New York Journal of Mathematics

New York J. Math. 14 (2008) 379–392.

v_{1}-periodic homotopy groups of the Dwyer–Wilkerson space

Martin Bendersky and Donald M. Davis

Abstract. The Dwyer–Wilkerson space DI(4) is the only exotic 2compact group. We compute v_{1}^{−1}π_{∗}(DI(4)), its v_{1}-periodic homotopy groups.

Contents

1. Introduction 379 2. Adams operations 381 3. Relationship with pseudosphere 384 4. Determination of KO^{∗}(Φ_{1}DI(4)) 387 References 391

1. Introduction

In [13], Dwyer and Wilkerson constructed a 2-complete space BDI(4), so named because its F_{2}-cohomology groups form an algebra isomorphic to the ring of rank-4 mod-2 Dickson invariants. Its loop space, called DI(4), has H^{∗}(DI(4); F_{2}) finite. In [14], they then defined a p-compact group to be a pair (X, BX), such that X = ΩBX (hence X is redundant), BX is connected and p-complete, and H^{∗}(X; F_{p}) is finite. In [1], Andersen and Grodal proved that (DI(4), BDI(4)) is the only simple 2-compact group not arising as the 2-completion of a compact connected Lie group.

The p-primary v_{1}-periodic homotopy groups of a topological space X, defined in [12] and denoted v_{1}^{−1}π_{∗}(X)_{(p)} or just v_{1}^{−1}π_{∗}(X) if the prime is clear, are a first approximation to the p-primary homotopy groups. Roughly, they are a localization of the portion of the actual homotopy groups detected

379^{ISSN 1076-9803/08}

Received June 7, 2007.

Mathematics Subject Classification. 55Q52, 57T20, 55N15.

Key words and phrases. v_{1}-periodic homotopy groups, p-compact groups, Adams operations, K-theory.

by p-local K-theory. In [11], the second author completed a 13-year project, often in collaboration with the first author, of determining v_{1}^{−1}π_{∗}(X)_{(p)} for all compact simple Lie groups and all primes p.

In this paper, we determine the 2-primary groups v_{1}^{−1}π_{∗}(DI(4)). Here and throughout, ν(−) denotes the exponent of 2 in the prime factorization of the absolute value of an integer.

Theorem 1.1. For any integer i, let e_{i} = min(21, 4 + ν(i − 90627)). Then

v_{1}^{−1}π_{8i+d}(DI(4)) ≈ \begin{cases}ℤ/2^{e_{i}} ⊕ ℤ/2 & d = 1 \\ ℤ/2^{e_{i}} & d = 2 \\ 0 & d = 3, 4 \\ ℤ/8 & d = 5 \\ ℤ/8 ⊕ ℤ/2 & d = 6 \\ ℤ/2 ⊕ ℤ/2 ⊕ ℤ/2 & d = 7, 8. \end{cases}

Since every v_{1}-periodic homotopy group is a subgroup of some actual homotopy group, this result implies that exp_{2}(DI(4)) ≥ 21, i.e., some homotopy group of DI(4) has an element of order 2^{21}. It would be interesting to know whether this bound is sharp.

Our proof involves studying the spectrum Φ_{1}DI(4) which satisfies

π_{∗}(Φ_{1}DI(4)) ≈ v_{1}^{−1}π_{∗}(DI(4)).

We will relate Φ_{1}DI(4) to the 2-completed K-theoretic pseudosphere T_{K/2} discussed in [8, 8.6]. We will prove the following surprising result, which was pointed out by Pete Bousfield.

Theorem 1.2. There is an equivalence of spectra

Φ_{1}DI(4) ≃ Σ^{725019}T_{K/2} ∧ M(2^{21}),

where M(2^{21}) is a mod 2^{21} Moore spectrum.

In Section 3, we will give the easy deduction of Theorem 1.1 from Theorem 1.2. As an immediate corollary of 1.2, we deduce that the 2^{21} bound on π_{∗}(Φ_{1}DI(4)) is induced from a bound on the spectrum itself.

Corollary 1.3. The exponent of the spectrum Φ_{1}DI(4) is 2^{21}; i.e., 2^{e}1_{Φ_{1}DI(4)} is null if and only if e ≥ 21.

v_{1}-periodic homotopy groups of DI(4) 381

In [5], Bousfield presented a framework that enables determination of the v_{1}-periodic homotopy groups of many simply-connected H-spaces X from their united K-theory groups and Adams operations. The intermediate step is KO^{∗}(Φ_{1}X). (All of our K^{∗}(−) and KO^{∗}(−)-groups have coefficients in the 2-adic integers ℤ̂_{2}, which we omit from our notation.) Our first proof of Theorem 1.1 used Bousfield’s exact sequence [5, 9.2] which relates v_{1}^{−1}π_{∗}(X) with KO^{∗}(Φ_{1}X), but the approach via the pseudosphere, which we present here, is stronger and more elegant. The insight for Theorem 1.2 is the observation that the two spectra have isomorphic Adams modules KO^{∗}(−).

