THE HATCHER–WALDHAUSEN MAP IS A SPECTRUM MAP

John Rognes

Abstract. We prove that the map G/O → ΩW h^{Diff}(∗) which was constructed by Waldhausen and proved to be a rational equivalence by Bökstedt, is in fact an infinite loop map if we use a multiplicative infinite loop space structure on the target. As an application we investigate the obstruction to improving Bökstedt’s two–primary results on splitting the étale K-theory space JK(ℤ) off from K(ℤ), to the unlooped space or spectrum level. Finally we determine the algebra of spectrum self maps of JK(ℤ) completed at two, and prove that two times Bökstedt’s splitting map is infinitely deloopable.

Introduction

In this paper we give a manifold model for Waldhausen’s A-theory space A(∗), geared towards studying the multiplicative spectrum structure induced by smash product of spaces on its one–component. This allows us to infinitely deloop the rational equivalence considered in [Wa2] and [Bö]

G/O ⟶ ΩW h^{Diff}(∗)

to a spectrum map, when W h^{Diff}(∗) is made into a spectrum using this multiplicative structure. See Corollary 1.5 for a precise statement. Here G/O is the classifying space for smooth normal invariants, or equivalently the homotopy fiber of the J-homomorphism BO → BG. ΩW h^{Diff}(∗) is a delooping of the stable smooth pseudoisotopy (or concordance) space of a point.

As an application we consider the problem of singly or infinitely delooping Bökstedt’s two–complete splitting of the loop space of his model for “étale K-theory of the integers,” ΩJK(ℤ), defined in [Bö], off from the loop space of algebraic Ktheory of the integers, ΩK(ℤ).

Let SG = Q(S^{0})_{1} and K(ℤ)_{1} denote the multiplicative spectra arising from smash product of spheres and tensor product of ℤ-modules respectively. We prove that if the usual map SG → K(ℤ)_{1} factors over the Adams e-invariant e: SG → J_{⊗}, in the stable category, then indeed JK(ℤ)_{1} splits off K(ℤ)_{1} as two–completed spectra. See Theorem 2.1. In more detail, a modification of Bökstedt’s argument provides the fibration sequence of spectra displayed horizontally below,

[figure: C_{⊗}; B fib(s); JK(ℤ)_{1}]

[figure: g; Φ]

K(ℤ)_{1}

Typeset by AMS-TEX

and the projection map factors through K(ℤ)_{1}. By assuming a null homotopy of the composite C_{⊗} → K(ℤ)_{1}, we obtain our splitting result.

In the course of lifting Bökstedt’s arguments to the spectrum level, we note some extensions of the theorems of Madsen, Snaith and Tornehave [MST] on degree zero spectrum level operations in real or complex K-theory, to operations of more general degree. See Proposition 2.2 and Corollary 2.3.

Using these results, we determine the algebra of spectrum self maps of a component of JK(ℤ), and prove that twice the section from ΩJK(ℤ) to ΩK(ℤ) deloops infinitely often. See Theorems 3.3 and 3.5.

The author thanks Marcel Bökstedt for proposing the problem of delooping the map G/O → ΩW h^{Diff}(∗), and for numerous extremely helpful and friendly discussions. Further thanks to Jørgen Tornehave and Ib Madsen for explanations, and for making possible a delightfully inspiring year’s stay at Århus University. Thanks also to John Klein for references to Ganea’s work.

§1. Multiplicative partitions

In this section we will construct the map G/O → ΩW h^{Diff}(∗) as a spectrum map, as the map of homotopy fibers of the horizontal maps in a homotopy commutative diagram of spectra :

BO  \overline{}BG

[figure]

A^{S}(∗)_{1}  \overline{}A(∗)_{1}

Here G is the monoid of stable self homotopy equivalences of spheres, A(∗)_{1} is the one–component of Waldhausen’s A-theory of a point, and A^{S}(∗)_{1} the corresponding space for stable A-theory [Wa1]. A(∗)_{1} comes equipped with a multiplicative infinite loop space structure, induced from smash product of spaces. As stabilization A(∗)_{1} → A^{S}(∗)_{1} is coherently compatible with smashing, A^{S}(∗)_{1} inherits a product from A(∗)_{1}. By [Wa3] the functor taking finite based sets to based spaces followed by stabilization induces a spectrum level equivalence SG = Q(S^{0})_{1} → A(∗)_{1} → A^{S}(∗)_{1}.

We will define the multiplicative infinite loop space structure on W h^{Diff}(∗) by setting W h_{⊗}^{Diff}(∗) to be the fiber of the trace map A(∗)_{1} → A^{S}(∗)_{1} ≅ SG.

We proceed to construct a model for the diagram above using a modified version of Waldhausen’s manifold models for the spaces related to A-theory [Wa2, §1]. The original manifold models admit a product up to homotopy, corresponding to a modified smash product of spaces, as indicated in [Wa3, p. 399]. It can be seen that the product given by Waldhausen’s construction will be associative up to homotopy. However, it is not clear how to systematically account for all higher coherence homotopies in this set up, which would be necessary for describing an infinite loop space structure on A(∗)_{1} and the related spaces mentioned above.

Similarly, it is not clear how to make Waldhausen’s construction using rigid tubes [Wa2, p. 159] of a crucial map BO → T^{Diff} into more than an H-map. The direct sum of subspaces, which gives the composition in a Grassmannian model for BO, will only correspond up to homotopy with the product on the manifold models given by smash product of spaces, and again it is difficult to see how to prove the existence of higher coherence homotopies in the original construction. On the other hand, this will be transparent in our model.

For simplicity we will only consider manifold models for A(X) when X = ∗ is a point, although similar constructions could be made for X an abelian topological monoid. We shall stably approximate Waldhausen’s partitions in D^{n} × I by partitions in S^{n} × I, and further by codimension zero submanifolds of S^{n+1} which are standardized near infinity. Here S^{n} × I approximates S^{n+1} by extending the boundary components to the origin and infinity, respectively. Our constructions will then be stably equivalent to the original ones.

Definitions 1.1. Let ℝ^{n} have the supremum metric ∥x⃗∥ = sup_{i} |x_{i}|, topologize S^{n} as the one–point compactification of ℝ^{n} based at infinity, and give S^{∞} the direct limit topology. For a (large) number δ we call the neighborhood of ∗ ∈ S^{∞} consisting of x⃗ with ∥x⃗∥ ≥ δ a δ-neighborhood of infinity. Let M′ ⊂ S^{n} be a compact codimension zero submanifold. For a sequence of positive numbers ϵ = (ϵ_{i}) we call the neighborhood M of M′ in S^{∞} consisting of x⃗ for which there exists a y⃗ ∈ M′ with |x_{i} − y_{i}| ≤ ϵ_{i} for all i an ϵ-thickening of M′. The thickenings of compact manifolds in S^{∞} which contain a standard neighborhood of infinity will be our partitions.

We now follow [Wa2, §1]. Let δ ≫ 0 be a fixed number. A partition is a subspace M ⊂ S^{∞} which is the union of (1) an ϵ-thickening of some compact codimension zero submanifold M′ ⊂ S^{n}, for some ϵ, M′ and S^{n}, and (2) the δ-neighborhood of the base point at infinity. This must be interpreted in a suitable category of manifolds, as explained in [Wa2, §1 and §6]. We will work with smooth manifolds with corners, i.e. the DIF F category.

