), and maps the last summand trivially. Thus π_{∗}(ℓ′) is a p-adic equivalence in all degrees ∗ = 2n + 1 with n ≢ 0 mod p − 1, and is trivial mod p in the remaining degrees. The first claim follows.

Now restrict to degrees ∗ < deg(β_{1}). By Definition 4.12 and Corollary 3.6 the homotopy groups of F and F′ agree in this range, and are concentrated in odd degrees ∗ ≢ −1 mod 2p − 2. Hence for n ≡ 0 mod p − 1 we have ℤ ≅ π_{2n±1}SU ≅ π_{2n±1} hofib(t) and π_{2n} hofib(t) = 0, so

π_{2n+1}(hofib(t); ℤ/p) ≅ π_{2n+1}(SU; ℤ/p) ≅ ℤ/p

and

π_{2n}(hofib(t); ℤ/p) = 0.

We now restrict attention to these degrees with n ≡ 0 mod p − 1. By the ℤ/p[v_{1}]- module structure the map

ℓ′: π_{2n+1}(SU; ℤ/p) ⟶^{≅} π_{2n+1}(hofib(t); ℤ/p) ⟶^{ℓ} π_{2n+1}(SU; ℤ/p)

is null. Thus the boundary map π_{2n+1}(SU; ℤ/p) → π_{2n}(hofib(ℓ); ℤ/p) is an isomorphism, from a free ℤ/p[v_{1}]-module on one generator. Since π_{2n}(BSG; ℤ/p) → π_{2n}(hofib(ℓ); ℤ/p) maps onto the generator in degree 2n = 2p − 2, it is also surjective throughout the range, by the ℤ/p[v_{1}]-module structure. Thus integrally ℤ/p ≅ π_{2n}BSG → π_{2n} hofib(ℓ) is a homomorphism inducing an isomorphism mod p, and hence it is a p-adic equivalence. This settles claim (ii).

Regarding claim (iii), there are no extension problems, because at least one of the groups π_{2n−1}F′ and π_{2n} hofib(ℓ′) is always zero in the initial range ∗ < deg(β_{1}). □

We can now assemble the pieces of the calculation. Let p be an odd prime, and let all spaces and groups be completed at p.

Theorem 5.4. (i) The homotopy groups of the homotopy fiber of the linearization map L: A(∗) → K(ℤ) satisfy

π_{∗} hofib(L) ≅ π_{∗} CokJ ⊕ π_{∗}B CokJ ⊕ π_{∗} hofib(ℓ).

(ii) When ∗ < deg(β_{p+1}) − 1 the homotopy groups of hofib(ℓ) fit into a short exact sequence

0 ⟶ π_{∗} hofib(ℓ′) ⟶ π_{∗} hofib(ℓ) ⟶ π_{∗−1}F′ ⟶ 0.

This extension is always trivial when ∗ < deg(β_{1}).

(iii) When ∗ < deg(β_{1}) the first term in (ii) satisfies

π_{∗} hofib(ℓ′) ≅ π_{∗}BJ_{⊗} ≅ \{ \begin{matrix}ℤ/p & if ∗ ≡ 0 mod 2p − 2, \\ 0 & otherwise. \end{matrix}

(iv) When ∗ < deg(β_{p+1}) the third term in (ii) fits into a short exact sequence

0 ⟶ π_{∗−1}F′ ⟶ π_{∗}BF ⟶ π_{∗} CokJ ⟶ 0.

When ∗ < deg(β_{1}) the right hand term is trivial and the left hand map is an isomorphism. In every degree ∗ the homotopy group π_{∗}BF is isomorphic to the torsion subgroup of π_{∗}Q(ΣℂP^{∞}).

Proof. Claim (i) follows by Dundas’ Theorem 0.6 and Proposition 2.10.

The short exact sequence of claim (ii) comes from Lemma 5.2, and when ∗ < deg(β_{1}) either π_{∗} hofib(ℓ′) or π_{∗−1}F′ is zero by Proposition 5.3(i) and Corollary 3.6.

The calculation in (iii) is the result of Proposition 5.3(ii).

The short exact sequence in (iv) comes from Corollary 4.15. Since β_{1} is the first nontrivial class in π_{∗} CokJ, it is clear that π_{∗} CokJ = 0 for ∗ < deg(β_{1}). By Segal’s Theorem 3.4 the torsion subgroup of π_{∗}Q(ℂP^{∞}) can be identified with π_{∗}F. □

Remark 5.5. This result reduces the determination of π_{∗} hofib(L) in the range ∗ < deg(β_{p+1}) − 1 to the identification of ℓ′: SU → SU, the calculation of the torsion subgroup π_{∗}BF in π_{∗}Q(ΣℂP^{∞}), the calculation of the surjection π_{∗}BF → π_{∗} CokJ, and the resolution of the extension problem in claim (ii).

Proof of Theorem 0.3. When ∗ < deg(β_{1}) = 2(p − 1)p − 2 the theorem above identifies π_{∗} hofib(L) with the sum of π_{∗}BF and π_{∗} hofib(ℓ′). The former groups are given in Corollary 3.6, the latter in Proposition 5.3(ii). Combining these results gives claim (i) of the theorem.