In several earlier e-mails, Bousfield explained to the authors how the results of [5] should enable us to determine KO^{∗}(Φ_{1}DI(4)). In Section 4, we present our account of these ideas of Bousfield. We thank him profusely for sharing his insights with us.

The other main input is the Adams operations in K^{∗}(BDI(4)). In [18], Osse and Suter showed that K^{∗}(BDI(4)) is a power series algebra on three specific generators, and gave some information toward the determination of the Adams operations. In private communication in 2005, Suter expanded on this to give explicit formulas for ψ^{k} in K^{∗}(BDI(4)). We are very grateful to him for sharing this information. In Section 2, we will explain these calculations and also how they then lead to the determination of KO^{∗}(Φ_{1}DI(4)).

2. Adams operations

In this section, we present Suter’s determination of ψ^{k} in K^{∗}(BDI(4)) and state a result, proved in Section 4, that allows us to determine KO^{∗}(Φ_{1}DI(4)) from these Adams operations.

Our first result, communicated by Suter, is the following determination of Adams operations in K^{∗}(BDI(4)). An element of K^{∗}(X) is called real if it is in the image of KO^{∗}(X) →^{c} K^{∗}(X).

Theorem 2.1 (Suter). There is an isomorphism of algebras

(2.2)  K^{∗}(BDI(4)) ≈ ℤ̂_{2}[[ξ_{8}, ξ_{12}, ξ_{24}]]

such that the generators are in K^{0}(−) and are real, ψ^{−1} = 1, and the matrices of ψ^{2} and ψ^{3} on the three generators, mod decomposables, are

Ψ^{2} ≡ (\begin{matrix}2^{4} & 0 & 0 \\ −2 & 2^{6} & 0 \\ 0 & −2 & 2^{14} \end{matrix}) , Ψ^{3} ≡ (\begin{matrix}3^{4} & 0 & 0 \\ −3^{3} & 3^{6} & 0 \\ 36/527 & −3^{5} · 41/17 & 3^{14} \end{matrix}) .

Proof. The subscripts of the generators indicate their “filtration,” meaning the dimension of the smallest skeleton on which they are nontrivial. A standard property of Adams operations is that if ξ has filtration 2r, then ψ^{k}(ξ) equals k^{r}ξ plus elements of higher filtration.

The isomorphism (2.2) is derived in [18, p. 184] along with the additional information that 4ξ_{24} − ξ_{12}^{2} has filtration 28, and

(2.3)  ξ_{12} = λ^{2}(ξ_{8}) + 8ξ_{8}
ξ_{24} = λ^{2}(ξ_{12}) + 32ξ_{12} + c_{1}ξ_{8}^{2} + c_{2}ξ_{8}^{3} + c_{3}ξ_{8}ξ_{12},

for certain explicit even coefficients c_{i}.

The Atiyah–Hirzebruch spectral sequence easily shows that ξ_{8} is real, since the 11-skeleton of BDI(4) equals S^{8}. Since λ^{2}(c(θ)) = c(λ^{2}(θ)), and products of real bundles are real, we deduce from (2.3) that ξ_{12} and ξ_{24} are also real. Since tc = c, where t denotes conjugation, which corresponds to ψ^{−1}, we obtain that the generators are invariant under ψ^{−1}, and hence so is all of K^{∗}(BDI(4)).

We compute Adams operations mod decomposables, writing ≡ for equivalence mod decomposables. Because 4ξ_{24} − ξ_{12}^{2} has filtration 28, we obtain

(2.4)  ψ^{k}(ξ_{24}) ≡ k^{14}ξ_{24}.

Here we use, from [18, p. 183], that all elements of K^{∗}(BDI(4)) of filtration greater than 28 are decomposable. Equation (2.4) may seem surprising, since ξ_{24} has filtration 24, but there is a class ξ_{28} such that 4ξ_{24} − ξ_{12}^{2} = ξ_{28}, and we can have ψ^{k}(ξ_{24}) ≡ k^{12}ξ_{24} + α_{k}ξ_{28} consistently with (2.4).

Using (2.3) and that ψ^{2} ≡ −2λ^{2} mod decomposables, we obtain

(2.5)  ψ^{2}(ξ_{8}) ≡ 2^{4}ξ_{8} − 2ξ_{12}
ψ^{2}(ξ_{12}) ≡ 2^{6}ξ_{12} − 2ξ_{24},

yielding the matrix Ψ^{2} in the theorem.