Define P = P(∗)_{•} to be the simplicial set with k-simplices the locally trivial families of partitions parametrized over a simplex Δ^{k}. The ϵ and M′ may vary through such a family, but there should exist a global (maximal) n for each simplex. P can be made into a simplicial category with a morphism from M to N if M ⊆ N, and we let hP denote the simplicial subcategory where the morphisms are (roughly) the homotopy equivalences. Precisely, the morphisms in hP are M → N such that M and N are thickenings of some M′ ⊆ N′ ⊂ S^{n}, with both inclusions

[figure: ∂M′; N′ − int(M′); ∂N′]

homotopy equivalences. Let hP_{k}^{m} be the full simplicial subcategory of hP where M has the homotopy type of a wedge of k m-spheres, or more precisely, the component of hP containing a thickening of an M′ ⊂ S^{n} obtained by adding k trivial m-handles to a δ-neighborhood of infinity. Let the simplicial set P_{k}^{m} be given by its objects.

We now discuss stabilization, the additive partial monoid structure on hP, and the coherently associative smash product.

Our thickened manifolds M ⊂ S^{∞} are already stabilized with respect to the containing dimension, but we can also stabilize (the homotopy type of) M itself by suspensions. Suppose M is a partition obtained by adding a δ-neighborhood of infinity to an ϵ-thickening of M′ ⊂ S^{n}. Let M′_{−} ⊂ ℝ^{n} be M′ with the point at infinity removed. Then the cartesian product M′_{−} × ℝ^{1} ⊂ ℝ^{n} × ℝ^{1} ≅ ℝ^{n+1} is a compact codimension zero submanifold away from a neighborhood of infinity, and the union of its ϵ-thickening and the δ-neighborhood of infinity is the stabilization of M. This process may actually better be viewed as smashing with S^{1}, as we shall see momentarily.

We can add two partitions if they are disjoint away from the δ-neighborhood of infinity, by forming their union. This provides an additive partial monoid structure on P (and similarly for hP), which we can use to group complete the nerve of P, as the inclusion of composable k-tuples of partitions into all k-tuples of partitions is a stable equivalence.

To multiply two partitions together, we will use an identification ℝ^{∞}×ℝ^{∞} ≅ ℝ^{∞}, say given by shuffling the coordinates : ((x_{i}), (y_{i})) ↦ (x_{1}, y_{1}, x_{2}, y_{2}, …). This pairing satisfies associativity and commutativity up to the usual coherent isomorphisms. Now suppose we are given two partitions M and N, which are ϵ_{M}- and ϵ_{N}-thickenings of M′ ⊂ S^{m} and N′ ⊂ S^{n} respectively. We shall define their product M ⊗ N, which has the homotopy type of M ∧ N, by thickening the cartesian product of M′_{−} and N′_{−} and adding a δ-neighborhood of infinity. Explicitly

M′ ∧ N′ ≅ (M′_{−} × N′_{−})_{+} ⊂ S^{∞} ∧ S^{∞}

is a compact subspace of S^{n} ∧ S^{m} ⊂ S^{∞} ∧ S^{∞} ≅ S^{∞}, which is a codimension zero submanifold of S^{n} ∧ S^{m} away from infinity. We shuffle the thickening distances ϵ_{M} and ϵ_{N} in the obvious way into ϵ, and let M ⊗ N be the union of the ϵ-thickening of M′ ∧ N′ with the standard δ-neighborhood of infinity.

n-fold suspension stabilization may now be realized as forming the ⊗-product with a thickening of S^{n} ⊂ S^{∞}.

We let P^{m} = ∐_{k≥0} P_{k}^{m}, and denote its nerve as a partial additive monoid by N_{Γ}(P^{m}). Consider the double square

(1.2)

[figure: colim_{m} P_{1}^{m}; colim_{m} P^{m}; ΩN_{Γ}(colim_{m} P^{m})]

[figure]

[figure: colim_{m} hP_{1}^{m}; colim_{m} hP^{m}; ΩN_{Γ}(colim_{m} hP^{m})]

where the direct limits are formed by suspensions. Recall the terminology from [Wa2, §1 and §3], where h\underline{P} _{k}^{m}(X × J^{n}) is the simplicial category of parametrized partitions of X × J^{n} × I obtained by adding k trivial m-handles to a standard picture near X × J^{n} × {0}, under homotopy equivalences. The notations obtained by omitting h, m or k are derived analogously to the cases for P above. The tube space T^{Diff} is equivalent to colim_{m,n} \underline{P} _{1}^{m}(D^{m+n}).

Lemma 1.3. There is a chain of homotopy equivalences linking the outer rectangle above to the homotopy cartesian square

T^{Diff}  \overline{}A^{S}(∗)

[figure]

BG  \overline{}A(∗).

This is the square of [Wa2, p. 159].

Proof. There are maps

hP_{k}^{m} \underline{}h\underline{P} _{k}^{m}(S^{n})  h\underline{P} _{k}^{m}(D^{n+1})

which become stable equivalences as n tends to infinity. Here we embed Waldhausen’s partitions of S^{n} × I into a partition of S^{∞} by thinking of S^{n} × I as an annulus missing neigborhoods of the origin and infinity, and then thickening. See also [Wa3, p. 400] for a discussion of this kind of rewriting.

These maps are sufficiently natural when k, m and n vary to assemble into the claimed chains of equivalences. The statement that the square is homotopy cartesian is [Wa2, Proposition 3.1]. □

We have thus recognized the homotopy types in the outer rectangle (1.2). The (coherently) associative and commutative ⊗-product is respected by the maps in that diagram, and gives rise to multiplicative infinite loop space structures on the leftmost spaces, and the one–components of the rightmost spaces. Hence these components of the outer rectangle of (1.2) form a square of infinite loop spaces.

We can also map BO into the model colim_{m} P_{1}^{m} for T^{Diff} as a spectrum map, using the additive (Whitney sum) infinite loop space structure on BO. To do this, chose a (small) number ϵ > 0, and also let ϵ denote the constant infinite sequence (ϵ, ϵ, …). Take as a model for BO(m) the Grassmannian space of m-planes in R^{∞}, and map a vector space V to its ϵ-thickening in R^{∞} ⊂ S^{∞}, with a δ-neighborhood of infinity added.

This map takes Whitney sum of vector spaces to the ⊗-product of partitions, and is thus compatible with stabilization in m. This is because the ϵ-thickening in the supremum metric of the Whitney sum V × W ⊂ ℝ^{∞} × ℝ^{∞} ≅ ℝ^{∞} is precisely the cartesian product of the ϵ-thickenings of V and W. The reason why we use the supremum metric on ℝ^{∞} is to make this map commute “on the nose” with these pairings.

This map is clearly homotopic to that of [Wa2, Proposition 3.2]. We have thus proved :

Theorem 1.4. There exists a homotopy commutative square of spectra

BO  \overline{}BG

[figure]

A^{S}(∗)_{1}  \overline{}A(∗)_{1}.