Regarding claim (ii), the classes in degrees 2n with n ≡ 0 mod p − 1 constitute the summand π_{∗} hofib(ℓ′), which is hit isomorphically from π_{2n}BSG by the proof of Proposition 5.3. These classes are in the delooped image of J, so map to zero through π_{2n}A(∗) → π_{2n} Wh^{Diff}(∗) by Corollary 3.3 of [33]. Hence they land in π_{2n}Q(S^{0}) ⊆ π_{2n}A(∗), but this group has no p-torsion in this range.

For claim (iii), the remaining classes map nontrivially to π_{∗−1}F′, which is a split summand in π_{∗} hofib(t) by Corollary 4.16, and thus in π_{∗}T C(∗) by (2.4). Thus the composite

π_{∗} hofib(L) ⟶ π_{∗}A(∗) ⟶^{trc} π_{∗}T C(∗)

maps these classes onto a direct summand in π_{∗}T C(∗) and thus also in π_{∗}A(∗), in this range. □

Proof of Corollary 0.4. When n ≡ 0 mod p−1 the classes of order p in π_{2n} hofib(L) map to zero in π_{2n}A(∗), and thus come from classes in K_{2n+1}(ℤ) through the connecting map. We claim that these classes are of infinite order. The connecting map factors as

ΩK(ℤ) ⟶^{trc} ΩT C(ℤ) ⟶ hofib(L).

The torsion classes in T C_{2n+1}(ℤ) come from the ImJ and B ImJ terms, which are hit by L and map to zero under the connecting map. Hence the classes detected in K_{2n+1}(ℤ) map to nonzero classes in π_{2n+1}SU, and are of infinite order.

The remaining classes are detected as the even-degree homotopy of hofib(t), which injects onto the torsion in π_{∗}Q(ΣℂP^{∞}), which in turn is a summand in π_{∗}Q(Σ_{+}ℂP^{∞}). This detecting map factors as

hofib(L) ⟶ A(∗) ⟶^{trc} T C(∗) ⟶ Q(Σ_{+}ℂP^{∞}).

The claim follows. □

6. A conjecture

We conclude with some speculations suggesting a simpler description of the homotopy fiber of the linearization map.

The map w: BSG → hofib(L) may be projected away from the terms CokJ and B CokJ to give a map BSG → hofib(ℓ). The target sits in the fiber sequence F′ → hofib(ℓ′) → hofib(ℓ), where ℓ′: SU → SU can be viewed as a K-theory operation. The map BSG → hofib(ℓ) is at least (2p − 1)-connected, and F′ is (4p−4)-connected, so on the level of homotopy groups there is a (2p−1)-connected homomorphism π_{∗}BSG → π_{∗} hofib(ℓ′). If we assume that this also holds on the space level, then the essential properties of the map ℓ′ can be determined.

Proposition 6.1. Suppose there is a (2p − 1)-connected map BSG → hofib(ℓ′). Then Ωℓ′ is homotopic to ψ^{k} − 1: BU → BU up to pre- and post-composition with homotopy equivalences of BU. Hence hofib(ℓ′) ≃ BJ_{⊗}.

Proof. The target is K-local to well below the connectivity of BSG, so this hypothetical map extends over Be_{⊗}: BSG → BJ_{⊗}. Thus there are horizontal fiber sequences

[figure: BU; c; ψ^{k}−1; BU; b; BJ_{⊗}; a; SU; BU; Ωℓ′; BU; hofib(ℓ′); SU]

with a vertical map a as indicated. Now K^{−1}(BU) = 0, so the composite BU → BJ_{⊗} → hofib(ℓ′) → SU is null-homotopic. Hence there is a lift b: BU → BU compatible with a. We may alter the choice of b by any map factoring through Ωℓ′, which is an equivalence on the first p − 2 of the p − 1 Adams summands in BU. Hence we may assume b is an equivalence on these summands. Let c be the induced map of homotopy fibers. Then we have a map of fiber sequences as above. The maps ψ^{k} − 1 and Ωℓ′ are equivalences on the first p − 2 Adams summands, so c inherits the same property from b. Finally consider the last Adams summand. By hypothesis the map a induces a mod p equivalence in degree 2p − 2. Hence so does b, and by v_{1}-periodicity the map b is a p-adic equivalence, also on the last Adams summand. Compatibility of c and b implies that c is also an equivalence on this final summand. For ψ^{k} − 1 does not admit any nontrivial factorizations through BU. Hence b and c are p-adic equivalences, and Ωℓ′ agrees with ψ^{k} − 1 up to such automorphisms. Thus hofib(ℓ′) ≃ BJ_{⊗}. □

We can now map CokJ and BSG into hofib(L). First CokJ has been identified as a term in a product splitting of hofib(L). Next the subspace B CokJ in BSG might imaginably map through w by an equivalence to the subspace B CokJ in hofib(L). Then the remaining BJ_{⊗} might map to hofib(ℓ) by the same map as hofib(ℓ′) does. Then only BF′, if it exists, remains as the infinite loop space cofiber. We express these observations as a conjectural fiber sequence

(6.2)  BSG × CokJ ⟶ hofib(L) ⟶ BF′.

If BF′ does not exist we can instead extend the sequence to the left by F′. By definition F′ and CokJ are related by the fiber sequence F′ → F → Ω CokJ. Hence hofib(L) seems to consist of BSG and the two components BF′ and CokJ of BF, reassembled in an untwisted fashion.

This ends our speculations.

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Fakultät für Mathematik, Universität Bielefeld, Bielefeld, Germany E-mail address: klein@mathematik.uni-bielefeld.de

Department of Mathematics, University of Oslo, Oslo, Norway E-mail address: rognes@math.uio.no