Applying ψ^{2}ψ^{3} = ψ^{3}ψ^{2} to ψ^{3}(ξ_{12}) ≡ 3^{6}ξ_{12} + γξ_{24} yields −2 · 3^{6} + 2^{14}γ = 2^{6}γ − 2 · 3^{14}, from which we obtain γ = −3^{5} · 41/17. Applying the same relation to ψ^{3}(ξ_{8}) = 3^{4}ξ_{8} + αξ_{12} + βξ_{24}, coefficients of ξ_{12} yield −2 · 3^{4} + α · 2^{6} = 2^{4}α − 2 · 3^{6} and hence α = −3^{3}. Now coefficients of ξ_{24} yield −2α + 2^{14}β = 2^{4}β − 2γ and hence β = 36/527. □

Let Φ_{1}(−) denote the functor from spaces to K/2_{∗}-local spectra described in [5, 9.1], which satisfies v_{1}^{−1}π_{∗}X ≈ π_{∗}τ_{2}Φ_{1}X, where τ_{2}Φ_{1}X is the 2-torsion part of Φ_{1}X. In Section 4, we will use results of Bousfield in [5] to prove the following result. Aspects of Theorem 2.1, such as K^{∗}(BDI(4)) being a power series algebra on real generators, are also used in proving this theorem.

Recall that KO^{∗}(−) has period 8.

Theorem 2.6. The groups KO^{i}(Φ_{1}DI(4)) are 0 if i ≡ 0, 1, 2 mod 8, and K^{0}(Φ_{1}DI(4)) = 0. Let M denote a free ℤ̂_{2}-module on three generators, acted on by ψ^{2} and ψ^{3} by the matrices of Theorem 2.1, with ψ^{−1} = 1. Let θ = \frac{1}{2}ψ^{2} act on M. Then there are exact sequences

0 → 2M →^{θ} 2M → KO^{3}(Φ_{1}DI(4)) → 0 → 0 → KO^{4}(Φ_{1}DI(4)) → M/2

→^{θ} M/2 → KO^{5}(Φ_{1}DI(4)) → M/2 →^{θ} M/2 → KO^{6}(Φ_{1}DI(4)) → M

→^{θ} M → KO^{7}(Φ_{1}DI(4)) → 0

and

0 → M →^{θ} M → K^{1}(ΦDI(4)) → 0.

For k = −1 and 3, the action of ψ^{k} in KO^{2j−1}(Φ_{1}DI(4)), KO^{2j−2}(Φ_{1}DI(4)), and K^{2j−1}(ΦDI(4)) agrees with k^{−j}ψ^{k} in adjacent M-terms.

In the remainder of this section, we use 2.1 and 2.6 to give explicit formulas for the Adams module KO^{i}(Φ_{1}DI(4)). A similar argument works for K^{∗}(Φ_{1}DI(4)). If g_{1}, g_{2}, and g_{3} denote the three generators of M, then the action of θ is given by

θ(g_{1}) = 8g_{1} − g_{2}
θ(g_{2}) = 2^{5}g_{2} − g_{3}
θ(g_{3}) = 2^{13}g_{3}.

Clearly θ is injective on M and 2M. We have

KO^{7}(Φ_{1}DI(4)) ≈ coker(θ|M) ≈ ℤ/2^{21}

with generator g_{1}; note that g_{2} = 2^{3}g_{1} in this cokernel, and then g_{3} = 2^{8}g_{1}. Similarly KO^{3}(Φ_{1}DI(4)) ≈ coker(θ|2M) ≈ ℤ/2^{21}. Also

KO^{4}(Φ_{1}DI(4)) ≈ ker(θ|M/2) = ℤ/2

with generator g_{3}, while KO^{6}(Φ_{1}DI(4)) ≈ coker(θ|M/2) = ℤ/2 with generator g_{1}. There is a short exact sequence

0 → coker(θ|M/2) → KO^{5}(Φ_{1}DI(4)) → ker(θ|M/2) → 0,

with the groups at either end being ℤ/2 as before. To see that this short

exact sequence is split, we use the map S^{7} →^{f} DI(4) which is inclusion of the bottom cell. The morphism f^{∗} sends the first summand of KO^{5}(Φ_{1}DI(4)) to one of the two ℤ/2-summands of KO^{5}(Φ_{1}S^{7}), providing a splitting homomorphism. Thus we have proved the first part of the following result.

Theorem 2.7. We have

K^{i}(Φ_{1}DI(4)) ≈ \begin{cases}0 & i = 0 \\ ℤ/2^{21} & i = 1, \end{cases}

KO^{i}(Φ_{1}DI(4)) ≈ \begin{cases}0 & i = 0, 1, 2 \\ ℤ/2^{21} & i = 3, 7 \\ ℤ/2 & i = 4, 6 \\ ℤ/2 ⊕ ℤ/2 & i = 5. \end{cases}

For k = −1 and 3, we have ψ^{k} = 1 on the ℤ/2’s, and on KO^{2j−1}(Φ_{1}DI(4)) with j even and K^{2j−1}(Φ_{1}DI(4)), ψ^{−1} = (−1)^{j} and

ψ^{3} = 3^{−j}(3^{4} − 3^{3} · 2^{3} + \frac{36}{527}2^{8}).

Completion of proof. To obtain ψ^{3} on the ℤ/2’s, we use the last part of Theorem 2.6 and the matrix Ψ^{3} of Theorem 2.1. If ψ^{3} is as in Ψ^{3}, then, mod 2, ψ^{3} − 1 sends g_{1} ↦ g_{2}, g_{2} ↦ g_{3}, and g_{3} ↦ 0. Thus ψ^{3} − 1 equals 0 on KO^{4}(Φ_{1}DI(4)) and KO^{6}(Φ_{1}DI(4)). Clearly ψ^{−1} = 1 on these groups.