Here BO = BO_{⊕} carries the additive infinite loop space structure from Whitney sum, while BG, A^{S}(∗)_{1} ≃ SG and A(∗)_{1} carry the multiplicative infinite loop space structures from smash product. The map BO → BG is the delooped Jhomomorphism, the map A^{S}(∗)_{1} → A(∗)_{1} is equivalent to the map induced by including the category of finite based sets into the category of finite spaces, and the map BG → A(∗)_{1} includes self homotopy equivalences of spheres into homotopy equivalences of more general spaces. □

Corollary 1.5. There exists a spectrum map G/O → ΩW h_{⊗}^{Diff}(∗) from the fiber of BO → BG to the fiber of A^{S}(∗)_{1} → A(∗)_{1}, which on underlying spaces agrees up to homotopy with Waldhausen’s map [Wa2, Corollary 3.4]. □

As Waldhausen remarks, A. Hatcher has constructed a similar space level map, which may be the same as this one. K. Igusa has also given a geometric description of such a map.

§2. Bökstedt’s model for étale K-theory

We apply the result of §1 to the question of whether Bökstedt’s two–complete splitting of the space ΩJK(ℤ) off from ΩK(ℤ) [Bö] can be lifted to the spectrum level. This turns out to reduce to the old problem of how the cokernel of J maps into K(ℤ); see Theorem 2.1.

The line of argument follows Bökstedt’s paper, with some modifications required to give spectrum level constructions. As all of the homotopies needed for Bökstedt’s splittings turn out to lift uniquely to the spectrum level, the added information content of a spectrum level splitting compared to a space level splitting may be viewed as lying entirely in the construction from §1, and the hypothesis about the cokernel of J.

We will work in the stable category for the remainder of the paper, and all spectra are implicitly completed at the prime two.

The spectrum JK(ℤ) can be defined as the homotopy fiber of the composite map of spectra

[figure: kO = ℤ × BO; \overline{ψ^{3} − 1}; BSpin; \overline{c}; BSU,]

where ψ^{3} is the Adams operation, and c complexification. Its one component JK(ℤ)_{1} has a multiplicative infinite loop space structure, given as the homotopy fiber of the composite :

[figure: BO_{⊗}; \overline{ψ^{3}/1}; BSpin_{⊗}; \overline{c}; BSU_{⊗}]

We use the notation f/g rather than f − g to denote the difference of two Hmaps f and g when we are thinking of the H-group structure of the target as a multiplicative one.

It will be clear later that the exponential cannibalistic equivalence ρ^{3} : BSO_{⊕} → BSO_{⊗} and the related map J_{⊕} → J_{⊗} [AP], [MST, Corollary 4.4] induce an equivalence of spectra from the additive zero component JK(ℤ)_{0} to JK(ℤ)_{1}, after passing to one–connected covers. The only difference between the spectra JK(ℤ)_{0} and JK(ℤ)_{1} is that BO(1) splits off the latter but not the former, just as with BO and BO_{⊗}. Our constructions will relate JK(ℤ)_{1} to K(ℤ)_{1} and A(∗)_{1} of the previous section, whence it will be natural for us to focus on the multiplicative model for the spectrum structure.

Let J_{⊕}, J_{⊗}, C_{⊕} and C_{⊗} be the additive and multiplicative connected image of J and cokernel of J spectra. There is a non–split fibration (see Theorem 3.4)

[figure: C_{⊗}; SG; \overline{e}; J_{⊗}]

and the usual map SG = Q(S^{0})_{1} → A(∗)_{1} → K(ℤ)_{1}.

Theorem 2.1. Let all spaces and spectra be completed at the prime two.

If the composite map C_{⊗} → SG → K(ℤ)_{1} is null homotopic as a map of spectra, then the natural map K(ℤ)_{1} → JK(ℤ)_{1} is a split surjection in the category of spectra.

If the space of space maps from C_{⊗} to K(ℤ)_{1} is contractible, then the natural map K(ℤ)_{1} → JK(ℤ)_{1} is a split surjection of spaces.

Note that Mitchell [Mit] has proved the space level version of the first hypothesis above, i.e. that the map SG → K(ℤ) of spaces factors through the image of J. The corresponding space level conclusion does not immediately follow, however.

Either of these hypotheses would hold if K(ℤ)_{1} were the connected cover of its K-localization, as predicted by the Lichtenbaum–Quillen conjecture [DF]. Imaginably, the second hypothesis could be proved by space level techniques not taking deloopings into account.

Before proving the theorem we give some preparatory extensions of the theorems of Madsen, Snaith and Tornehave [MST], which follow from Adams’ presentation in [Ad, Chapter 6].

The original theorems of Madsen, Snaith and Tornehave precisely describe the (degree zero) spectrum maps BSO → BSO and BSU → BSU, which we think of as K-theory operations. In particular such maps are detected on rational homotopy groups. (See [Ad, 6.4.2] and [Ma, V.2.9].) Also, as we are considering the two– complete case, they can all be written as the sum of a scalar times the identity map, plus a map factoring through the Adams operation ψ^{3} − 1, in both the real and complex cases [MST, 2.3].

Note that by the Hurewicz theorem, any spectrum map X → Y has a unique lifting up to homotopy to the k-connected cover of Y, where k is the connectivity of X.

We will use η to denote the essential spectrum map S^{1} → S^{0}, or the map BY → Y induced by smashing with η for any spectrum Y, or even the unique lift of this η mapping into the k-connected cover of Y, where k is the connectivity of BY.

Proposition 2.2. There are no essential spectrum maps BSO → KO in degrees congruent to 3, 5, 6 or 7 modulo 8.

All spectrum maps BSO → KO in degrees congruent to 1 or 2 modulo 8 factor through η or η^{2} respectively. In view of the Hurewicz theorem, the same conclusion applies to spectrum maps BSO → Spin or BBSO → Spin.

Let [X, Y]_{n} denote the homotopy classes of spectrum maps of degree n from X to Y, when X and Y are spectra, and let [X, Y] = [X, Y]_{0}.

Proof. The first part is immediate from the universal coefficient theorem [Ad, 6.4.7]

[BSO, KO]_{∗} ≅ Hom_{π_{∗}KO}(KO_{∗}BSO, π_{∗}KO),

the fact that KO_{∗}BSO is a free π_{∗}KO-module on (countably many) generators in degree zero [Ad, p. 162], and the vanishing of π_{∗}KO in the degrees mentioned.

For the second part, note that left composition with η^{i} for i = 1 or 2 defines a natural transformation of degree i of both sides of the formula above. As multiplication by η^{i} maps π_{0}KO onto π_{i}KO, η_{∗}^{i}: [BSO, KO]_{0} → [BSO, KO]_{i} is onto, and the result follows. □

Corollary 2.3. There are no essential spectrum maps BSO → SU or SU → BSO. A spectrum map f: BBSO → SU or SU → BBSO is null homotopic if and only if π_{∗}(f; ℚ) = 0; i.e. such maps are detected rationally.

Proof. For the first part map BSO to or from SU in the fibration sequence [Mil, §24] or [Ma, Proof of V.5.15] :

[figure: BSO; \overline{η}; Spin; \overline{c}; SU; BBSO]

The result then follows from a little chase using Proposition 2.2.

The second part is handled similarly, using Proposition 2.2 and the fact that a spectrum map BBSO → BBSO which is trivial on rational homotopy is null homotopic. □

In the remainder of this section we prove Theorem 2.1. Bökstedt constructs his section ΩJK(ℤ) → ΩK(ℤ) by establishing a homotopy commutative diagram on the (looped) space level, which is similar to the lower part of our diagram (2.8) displayed after the proof of Lemma 2.7. However, his diagram has vertical morphisms mapping upwards, formed using a non–deloopable section to the unit e: SG → J_{⊗} obtained from a solution to the Adams conjecture. We instead establish (2.8) as a diagram on the spectrum level.

We then turn to the extension problem indicated in the introduction, and show how to reduce it to either of the hypotheses of the theorem, in the spectrum and space level cases respectively. With the notation of (2.8), the problem is to factor g: B fib(s) → K(ℤ)_{1} through B fib(s) → JK(ℤ)_{1}, which can be achieved if we know enough about how the composite C_{⊗} → B fib(s) → K(ℤ)_{1} is null homotopic.

Given this construction of a map h: JK(ℤ)_{1} → K(ℤ)_{1}, the necessary adaption of Bökstedt’s space level result shows that h is a section to the natural map Φ, by an argument which we explain at the end of this section.

To construct (2.8), first recall the null homotopy of the composite BSO → SG → K(ℤ)_{1} :

BSO \overline{Bj} \overline{}BSG

[figure: s; i_{B}]

[figure: SG; p_{Q}; \overline{i_{Q}}; A(∗)_{1}; \overline{p_{W}}; i_{W}; W h_{⊗}^{Diff}(∗)]

l

K(ℤ)_{1}

The commuting square is from Theorem 1.4, and the multiplicative splitting of A(∗)_{1} into SG and W h_{⊗}^{Diff}(∗) from [Wa1] and [Wa3]. l denotes the linearization map.

Lemma 2.4. The composite l ∘ i_{Q} ∘ s: BSO → SG → K(ℤ)_{1} is null homotopic as a map of spectra.

Proof. Bökstedt’s argument applies unchanged : s is homotopic to the composite p_{Q}∘i_{B}∘Bj, and l∘i_{B} is null homotopic. Thus l∘i_{Q}∘s is homotopic to l∘i_{W} ∘p_{W} ∘i_{B}∘ Bj (up to sign), which factors through the null map SG → A(∗)_{1} → W h_{⊗}^{Diff}(∗). □

Next we consider the map s: BSO → SG. Bökstedt and Waldhausen proved that the composite p_{Q} ∘ i_{B}: BSG → A(∗)_{1} → SG is homotopic to multiplication by η as a map of spaces [BW]. Presumably this also is true on the spectrum level, and could plausibly be proved by extending the methods of §1. We will however only need, and prove, the following weaker statements.

Let ζ_{⊗}: Spin_{⊗} → J_{⊗} be the induced map in the Puppe fibration sequence generated by the fibration J_{⊗} → BO_{⊗} → BSpin_{⊗} defining J_{⊗}. So ζ_{⊗} ∘ Ω(ψ^{3}/1) ≃ ∗. Let t_{⊗}: BSO_{⊗} → J_{⊗} be the composite t_{⊗} = ζ_{⊗} ∘ η, and similarly for t_{⊕}.

Lemma 2.5. The only essential spectrum map f: BSO_{⊗} → J_{⊗} is t_{⊗}, which is nontrivial on homotopy. Hence spectrum maps BSO_{⊗} → J_{⊗} are detected on underlying spaces.

Corollary 2.6. The composite e ∘ s: BSO → SG → J_{⊗} is homotopic as a map of spectra to the composite

[figure: BSO; \overline{Bj}; BSG; \overline{η}; SG; \overline{e}; J_{⊗}.]

Proof. The two maps are homotopic on the space level by Bökstedt and Waldhausen’s result quoted above, hence on the spectrum level by the lemma. □

Proof of Lemma 2.5. Consider the diagram :

BSO_{⊗} ^{ϕ} BSO_{⊗}

[figure: η; f]

[figure: O_{⊗}; \overline{Ω(ψ^{3}/1)}; Spin_{⊗}; \overline{ζ_{⊗}}; J_{⊗}; BO_{⊗}]

As f is trivial on rational homotopy, and BO_{⊗} ≅ BSO_{⊗} × BO(1), the composite of f into BO_{⊗} is null homotopic. Hence there exists a lift of f into Spin_{⊗}, which factors through η by Proposition 2.2. Thus there exists a lifting ϕ, which in additive notation can be written as a series

ϕ ≅ ϵ(ϕ) + ∑_{r,i} a_{r,i}(ψ^{2^{r}+i} − ψ^{i})

by [MST, Lemmas 2.2 and 2.11]. Here a_{r,i} is only nonzero for certain odd i, and ϵ(ϕ) is a scalar multiple of the identity map. Each of the terms on the right factor through ψ^{3}/1, and as ψ^{3}/1 commutes with η, they contribute nothing when composed with ζ_{⊗} [Ro]. Explicitly, ζ_{⊗} ∘ η ∘ (ψ^{3}/1) ≃ ζ_{⊗} ∘ Ω(ψ^{3}/1) ∘ η ≃ ∗. Thus we may make use of the choice in selecting a lifting ϕ of f, and can indeed take ϕ to be a scalar multiple ϵ(ϕ) of the identity. The lemma follows, as η and thus t_{⊗} has order two. □

Recall from [Bö] the diagram of fibration sequences in its multiplicative version :

[figure: BSO_{⊗}; =; ==; BSO_{⊗}; ∗; BBSO_{⊗}]

[figure: η; t_{⊗}; η]

[figure: Spin_{⊗}; \overline{ζ_{⊗}}; J_{⊗}; BO_{⊗}; \overline{ψ^{3}/1}; BSpin_{⊗}] ∥
[figure: c; i_{⊗}; ∥; c]

[figure: SU_{⊗}; JK(ℤ)_{1}; BO_{⊗}; BSU_{⊗}]

The map i_{⊗} is defined by this diagram. To complete the construction of the homotopy commutative diagram (2.8) we need the following :

Lemma 2.7. The composites t_{⊗} ∘ ρ^{3} and e ∘ s are homotopic as maps of spectra.