To see that ψ^{k}−1 is 0 on KO^{5}(Φ_{1}DI(4)), we use the commutative diagram

0 ⟶ ℤ/2 ⟶^{i} KO^{5}(Φ_{1}DI(4)) ⟶^{ρ} ℤ/2 ⟶ 0
≈↓  ^{f^{∗}}↓  ^{0}↓
0 ⟶ ℤ/2 ⟶ KO^{5}(Φ_{1}S^{7}) ⟶ ℤ/2 ⟶ 0.

We can choose generators G_{1} and G_{2} of KO^{5}(Φ_{1}DI(4)) ≈ ℤ/2 ⊕ ℤ/2 so that G_{1} ∈ im(i), ρ(G_{2}) ≠ 0, and f^{∗}(G_{2}) = 0. Since ψ^{k} − 1 = 0 on the ℤ/2’s on either side of KO^{5}(Φ_{1}DI(4)), the only way that ψ^{k} − 1 could be nonzero on KO^{5}(Φ_{1}DI(4)) is if (ψ^{k} − 1)(G_{2}) = G_{1}. However this yields the contradiction

0 = (ψ^{k} − 1)f^{∗}G_{2} = f^{∗}(ψ^{k} − 1)G_{2} = f^{∗}(G_{1}) ≠ 0.

On KO^{2j−1}(Φ_{1}DI(4)) with j even and K^{2j−1}(Φ_{1}DI(4)), ψ^{3} sends the generator g_{1} to

3^{−j}(3^{4}g_{1} − 3^{3}g_{2} + \frac{36}{527}g_{3}) = 3^{−j}(3^{4} − 3^{3} · 2^{3} + \frac{36}{527}2^{8})g_{1},

and ψ^{−1}(g_{1}) = (−1)^{j}g_{1} by Theorem 2.6. □

3. Relationship with pseudosphere

In this section, we prove Theorems 1.2 and 1.1.

Following [8, 8.6], we let T = S^{0} ∪_{η} e^{2} ∪_{2} e^{3}, and consider its K/2localization T_{K/2}. The groups π_{∗}(T_{K/2}) are given in [8, 8.8], while the Adams module is given by

K^{i}(T_{K/2}) = \begin{cases}ℤ̂_{2} & i even, with ψ^{k} = k^{−i/2} \\ 0 & i odd; \end{cases}

KO^{i}(T_{K/2}) = \begin{cases}ℤ̂_{2} & i ≡ 0 mod 4, with ψ^{k} = k^{−i/2} \\ ℤ/2 & i = 2, 3, with ψ^{k} = 1 \\ 0 & i = 1, 5, 6, 7. \end{cases}

Bousfield calls this the 2-completed K-theoretic pseudosphere. Closely related spectra have been also considered in [15] and [4].

Let M(n) = S^{−1} ∪_{n} e^{0} denote the mod n Moore spectrum. Then, for e > 1 and k odd,

K^{i}(T_{K/2} ∧ M(2^{e})) = \begin{cases}ℤ/2^{e} & i even, with ψ^{k} = k^{−i/2} \\ 0 & i odd; \end{cases}

(3.1) KO^{i}(T_{K/2} ∧ M(2^{e})) = \begin{cases}ℤ/2^{e} & i ≡ 0 mod 4, with ψ^{k} = k^{−i/2} \\ ℤ/2 & i = 1, 3, with ψ^{k} = 1 \\ ℤ/2 ⊕ ℤ/2 & i = 2, with ψ^{k} = 1 \\ 0 & i = 5, 6, 7. \end{cases}

Proof. Let Y = T_{K/2} ∧ M(2^{e}). Most of (3.1) is immediate from the exact sequence

→^{2^{e}} KO^{i}(T_{K/2}) → KO^{i}(Y) → KO^{i+1}(T_{K/2}) →^{2^{e}} .

To see that KO^{2}(Y) = ℤ/2 ⊕ ℤ/2 and not ℤ/4, one can first note that

(3.2)  M(2^{e}) ∧ M(2) ≃ Σ^{−1}M(2) ∨ M(2).

The exact sequence

(3.3)  KO^{2}(Y) →^{2} KO^{2}(Y) → KO^{2}(Y ∧ M(2)) → KO^{3}(Y) →^{2}

implies that if KO^{2}(Y) = ℤ/4, then |KO^{2}(Y ∧ M(2))| = 4. However, by (3.2),

(3.4) KO^{2}(Y ∧ M(2)) ≈ KO^{2}(T_{K/2} ∧ M(2)) ⊕ KO^{3}(T_{K/2} ∧ M(2)).