Proof. We use

[figure: BSO_{⊗}; \overline{η}; Spin_{⊗}; \overline{ζ_{⊗}}; J_{⊗}] [figure]
ρ^{3}  Ωρ^{3}  [figure: e]

[figure: BSO; \overline{η}; Spin; SO]

[figure: Bj; j]

BSG  \overline{η}  \overline{}SG

which commutes by naturality of multiplication by η and the J-theory diagram of [Ma, p. 107] :

[figure: J_{⊕}; BO; \overline{ψ^{3} − 1}; BSpin] ∥
α∨  γ∨  ∥

[figure: Spin; \overline{j}; SG; SG/Spin; BSpin; \overline{Bj}; BSG]

[figure: Ωρ^{3}; e; f; ρ^{3}]

[figure: Spin_{⊗}; \overline{ζ_{⊗}}; J_{⊗}; BO_{⊗}; \overline{ψ^{3}/1}; BSpin_{⊗}]

where α and γ cannot be chosen as H-maps. The lower left square can be recognized in the diagram above. □

We proceed to try to produce a map from JK(ℤ)_{1} to K(ℤ)_{1}, not by using the non–deloopable splitting α, but by lifting the map from the (spectrum level) cofiber g: B fib(s) → K(ℤ)_{1} over B fib(s) → JK(ℤ)_{1}. We have constructed the following diagram in the stable category

(2.8)  [figure: ∗; C_{⊗}; = = =; = = = =; C_{⊗}; BSO; \overline{s}; SG; B fib(s); \overline{g}; K(ℤ)_{1}; ρ^{3}; ≃; e; h; BSO_{⊗}; \overline{t_{⊗}}; J_{⊗}; \overline{i_{⊗}}; JK(ℤ)_{1}]

where the spectrum map h: JK(ℤ)_{1} → K(ℤ)_{1} exists as a lifting if and only if the composite C_{⊗} → B fib(s) → K(ℤ)_{1} is null homotopic. This is the same map as in the hypothesis of Theorem 2.1. The left part of the diagram is a square of fibrations, and the map g: B fib(s) → K(ℤ)_{1} is given by the null homotopy of Lemma 2.4.

We note that Mitchell’s result that C_{⊗} → K(ℤ)_{1} is null homotopic on the space level only suffices to define a map from the mapping cone (space level cofiber) of C_{⊗} → B fib(s) to K(ℤ)_{1}, not from the spectrum level cofiber JK(ℤ)_{1}.

However, by “Ganea theory” [Gan] there exists a tower of obstructions to this space level lifting extension problem, lying in the group of homotopy classes of space level maps C_{⊗} ∗ ΩJK(ℤ)_{1} ∗ ⋯ ∗ ΩJK(ℤ)_{1} to K(ℤ)_{1}. Here ∗ denotes the join of spaces, and there are one or more factors ΩJK(ℤ)_{1}. To see this, consider the tower of spaces

⋮  ⋮

[figure: C_{⊗} ∗ ΩJK(ℤ)_{1} ∗ ΩJK(ℤ)_{1}; \overline{i_{2}}; C(i_{1}); JK(ℤ)_{1}] ∥
∥

[figure: C_{⊗} ∗ ΩJK(ℤ)_{1}; \overline{i_{1}}; C(i_{0}); JK(ℤ)_{1}] ∥
∥

[figure: C_{⊗}; \overline{i_{0}}; B fib(s); JK(ℤ)_{1}]

g

K(ℤ)_{1}

where the rows are fibrations and C(i_{s}) the mapping cone on i_{s}. The map g extends to JK(ℤ)_{1} precisely if all the maps from the fibers to K(ℤ)_{1} are null homotopic. Now for any pointed space X, the set of homotopy classes of maps C_{⊗} ∗X → K(ℤ)_{1} is the same as the set of homotopy classes of maps ΣX → Map(C_{⊗}, K(ℤ)_{1}). Hence the second part of Theorem 2.1 follows if the space Map(C_{⊗}, K(ℤ)_{1}) is contractible, as in [Bö].

It appears unlikely that Mitchell’s argument will extend to resolve these higher obstruction questions. Let us therefore make the following assumption :

Hypothesis 2.9. The composite spectrum map C_{⊗} → SG → K(ℤ)_{1} is null homotopic.

Granted this, a lifting h: JK(ℤ)_{1} → K(ℤ)_{1} of g exists. Bökstedt defines a ring spectrum map Φ: K(ℤ) → JK(ℤ). In fact there exists a commutative square of rings

ℤ  \overline{}ℝ

[figure]

ℤ_{3}  \overline{}ℂ

where ℤ_{3} denotes the three–adic integers, and a two–complete equivalence K(ℤ_{3}) → K(𝔽_{3}) induced by the residue field homomorphism [Gab], which composes with a Brauer lift K(𝔽_{3}) → K(ℂ) to give the map on K-theory induced by the lower horizontal arrow. JK(ℤ) is a cover of the pullback P B of the induced square

[figure: K(ℤ); P B; K(ℝ)]

[figure]

[figure: K(𝔽_{3}); ≃; K(ℤ_{3}); K(ℂ)]

and Φ: K(ℤ) → JK(ℤ) is a lift of the map K(ℤ) → P B.

Lemma 2.10. There is a homotopy commutative diagram of spectra :

SG \overline{l ∘ i_{Q}}\overline{}K(ℤ)_{1}

[figure: e; Φ]

[figure: J_{⊗}; \overline{i_{⊗}}; JK(ℤ)_{1}; BO_{⊗}]

Presumably there is a similar result for the additive spectra. This diagram fits into the middle lower square of the J-theory diagram, as displayed in the proof of Lemma 2.7.

Proof. The right triangle commutes by the definition of Φ. We next consider the left square.

As in [Bö, Proof of 2.1] we have a covering ℤ → JK(ℤ)_{1} → P B_{1}, and as there are no nontrivial homomorphisms π_{1}(SG) ≅ ℤ/2 → ℤ it suffices to compare maps into P B_{1}. This in turn reduces to comparing maps into BO_{⊗} and K(ℤ_{3})_{1} ≅ K(𝔽_{3})_{1}, as well as the homotopy linking the composites into BU_{⊗}.

Both composites SG → K(𝔽_{3})_{1} agree with the map given by taking a finite set to the 𝔽_{3}-vector space it generates, as is seen by the “discrete model” for J_{⊗} described e.g. in [Ma, VIII.3.1].

The composites SG → BO_{⊗} are homotopic on the underlying space level by Bökstedt’s argument, and thus on the spectrum level by [Li] or [Ma, V.7.9].

It remains to compare the homotopies of maps into BU_{⊗}, or equivalently a lifting SG → U_{⊗}. Now there is only one such, the null map, which follows from the K-theoretic equivalence SG → J_{⊗} [HS] and the lemma below. □

Lemma 2.11. There are no essential spectrum maps f: J_{⊗} → U_{⊗}.