Also, there is a cofiber sequence

(3.5)  T ∧ M(2) → Σ^{−1}A_{1} → Σ^{5}M(2),

where H^{∗}(A_{1}; F_{2}) is isomorphic to the subalgebra of the mod 2 Steenrod algebra generated by Sq^{1} and Sq^{2}, and satisfies KO^{∗}(A_{1}) = 0. Thus

KO^{i}(T_{K/2} ∧ M(2)) ≈ KO^{i}(Σ^{4}M(2)) ≈ \begin{cases}ℤ/4 & i = 2 \\ ℤ/2 & i = 3, \end{cases}

so that |KO^{2}(Y ∧M(2))| = 8, contradicting a consequence of the hypothesis that KO^{2}(Y) = ℤ/4.

We conclude the proof by showing that, for odd k, ψ^{k} = 1 on KO^{2}(Y). First note that ψ^{k} = 1 on KO^{∗}(M(2)). This follows immediately from the Adams operations on the sphere, except for ψ^{k} on KO^{−2}(M(2)) ≈ ℤ/4. This is isomorphic to K̃O(RP^{2}), where ψ^{k} = 1 is well-known. Now use (3.13) to deduce that ψ^{k} = 1 on KO^{∗}(T_{K/2} ∧ M(2)), and then (3.4) to see that ψ^{k} = 1 on KO^{2}(Y ∧ M(2)). Finally, use (3.3) to deduce that ψ^{k} = 1 on KO^{2}(Y). □

Comparison of 2.7 and (3.1) yields an isomorphism of graded abelian groups

(3.6)  KO^{∗}(Σ^{8L+3}T_{K/2} ∧ M(2^{21})) ≈ KO^{∗}(Φ_{1}DI(4))

for any integer L. We will show that if L = 90627, then the Adams operations agree too. By [9, 6.4], it suffices to prove they agree for ψ^{3} and ψ^{−1}.

Note that one way of distinguishing a K-theoretic pseudosphere from a sphere is that in KO^{∗}(sphere) (resp. KO^{∗}(pseudosphere)) the ℤ/2-groups are in dimensions 1 and 2 less than the dimensions in which ψ^{3} ≡ 1 mod 16 (resp. ψ^{3} ≡ 9 mod 16), and similarly after smashing with a mod 2^{e} Moore spectrum. Since 3^{4} − 6^{3} + \frac{36}{527}2^{8} ≡ 9 mod 16, the ℤ/2-groups in KO^{∗}(Φ_{1}DI(4)) occur in dimensions 1, 2, and 3 less that the dimension in which ψ^{3} ≡ 9 mod 16, and so Φ_{1}DI(4) should be identified with a suspension of T_{K/2} ∧ M(2^{21}) and not S_{K/2} ∧ M(2^{21}).

In KO^{4t−1}(Σ^{8L+3}T_{K/2} ∧ M(2^{21})), ψ^{3} = 3^{−2(t−2L−1)} and ψ^{−1} = 1. Thus if L satisfies

(3.7)  3^{4L+2} ≡ 3^{4} − 6^{3} + \frac{36}{527}2^{8} mod 2^{21},

then KO^{∗}(Σ^{8L+3}T_{K/2} ∧ M(2^{21})) and KO^{∗}(Φ_{1}DI(4)) will be isomorphic Adams modules. Maple easily verifies that (3.7) is satisfied for L = 90627.

A way in which this number L can be found begins with the mod 2^{18} equation

∑_{i=1}^{6} \binom{2L−1}{i}8^{i−1} ≡ \frac{3^{4L−2} − 1}{8} ≡ \frac{1}{9}(\frac{2^{7}}{527} − 3) ≡ 192725,

where we use Maple at the last step. This easily implies L ≡ 3 mod 8, and so we let L = 8b + 3. Again using Maple and working mod 2^{18} we compute

∑_{i=1}^{6} \binom{16b+5}{i}8^{i−1} − 192725 ≡ 2^{10}u_{0} + 2^{4}u_{1}b + 2^{10}u_{2}b^{2} + 2^{17}b^{3},

with u_{i} odd. Thus we must have b ≡ 64 mod 128, and so L ≡ 515 mod 2^{10}. Several more steps of this type lead to the desired value of L.

Thus, in the terminology of 4.3, we have proved the following result.

Proposition 3.8. If L = 90627, then there is an isomorphism of Adams modules

K_{CR}^{∗}(Σ^{8L+3}T_{K/2} ∧ M(2^{21})) ≈ K_{CR}^{∗}(Φ_{1}DI(4)).

Theorem 1.2 follows immediately from this using the remarkable [8, 5.3], which says, among other things, that K/2-local spectra X having some K^{i}(X) = 0 are determined up to equivalence by the Adams module K_{CR}^{∗}(X). Theorem 1.1 follows immediately from Theorem 1.2 and the following result.

Proposition 3.9. For all integers i,

π_{8i+d}(T_{K/2} ∧ M(2^{21})) ≈ \begin{cases}ℤ/2 ⊕ ℤ/2^{min(21,ν(i)+4)} & d = −2 \\ ℤ/2^{min(21,ν(i)+4)} & d = −1 \\ 0 & d = 0, 1 \\ ℤ/8 & d = 2 \\ ℤ/2 ⊕ ℤ/8 & d = 3 \\ ℤ/2 ⊕ ℤ/2 ⊕ ℤ/2 & d = 4, 5. \end{cases}

Proof. For the most part, these groups are immediate from the groups π_{∗}(T_{K/2}) given in [8, 8.8] and the exact sequence

(3.10)  ⟶^{2^{21}} π_{j+1}(T_{K/2}) → π_{j}(T_{K/2} ∧ M(2^{21})) → π_{j}(T_{K/2}) ⟶^{2^{21}} .

All that needs to be done is to show that the following short exact sequences, obtained from (3.10), are split.

0 → ℤ/2 → π_{8i+3}(T_{K/2} ∧ M(2^{21})) → ℤ/8 → 0 (3.11)

0 → ℤ/2 ⊕ ℤ/2 → π_{8i+4}(T_{K/2} ∧ M(2^{21})) → ℤ/2 → 0
0 → ℤ/2 → π_{8i+5}(T_{K/2} ∧ M(2^{21})) → ℤ/2 ⊕ ℤ/2 → 0
0 → ℤ/2^{min(21,ν(i)+4)} → π_{8i−2}(T_{K/2} ∧ M(2^{21})) → ℤ/2 → 0.

Let Y = T_{K/2} ∧ M(2^{21}). We consider the exact sequence for π_{∗}(Y ∧ M(2)),

(3.12)  →^{2} π_{i+1}(Y) → π_{i}(Y ∧ M(2)) → π_{i}(Y) →^{2} .

If the four sequences (3.11) are all split, then by (3.12) the groups π_{8i+d}(Y ∧ M(2)) for d = 2, 3, 4, 5, −2 have orders 2^{3}, 2^{5}, 2^{6}, 2^{5}, and 2^{3}, respectively, but if any of the sequences (3.11) fails to split, then some of the orders |π_{8i+d}(Y ∧ M(2))| will have values smaller than those listed here.

By (3.2),

π_{i}(Y ∧ M(2)) ≈ π_{i+1}(T_{K/2} ∧ M(2)) ⊕ π_{i}(T_{K/2} ∧ M(2)).

By (3.5), since localization preserves cofibrations and (A_{1})_{K/2} = ∗, there is an equivalence

(3.13)  Σ^{4}M_{K/2} ≃ T_{K/2} ∧ M(2),

and hence

(3.14)  π_{i}(Y ∧ M(2)) ≈ π_{i−3}(M_{K/2}) ⊕ π_{i−4}(M_{K/2}).

By [10, 4.2],

π_{8i+d}(M_{K/2}) = \begin{cases}0 & d = 4, 5 \\ ℤ/2 & d = −2, 3 \\ ℤ/2 ⊕ ℤ/2 & d = −1, 2 \\ ℤ/4 ⊕ ℤ/2 & d = 0, 1. \end{cases}

This is the sum of two “lightning flashes,” one beginning in 8d − 2 and the other in 8d − 1. Substituting this information into (3.14) yields exactly the orders which were shown in the previous paragraph to be true if and only if all the exact sequences (3.11) split. □

4. Determination of KO^{∗}(Φ_{1}DI(4))

In this section, we prove Theorem 2.6, which shows how ψ^{k} in K^{∗}(BDI(4)) leads to the determination of KO^{∗}(Φ_{1}DI(4)). Our presentation here follows suggestions in several e-mails from Pete Bousfield.

The first result explains how KO^{∗}(BDI(4)) follows from K^{∗}(BDI(4)).

Theorem 4.1. There are classes g_{8}, g_{12}, and g_{24} in KO^{0}(BDI(4)) such that c(g_{i}) = ξ_{i}, with ξ_{i} as in 2.1, and

KO^{∗}(BDI(4)) ≈ KO^{∗}[[g_{8}, g_{12}, g_{24}]].

The Adams operations ψ^{2} and ψ^{3} mod decomposables on the basis of g_{i}’s are as in 2.1.

Proof. In [2, 2.1], it is proved that if there is a torsion-free subgroup F^{∗} ⊂ KO^{∗}(X) such that F^{∗} ⊗ K^{∗}(pt) → K^{∗}(X) is an isomorphism, then so is F^{∗} ⊗ KO^{∗}(pt) → KO^{∗}(X). The proof is a Five Lemma argument using exact sequences in [17, p. 257]. Although the result is stated for ordinary (not 2-completed) KO^{∗}(−), the same argument applies in the 2-completed context. If F^{∗} is a multiplicative subgroup, then the result holds as rings. Our result then follows from 2.1, since the generators there are real. A similar proof can be derived from [5, 2.3]. □

Next we need a similar sort of result about KO^{∗}(DI(4)). We could derive much of what we need by an argument similar to that just used, using the result of [16] about K^{∗}(DI(4)) as input. However, as we will need this in a specific form in order to use it to draw conclusions about KO^{∗}(Φ_{1}DI(4)), we begin by introducing much terminology from [5].