Proof. Consider maps of the fibration sequence below to U_{⊗} :

[figure: Spin_{⊗}; \overline{ζ_{⊗}}; J_{⊗}; BO_{⊗}]

J_{⊗} is rationally a point, so f ∘ζ_{⊗} is null on rational homotopy. By the second part of Corollary 2.3 (desuspended two degrees) it follows that f extends over J_{⊗} → BO_{⊗}. The extension BO_{⊗} → U_{⊗} is trivial, because of the first part of Corollary 2.3 and the absence of maps BO(1) → U_{⊗} [AH] or BSO → U(1). □

We may now prove Theorem 2.1. Let P X denote the contractible path space of a based space X. Again adapting Bökstedt’s proof, we consider the diagram of vertical fibrations

[figure: BSO_{⊗}; \begin{matrix}ρ^{3} \\ ≃ \end{matrix}; BSO; P K(ℤ)_{1}; \overline{≃}; P JK(ℤ)_{1}]

[figure: t_{⊗}; s]

[figure: J_{⊗}; e; SG; K(ℤ)_{1}; \overline{Φ}; JK(ℤ)_{1}] ∥  ∥
[figure: i_{⊗}; ∥; ∥]

[figure: JK(ℤ)_{1}; B fib(s); K(ℤ)_{1}; \overline{Φ}; JK(ℤ)_{1}]

where there exists unique lifting maps from the left to the right by the K-acyclicity of C_{⊗} = fib(e) and the fact that JK(ℤ)_{1} is the connected cover of its K-localization. If we assume Hypothesis 2.9 the bottom lifting map may be taken to be Φ ∘ h. By Lemma 2.10 the middle lifting map is homotopic to i_{⊗}: J_{⊗} → JK(ℤ)_{1}, and the three horizontal fibers form a fibration

[figure: BSO_{⊗}; \overline{a}; BSO_{⊗}; fib(Φ ∘ h)]

where t_{⊗} ∘ a ≃ t_{⊗}. By [Bö, Lemma 1.10], the map a is a homotopy equivalence of spaces, and thus of spectra. This again uses essentially that we are in a two– complete situation. Hence fib(Φ∘h) is contractible and Φ is a split surjection. This completes the proof of Theorem 2.1.

Remark 2.12. Bökstedt, Hsiang and Madsen [BHM] have constructed a cyclotomic trace map trc: K(ℤ) → T C(ℤ, p) for each prime p. Suppose, as follows for p odd from Hypothesis 4.15 in [BM], that T C(ℤ, p) is the connected cover of its Klocalization. Then the analog of Hypothesis 2.9 would be satisfied for maps into T C(ℤ, p), and the constructions above may provide a factorization of trc through Φ: K(ℤ)_{1} → JK(ℤ)_{1}. Such a factorization would exist by Thomason’s theorem [Th] and the K-locality hypothesis cited.

Remark 2.13. Let G/O_{⊗} ≃ BSO_{⊗} × C_{⊗} be defined as the pullback in a square

G/O_{⊗}  \overline{}SG

[figure: e]

[figure: BSO_{⊗}; \overline{η}; Spin_{⊗}; \overline{ζ_{⊗}}; J_{⊗}.]

Miller and Priddy defined G/O_{⊕} as the pullback in the analogous additive diagram, where there is no infinite loop section Spin_{⊕} → Q(S^{0})_{0}. They conjectured that G/O_{⊕} ≃ G/O as infinite loop spaces [MP, Conjecture A, p. 339]. In view of the splitting above, and Theorem 3.4 below, G/O_{⊗} cannot be equivalent to either of these.

Thus there are distinct fibration sequences

G/O_{⊕} ⟶ Q(S^{0})_{0} ⟶ JK(ℤ)_{0}
G/O_{⊗} ⟶ SG ⟶ JK(ℤ)_{1}

related to the fundamental E_{∞}-ring spectrum map Q(S^{0}) → K(ℤ).

The composite Φ ∘ l: A(∗)_{1} → K(ℤ)_{1} → JK(ℤ)_{1} induces a map ΩW h_{⊗}^{Diff}(∗) → G/O_{⊗} on the fibers of the natural maps from SG into A(∗)_{1} or JK(ℤ)_{1} respectively. The composite G/O → ΩW h_{⊗}^{Diff}(∗) → G/O_{⊗} is a rational equivalence, but fails to be an equivalence on the K-acyclic part, as is seen by mapping C_{⊗} into this sequence.

§3. Twice a splitting map

In this section we shall consider the spectrum self maps of JK(ℤ). All spectra are still completed at two, and we continue to work in the stable category.

We will compare the spectrum structures on the additive and multiplicative components JK(ℤ)_{0} and JK(ℤ)_{1}, and note that their one–connected coverings J̃K(ℤ)_{0} and J̃K(ℤ)_{1} are equivalent as spectra. We use J̃K(ℤ) to denote the additive version, as a representative for either.

In Proposition 3.2 we prove that spectrum self maps of J̃K(ℤ) are detected on homotopy groups. As an application we determine the algebra of such spectrum self maps, as the quotient algebra of the algebra End(BSU) of spectrum self maps of BSU determined in [MST], by the closed two–sided ideal generated by an operation (ψ^{3} − 1)(1 + ψ^{−1}). See Theorem 3.3.

In Theorem 3.4 we give a proof due to Tornehave for the fact that the cofiber map J_{⊗} → BC_{⊗} corresponding to the fibration sequence C_{⊗} → SG → J_{⊗} has order precisely two as a spectrum map. Hence twice the identity map of J_{⊗} factors through e: SG → J_{⊗}, and an explicit factorization can be constructed using Friedlander’s spectrum level solution to the complex Adams conjecture.

In a similar manner we can construct a spectrum map 2h: JK(ℤ)_{1} → K(ℤ)_{1}. By the results above, this provides a unique infinite delooping of twice Bökstedt’s splitting map

Ωh: ΩΩ^{∞}JK(ℤ) ⟶ ΩΩ^{∞}K(ℤ).

See Theorem 3.5.

Technically it is easiest to work with the one–connected coverings of the spectra of interest in all these considerations. Finally, in Proposition 3.6 we remark on the modifications needed to consider self maps of a full component of JK(ℤ).

We use the following notation. A subscript denotes a component, and a tilde denotes the one–connected cover. BSpinU is the fiber of the essential composite BU → K(ℤ, 2) → K(ℤ/2, 2). J̃U is thus the fiber of ψ^{3} − 1: BU → BSpinU. J̃K(ℤ) fits into fibration sequences J̃K(ℤ) → BSO → BSU and BSO → J̃ → J̃K(ℤ). There are similar multiplicative sequences.

Proposition 3.1.

The cannibalistic equivalences ρ^{3} : BSO → BSO_{⊗} and ρ^{3} : BSU → BSU_{⊗} induce a homotopy equivalence, also denoted ρ^{3}, between the one–connected covers of JK(ℤ)_{0} and JK(ℤ)_{1}.

The two extensions J̃K(ℤ) → JK(ℤ)_{0} → BO(1) and J̃K(ℤ) → JK(ℤ)_{1} → BO(1) realize the two elements in [O(1), J̃K(ℤ)] ≅ ℤ/2. Only the latter extension is split.

There are no essential spectrum maps BO(1) → J̃K(ℤ).

Proof. The cannibalistic equivalences ρ^{3} : BSO → BSO_{⊗} and ρ^{3} : BSU → BSU_{⊗} commute with the relevant Adams operations and complexification, thus inducing a homotopy equivalence on the homotopy fibers of the horizontal composites in the diagram below. These are the one–connected covers of JK(ℤ)_{0} and JK(ℤ)_{1}.

[figure: BSO; \overline{ψ^{3} − 1}; BSpin; \overline{c}; BSU]

[figure: ≃; ρ^{3}; ≃; ρ^{3}; ≃; ρ^{3}]

[figure: BSO_{⊗}; \overline{ψ^{3}/1}; BSpin_{⊗}; \overline{c}; BSU_{⊗}]

Alternatively one may use the composite BSO → Spin → J̃_{⊕} and the corresponding map from the multiplicative case, and see commutation from the J-theory diagram.