The study of united K-theory begins with two categories, which will then be endowed with additional structure. We begin with a partial definition of each, and their relationship. For complete details, the reader will need to refer to [5] or an earlier paper of Bousfield.

Definition 4.2 ([5, 2.1]). A CR-module M = {M_{C}, M_{R}} consists of ℤgraded 2-profinite abelian groups M_{C} and M_{R} with continuous additive operations

\begin{matrix}M_{C}^{∗} & ⟶_{≈}^{B} & M_{C}^{∗−2},  M_{R}^{∗} & ⟶_{≈}^{B_{R}} & M_{R}^{∗−8},  M_{C}^{∗} & ⟶_{≈}^{t} & M_{C}^{∗}, \\ M_{R}^{∗} & ⟶^{η} & M_{R}^{∗−1},  M_{R}^{∗} & ⟶^{c} & M_{C}^{∗},  M_{C}^{∗} & ⟶^{r} & M_{R}^{∗}, \end{matrix}

satisfying 15 relations, which we will mention as needed.

We omit the descriptor “2-adic,” which Bousfield properly uses, just as we omit writing the 2-adic coefficients ℤ̂_{2} which are present in all our Kand KO-groups.

Example 4.3. For a spectrum or space X, the united 2-adic K-cohomology

K_{CR}^{∗}(X) := {K^{∗}(X), KO^{∗}(X)}

is a CR-module, with complex and real Bott periodicity, conjugation, the Hopf map, complexification, and realification giving the respective operations.

Definition 4.4. A Δ-module N = {N_{C}, N_{R}, N_{H}} is a triple of 2-profinite abelian groups N_{C}, N_{R}, and N_{H} with continuous additive operations

N_{C} ⟶_{≈}^{t} N_{C},  N_{R} ⟶^{c} N_{C},  N_{C} ⟶^{r} N_{R},
N_{H} ⟶^{c′} N_{C},  N_{C} ⟶^{q} N_{H}

satisfying nine relations.

Example 4.5. For a CR-module M and an integer n, there is a Δ-module Δ^{n}M = {M_{C}^{n}, M_{R}^{n}, M_{R}^{n−4}} with c′ = B^{−2}c and q = rB^{2}. In particular, for a space X and integer n, there is a Δ-module K_{Δ}^{n}(X) := Δ^{n}K_{CR}^{∗}(X).

Now we add additional structure to these definitions.

Definition 4.6 ([5, 4.3,6.1]). A θΔ-module is a Δ-module N together with homomorphisms N_{C} →^{θ} N_{C}, N_{R} →^{θ} N_{R}, and N_{H} →^{θ} N_{R} satisfying certain relations listed in [5, 4.3]. An Adams Δ-module is a θΔ-module N together

with Adams operations N ⟶_{≈}^{ψ^{k}} N for odd k satisfying the familiar properties.

Example 4.7. In the notation of Example 4.5, K_{Δ}^{−1}(X) is an Adams Δmodule with θ = −λ^{2}.

Definition 4.8 ([5, 2.6,3.1,3.2]). A special ϕCR-algebra {A_{C}, A_{R}} is a CRmodule with bilinear A_{C}^{m} × A_{C}^{n} → A_{C}^{m+n} and A_{R}^{m} × A_{R}^{n} → A_{R}^{m+n} and also

A_{C}^{0} →^{ϕ} A_{R}^{0} and A_{C}^{−1} →^{ϕ} A_{R}^{0} satisfying numerous properties.

Remark 4.9. The operations ϕ, which were initially defined in [7], are less familiar than the others. Two properties are cϕa = t(a)a and ϕ(a + b) = ϕa + ϕb + r(t(a)b) for a, b ∈ A_{C}^{0}. For a connected space X, K_{CR}^{∗}(X) is a special ϕCR-algebra.

The following important lemma is taken from [5].

Lemma 4.10 ([5, 4.5,4.6]). For any θΔ-module M, there is a universal special ϕCR-algebra L̂M. This means that there is a morphism M →^{α} L̂M such that any morphism from M into a ϕCR-algebra factors as α followed by a unique ϕCR-algebra morphism. There is an algebra isomorphism Λ̂M_{C} → (L̂M)_{C}, where Λ̂(−) is the 2-adic exterior algebra functor.

In [5, 2.7], Bousfield defines, for a CR-algebra A, the indecomposable quotient Q̂A. We apply this to A = K_{CR}^{∗}(BDI(4)), and consider the Δmodule Q̂K_{Δ}^{0}(BDI(4)), analogous to [5, 4.10]. We need the following result, which is more delicate than the K_{Δ}^{−1}-case considered in [5, 4.10].

Lemma 4.11. With θ = −λ^{2}, the Δ-module Q̂K_{Δ}^{0}(BDI(4)) becomes a θΔmodule.

Proof. First we need that θ is an additive operation. In [7, 3.6], it is shown that θ(x + y) = θ(x) + θ(y) − xy if x, y ∈ KO^{n}(X) with n ≡ 0 mod 4. The additivity follows since we are modding out the product terms. (In the case n ≡ −1 mod 4 considered in [5, 4.10], the additivity of θ is already present before modding out indecomposables.)