From [AH] there are no spectrum maps from the Eilenberg–MacLane spectra K(ℤ, n) or K(ℤ/2, n) to the periodic spectra K or KO. From the Hurewicz theorem and knowledge of the Steenrod algebra it follows easily that [O(1), SU] ≅ 0, [O(1), BSO] ≅ ℤ/2, and [O(1), BSU] ≅ ℤ/2, with the two elements in [O(1), BSO] represented by the extensions BSO → BO → BO(1) and BSO_{⊗} → BO_{⊗} → BO(1), where only the latter is split.

From the extension SU → J̃K(ℤ) → BSO we find that [O(1), J̃K(ℤ)] is the kernel of the homomorphism

ℤ/2 ≅ [O(1), BSO] ⟶ [O(1), BSU] ≅ ℤ/2.

The map is induced by left composition with c ∘ (ψ^{3} − 1): BSO → BSpin → BSU, where ψ^{3} − 1 extends through BO. So the image of the nonzero element factors through the null homotopic composite O(1) → BSO → BO, hence is zero.

Thus [O(1), J̃K(ℤ)] ≅ ℤ/2. The proof of [BO(1), J̃K(ℤ)] ≅ 0 is similar but easier.

It only remains to note that JK(ℤ)_{0} → BO(1) is not split as it factors JK(ℤ)_{0} → BO → BO(1), while JK(ℤ)_{1} → BO(1) is split via the inclusion of 1 × 1-matrices BO(1) → K(ℤ)_{1} → JK(ℤ)_{1}. □

Let v be the self map of J̃K(ℤ) making the diagram below commute

[figure: J̃K(ℤ); BSO; \overline{ψ^{3} − 1}; BSpin; \overline{c}; BSU]

[figure: v; 2; 2; r; 1 + ψ^{−1}]

[figure: J̃K(ℤ); BSO; \overline{ψ^{3} − 1}; BSpin; \overline{c}; BSU]

where ψ^{−1} : BSU → BSU denotes complex conjugation. r ∘ c = 2, while c ∘ r = 1 + ψ^{−1}.

Proposition 3.2. The rationally trivial spectrum maps f: J̃K(ℤ) → J̃K(ℤ) are precisely the two–adic scalar multiples of v, all of which are distinct. These are all detected on homotopy groups. Hence f is null homotopic if and only if π_{∗}(f) = 0.

Proof. Consider the extension SU → J̃K(ℤ) → BSO. Since there are no essential maps SU → BSO (Corollary 2.3), f fits into a ladder

[figure: SU; J̃K(ℤ); BSO]

[figure: f]

[figure: SU; J̃K(ℤ); BSO]

which we may extend to the left into a diagram

[figure: SO; \overline{ψ^{3} − 1}; Spin; \overline{c}; SU; J̃K(ℤ)]

[figure: x; y; u; z; f]

[figure: SO; \overline{ψ^{3} − 1}; Spin; \overline{c}; SU; J̃K(ℤ)]

[figure: η]

BSO  BBSO

which we know commutes if we omit the maps labeled y and u. The bottom part of the diagram is the fibration sequence arising from U/O ≃ B(ℤ × BO).

Let us first assume that f is rationally trivial. Then z: SU → SU is rationally trivial in degrees congruent to 1 modulo 4, since SU → J̃K(ℤ) is a rational equivalence in these degrees. So the composite

[figure: SU; \overline{z}; SU; BBSO]

is rationally trivial, hence null homotopic by Corollary 2.3. Thus there exists a lifting u: SU → Spin factoring z, with c ∘ u = z. u is unique as there are no maps SU → BSO. We define y = u ∘ c: Spin → Spin.

We claim that the left square commutes. The composites (ψ^{3} − 1) ∘ x and y ∘ (ψ^{3} − 1) become homotopic when continued through c to SU. As there are no choices of liftings SO → BSO, they are also homotopic already as maps into Spin. This proves the claim, so the whole diagram above commutes.

Next we claim that any map SU → Spin extends over realification r: SU → Spin. From the homotopy equivalence from real Bott periodicity, ΩO ≃ O/U, we obtain a fibration sequence

[figure: B^{5}(ℤ × BO); SU; \overline{r}; Spin]

and by Proposition 2.2 there are no essential spectrum maps B^{5}(ℤ × BO) → Spin, whence such a factorization always exists (uniquely).

In particular u factors as a composite w ∘ r for some real operation w: Spin → Spin, which we may write as the sum of a scalar multiple of the identity, and an operation factoring through ψ^{3} − 1.

With choices of x and z such that u factors through ψ^{3} − 1, the induced map f is null homotopic. (This follows from a formal chase.) Hence we may alter w by the summand factoring through ψ^{3} − 1 without altering f. So we may assume u is a scalar multiple nr of the realification map. But then f is precisely nv, where v was defined before the statement of the theorem.

Finally we note that nv is nontrivial on some homotopy groups for n different from zero. In degree 8i − 1 the group JK_{8i−1}(ℤ) is cyclic of order 3^{4i} − 1, which has arbitrarily large two–valuation for suitable i. (To see this, note that c∘(ψ^{3} −1) induces multiplication by this scalar from π_{8i}(BSO) ≅ ℤ to π_{8i}(BSU) ≅ ℤ.) Also π_{8i−1}(nv) is multiplication by 2n, which thus is nonzero for i such that the two– valuation of 2n is less than that of 3^{4i} − 1. Hence the spectrum self maps of J̃K(ℤ) are detected on homotopy groups. □

Let End(BSU) be the algebra of homotopy classes of spectrum maps BSU → BSU. Its elements are determined in [MST, 2.3 and Theorem 2.9] as the series

ϵ(ϕ) + ∑_{r≥0} ∑_{i≥0} a_{r,i}(ψ^{2^{r}+i} − ψ^{i})

where ϵ(ϕ) is a scalar multiple of the identity, each a_{r,i} is a two–adic integer, and a_{r,i} = 0 if i ≥ 2^{r+1} − 2^{r} or if i is even.

Let End(J̃K(ℤ)) be the algebra of homotopy classes of spectrum maps J̃K(ℤ) → J̃K(ℤ).

Theorem 3.3. End(J̃K(ℤ)) is the quotient algebra of End(BSU) by the closed two–sided ideal generated by (ψ^{3} − 1)(1 + ψ^{−1}).

Proof. With the notation of the preceding proof, every spectrum map f: J̃K(ℤ) → J̃K(ℤ) is determined by two spectrum maps z: SU → SU and x: SO → SO, where in turn z determines x rationally, hence up to homotopy. Homotopy classes of spectrum self maps of SU correspond bijectively to homotopy classes of spectrum self maps of BSU under (de)looping, and similarly in the real case.

The z which determine a null homotopic f are precisely those which factor through c ∘ (ψ^{3} − 1), i.e. the ideal in End(BSU) generated by the composite c ∘ (ψ^{3} − 1) ∘ r. The result then follows from the relation c ∘ r = 1 + ψ^{−1}. □

J. Tornehave proved the following result, following a discussion with M. Bökstedt where the author played a minor part.

Theorem 3.4. (Tornehave) The cofiber map J_{⊗} → BC_{⊗} of e: SG → J_{⊗} has order precisely two as a spectrum map.