There are five additional properties which must be satisfied by θ. That θcx = cθx and θtz = tθz are easily obtained from [7, 3.4]. That θc′y = cθy follows from [8, 6.2(iii),6.4]. That θqz = θrz follows from preceding [7, 3.10] by c, which is surjective for us. Here we use that rc = 2 and q = rB^{2}. Finally, θrz = rθz for us, since c is surjective; here we have used the result ϕcx = 0 given in [5, 4.3]. □

Now we obtain the following important description of the CR-algebra K_{CR}^{∗}(DI(4)).

Theorem 4.12. There is a morphism of θΔ-modules

Q̂K_{Δ}^{0}(BDI(4)) → K̃_{Δ}^{−1}(DI(4))

which induces an isomorphism of special ϕCR-algebras

L̂(Q̂K_{Δ}^{0}(BDI(4))) → K_{CR}^{∗}(DI(4)).

Proof. The map ΣDI(4) = ΣΩBDI(4) → BDI(4) induces a morphism

K_{Δ}^{0}(BDI(4)) → K_{Δ}^{−1}(DI(4))

which factors through the indecomposable quotient Q̂K_{Δ}^{0}(BDI(4)). In [16, 1.2], a general result is proved which implies that K^{∗}(DI(4)) is an exterior algebra on elements of K^{1}(DI(4)) which correspond to the generators of the power series algebra K^{∗}(BDI(4)) under the above morphism followed by the Bott map. Thus our result will follow from [5, 4.9], once we have shown that the θΔ-module M := Q̂K_{Δ}^{0}(BDI(4)) is robust([5, 4.7]). This requires that M is profinite, which follows as in the remark following [5, 4.7], together with two properties regarding ϕ, where ϕz := θrz − rθz for z ∈ M_{C}. In our case, c is surjective, and so ϕ = 0 as used in the previous proof.

One property is that M is torsion-free and exact. This follows from the Bott exactness of the CR-module K_{CR}^{∗}(BDI(4)) noted in [5, 2.2], and [5, 5.4], which states that, for any n, the Δ-module Δ^{n}N associated to a Bott exact CR-module N with N_{C}^{n} torsion-free and N_{C}^{n−1} = 0 is torsion-free. The other property is ker(ϕ) = cM_{R} + c′M_{H} + 2M_{C}. For us, both sides equal M_{C} since c is surjective and ϕ = 0. □

Our Theorem 2.6 now follows from [5, 9.5] once we have shown that the Adams Δ-module M := Q̂K_{Δ}^{0}(BDI(4)) is “strong.” ([5, 7.11]) This result ([5, 9.5]) requires that the space (here DI(4)) be an H-space (actually K/2_{∗}- durable, which is satisfied by H-spaces) and that it satisfies the conclusion of our 4.12. It then deduces that KO^{∗}(Φ_{1}DI(4)) fits into an exact sequence which reduces to ours provided M_{R} = M_{C} and M_{H} = 2M_{C}. These equalities are implied by Q̂K_{Δ}^{0}(BDI(4)) being exact, as was noted to be true in the previous proof, plus t = 1 and c surjective, as were noted to be true in 2.1. Indeed, the exactness property, ([5, 4.2]), includes that cM_{R} + c′M_{H} = ker(1 − t) and cM_{R} ∩ c′M_{H} = im(1 + t). Another perceptible difference is that Bousfield’s exact sequence is in terms of M := M/ϕ, while ours involves M, but these are equal since, as already observed, ϕ = 0 since c is surjective.

Note also that the Adams operations in M in the exact sequence of [5, 9.5], which reduces to that in our 2.6, are those in the Adams Δ-module Q̂K_{Δ}^{0}(BDI(4)), which are given in our 2.1. The morphism θ in [5, 9.5] or our 2.6 is \frac{1}{2}ψ^{2}, since this equals −λ^{2} mod decomposables.

Finally, we show that our M is strong. One of the three criteria for being strong is to be robust, and we have already discussed and verified this. The second requirement for an Adams Δ-module to be strong is that it be “regular.” This rather technical condition is defined in [5, 7.8]. In [5, 7.9], a result is proved which immediately implies that K̃_{Δ}^{−1}(DI(4)) is regular. By 4.12, our M injects into K̃_{Δ}^{−1}(DI(4)), and so by [5, 7.10], which states that a submodule of a regular module is regular, our M is regular.

The third requirement for M to be strong is that it be ψ^{3}-splittable ([5, 7.2]), which means that the quotient map M → M/ϕ has a right inverse. As we have noted several times, we have ϕ = 0, and so the identity map serves as a right inverse to the identity map. This completes the proof that our M is strong, and hence that [5, 9.5] applies to DI(4) to yield our Theorem 2.6.

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Hunter College, CUNY, NY, NY 01220 mbenders@hunter.cuny.edu

Lehigh University, Bethlehem, PA 18015 dmd1@lehigh.edu

This paper is available via http://nyjm.albany.edu/j/2008/14-17.html.