Proof. At odd primes, the map is null homotopic, while at two it is not. Localized at two we have a diagram of spectra

[figure: C_{⊗}; SG; \overline{e}; J_{⊗}; BC_{⊗}]

[figure]

[figure: BSO; SG/Spin; \overline{f}; BO_{⊗}] [figure]
[figure: c] γ_{ℂ}  ρ^{3}

BU  \overline{r} \overline{}BSO

where γ_{ℂ} is Friedlander’s spectrum level lifting of a solution to the Adams conjecture to BU [Fr, Theorem 9.2]. The composite r ∘ c: BSO → BSO is homotopic to twice the identity, and the composite SG/Spin → BO_{⊗} → BC_{⊗} is null homotopic. Thus BSO → BC_{⊗} on the right side has order two. ρ^{3} maps as an equivalence to BSO_{⊗}, so BSO_{⊗} → BC_{⊗} has order two, and it remains to prove the same for the factor BO(1) → BO_{⊗} → BC_{⊗}. But this is obvious, as the identity map of BO(1) has order two. Consequently J_{⊗} → BO_{⊗} → BC_{⊗} also has order two. □

We now turn to infinitely delooping twice Bökstedt’s section. First note that the Adams operations commute with realification when restricted to the additive zero component BU. In particular we have horizontal fibration sequences in the diagram below,

[figure: J̃; BSO; \overline{ψ^{3} − 1}; BSpin]

[figure: c; c; c]

[figure: J̃U; BU; \overline{ψ^{3} − 1}; BSpinU]

[figure: r; r; r]

[figure: J̃; BSO; \overline{ψ^{3} − 1}; BSpin]

where the vertical composites are multiplication by two.

Next, Friedlander’s solution to the complex Adams conjecture gives spectrum maps α_{ℂ}: J̃U → S̃G and γ_{ℂ}: BU → S̃G/Spin, making the following diagram commute :

[figure: J̃U; BU; \overline{ψ^{3} − 1}; BSpinU]
[figure: α_{ℂ}; r; γ_{ℂ}; r; r; r]
[figure: J̃; BSO; \overline{ψ^{3} − 1}; BSpin]
[figure: id]
[figure: S̃G; S̃G/Spin; \overline{ψ^{3} − 1}; BSpin]
[figure: e; f; σ^{3}; ρ^{3}; ρ^{3}]
[figure: J̃_{⊗}; BSO_{⊗}; \overline{ψ^{3} − 1}; BSpin_{⊗}]

On the space level, α_{ℂ} ∘ c ≃ α ∘ 2: J̃ → S̃G, where α is a (non–deloopable) lifting of a solution to the real Adams conjecture. Hence we obtain the diagram of spectra below. The left side consists of horizontal fibration sequences induced by maps assembled from the two preceding diagrams. The lower right hand triangle commutes by Lemma 2.10.

[figure: BSO; \overline{t_{⊕}}; J̃; J̃K(ℤ)_{0}]

[figure: 2 = r ∘ c; α_{ℂ} ∘ c; 2h ∘ ρ^{3}]

[figure: BSO; \overline{s}; S̃G; B fib(s); \overline{g}; K̃(ℤ)_{1}]

[figure: ρ^{3}; ≃; e; Φ]

[figure: BSO_{⊗}; \overline{t_{⊗}}; J̃_{⊗}; J̃K(ℤ)_{1}]

The map denoted 2h∘ρ^{3} is defined as the composite making the right part of the diagram commute. We can factor it uniquely through the equivalence ρ^{3} : J̃K(ℤ)_{0} → J̃K(ℤ)_{1} to obtain a spectrum map

2h: J̃K(ℤ)_{1} ⟶ K̃(ℤ)_{1},

with some abuse of notation.

In view of the splittings JK(ℤ)_{1} ≃ J̃K(ℤ)_{1} × BO(1) and K(ℤ)_{1} ≃ K̃(ℤ)_{1} × BO(1), it is clear that 2h can be extended to the full component JK(ℤ)_{1}, mapping arbitrarily on π_{1}. Hence we have proved :

Theorem 3.5. There is a spectrum map 2h: JK(ℤ)_{1} → K(ℤ)_{1} whose underlying map on looped spaces

ΩΩ^{∞}(2h): ΩJK(ℤ)_{1} ⟶ ΩK(ℤ)_{1}

is homotopic to twice Bökstedt’s section [Bö, 1.6] to the natural map Φ: K(ℤ) → JK(ℤ). The composite Φ∘2h is homotopic to twice a homotopy equivalence JK(ℤ)_{1} → JK(ℤ)_{1}. □

It remains to discuss which spectrum self maps of J̃K(ℤ) extend to spectrum maps JK(ℤ)_{i} → JK(ℤ)_{j} for i, j = 0, 1. We can write such a self map f as the sum of a scalar multiple ϵ(f) times the identity, plus an operation factoring through ψ^{3} − 1.

Proposition 3.6. A spectrum self map f of J̃K(ℤ) extends to a spectrum map f′: JK(ℤ)_{i} → JK(ℤ)_{j}, unless i = 0, j = 1 and ϵ(f) is a two–adic unit.

If j = 0, such an extension is unique. If j = 1 there are two possible extensions, distinguished by the induced map on π_{1}. In particular, spectrum maps JK(ℤ)_{i} → JK(ℤ)_{j} are detected on homotopy groups.

Proof. Consider the following diagram, where initially only f is given. An extension f′ of f exists precisely if of one of the two possible maps f′′: O(1) → O(1) makes the left square commute.

[figure: O(1); J̃K(ℤ); JK(ℤ)_{i}; BO(1)]

[figure: f′′; f; f′]

[figure: O(1); J̃K(ℤ); JK(ℤ)_{j}; BO(1)]

If i = 1 we can take f′′ to be trivial and have the left square commute, so an extension f′ always exists. If i = 0, we note that the summand of f factoring through ψ^{3} − 1 also factors through some map JK(ℤ)_{i} → J̃K(ℤ), because ψ^{3} − 1 acts trivially on π_{∗} in the lowest degrees. Hence the extension problem is equivalent to the one where we replace f by the scalar multiple ϵ(f) of the identity. Then an f′′ making the left square commute exists precisely if j = 0, or if ϵ(f) is divisible by two. This completes the characterization of when the extension f′ exists.

For uniqueness, suppose f ≃ ∗ and j = 0. Then f′ factors through a map BO(1) → JK(ℤ)_{0}, which must be null homotopic, since otherwise the composite BO(1) → JK(ℤ)_{0} → BO(1) would provide an impossible splitting. Hence f′ ≃ ∗.

On the other hand, if j = 1 we are free to alter an extension f′: JK(ℤ)_{i} → JK(ℤ)_{1} by the essential composite JK(ℤ)_{i} → BO(1) → JK(ℤ)_{1}. This gives two distinct extensions, which are distinguished by their effect on π_{1}. □

References

[Ad] J. F. Adams, Infinite loop spaces, Ann. of Math. Studies, vol. 90, Princeton University Press, 1978.

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Department of Mathematics, University of Oslo, Postboks 1053, Blindern, 0316 22 JOHN ROGNES Oslo, Norway.