REAL CONNECTIVE K-THEORY AND THE QUATERNION GROUP

DILIP BAYEN AND ROBERT R. BRUNER

January, 1994

Revised April, 1995

Abstract. Let ko be the real connective K theory spectrum. We compute ko_{∗}BG and ko^{∗}BG for groups G whose Sylow 2-subgroup is quaternion of order 8. Using this we compute the coefficients t(ko)_{∗}^{G} of the G fixed points of the Tate spectrum t(ko) for G = Sl_{2}(3) and G = Q_{8}. The results provide a counterexample to the optimistic conjecture of Greenlees and May [9, Conj. 13.4], by showing, in particular, that t(ko)^{G} is not a wedge of Eilenberg-Maclane spectra, as occurs for groups of prime order.

1. Introduction

Let G be a group whose Sylow 2-subgroup is quaternion of order 8. The real connective K-homology of the classifying space BG is of interest because a number of geometric invariants of manifolds with fundamental group G lie in ko_{∗}BG. In particular, Stolz shows in [14] that for fundamental groups which are finite 2groups the question of positive scalar curvature reduces to ko_{∗}(BG). Using this, the Gromov-Lawson conjecture has been settled for manifolds whose fundamental group has periodic cohomology [5, 6, 7, 15].

Further, the periodic K-theory of BG is determined by the real, complex and quaternionic representations of G. The connective K-theory provides more refined invariants of G whose algebraic meaning would be interesting to discern.

The approach we use applies quite generally to the computation of the kohomology and cohomology of BG, and is as follows. We localize at 2 throughout, let H^{∗} be H^{∗}( ; Z/2), and let A and A(1) be the mod 2 Steenrod algebra and its subalgebra generated by Sq^{1} and Sq^{2}.

(1) Compute the periodic K-theory KO_{∗}BG in terms of the representation ring of G, using the result of Stolz and Rosenberg quoted as Theorem 2 below.

(2) Decompose BG stably using the results of Martino and Priddy quoted as Theorem 1.

1991 Mathematics Subject Classification. 19L41, 19L47, 19L64, 55N15, 55R35, 55Q91, 55M05.

Key words and phrases. quaternion group, classifying space, connective K-theory, Tate cohomology.

(3) Describe the corresponding summands of H^{∗}BG as A(1)-modules.

(4) Compute the E_{2}-term of the Adams spectral sequence

Ext_{A}(H^{∗}(ko ∧ X), Z/2) ≅ Ext_{A(1)}(H^{∗}X, Z/2) ⟹ ko_{∗}X

for each summand X of BG.

(5) Determine the differentials and extensions in the Adams spectral sequence of each summand using the Bockstein spectral sequence of ko ∧ X and the fact that, after inverting the Bott element, ko_{∗} becomes KO_{∗}, which is known by steps 1 and 2.

(6) Write X as a convenient direct limit, X = lim_{→} X_{n}, compute each ko^{∗}X_{n} using S-duality, and compute ko^{∗}X = lim_{←} ko^{∗}X_{n}.

In the last section we provide some background information on Tate theories and compute t(ko)_{∗}^{G} using the norm sequence

⋯ → ko_{n}BG_{+} → ko^{−n}BG_{+} → t(ko)_{n}^{G} → ko_{n−1}BG_{+} → · · · .

The authors wish to thank Stephan Stolz for numerous helpful conversations. and for interesting us in these matters, and Peter Gilkey for sharing his calculations of the eta invariant at a crucial moment.

Some of the results here are a part of the first author’s thesis [DB].

2. Decomposition of BG and group representations of Q_{8}

Theorem 1 (J. Martino and S. Priddy [10]). Let G be a finite group with Sylow 2-subgroup the quaternion group Q_{8}. Then the complete 2-local stable decomposition of BG into indecomposable summands is either

(1) BG ≃ BSl_{2}(3), or

(2) BG ≃ BSl_{2}(3) ∨ Σ^{−1}BS^{3}/BN ∨ Σ^{−1}BS^{3}/BN,

where N is the normalizer of a maximal torus in S^{3}.

Note that BQ_{8} itself is the larger of these. Since Q_{8} acts freely on the 3-sphere, its cohomology is periodic with period 4. Explicitly,

H^{∗}(BQ_{8}) = Z/2[x, y, w]/(x^{2} + xy + y^{2}, x^{2}y + xy^{2}),
|x| = |y| = 1, and |w| = 4, and
H^{∗}(BSl_{2}(3)) = Z/2[w] ⊗ ∧(x̃), |w| = 4, \left|{ \atop x̃}\right| = 3.

The inclusion of the Sylow 2-subgroup Q_{8} → Sl_{2}(3) induces an isomorphism in cohomology in dimensions 3 and 4 mod 4, with x̃ mapping to x^{2}y = xy^{2}. The two copies of Σ^{−1}BS^{3}/BN are embedded as

{x^{ϵ}w^{k} | ϵ = 1, 2 and k ≥ 0}

and

{y^{ϵ}w^{k} | ϵ = 1, 2 and k ≥ 0}.

The Bocksteins are given by β_{1}x = x^{2}, β_{1}y = y^{2}, and β_{3}x̃ = w by [13]. Thus, Σ^{−1}BS^{3}/BN is composed of mod 2 Moore spaces and BSl_{2}(3) is composed of mod 8 Moore spaces.

The real representations of Q_{8} determine the periodic real K-theory of BQ_{8} by the following result.

Theorem 2 (S. Stolz and J. Rosenberg [15]). Let G be a finite p-group. Then KO_{∗}(BG) splits as a KO_{∗}-module into the direct sum of copies of

\begin{aligned}& 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & (mod 8) \\ (ℝ) & 0 & (Z/2) & (Z/2) & Z/p^{∞} & 0 & 0 & 0 & Z/p^{∞} \\ (ℂ) & 0 & Z/p^{∞} & 0 & Z/p^{∞} & 0 & Z/p^{∞} & 0 & Z/p^{∞} \\ (ℍ) & 0 & 0 & 0 & Z/p^{∞} & 0 & (Z/2) & (Z/2) & Z/p^{∞} \end{aligned}

one for each nontrivial irreducible real, complex, or quaternionic representation, respectively. Here (Z/2) is Z/2 if p = 2 and is 0 otherwise.

Note that, when p = 2, these three KO_{∗}-modules are KO_{∗}(RP^{∞}), KU_{∗}(RP^{∞}), and KSp_{∗}(RP^{∞}).

Q_{8} has three nontrivial 1-dimensional representations and one irreducible 2dimensional representation. By the Frobenius-Schur indicator, the 1-dimensional representations are all real, and the 2-dimensional representation is quaternionic. (Equivalently, we could observe that the Wedderburn decomposition of the real group ring ℝ[Q_{8}], is ℝ^{4} ⊕ ℍ.) Comparing the splittings of KO_{∗}BQ_{8} given by the preceding two theorems, the following corollary is immediate.

Corollary 3. As KO_{∗}-modules,

KO_{∗}(Σ^{−1}BS^{3}/BN) ≅ KO_{∗}(RP^{∞})
KO_{∗}(BSl_{2}(3)) ≅ KO_{∗}(RP^{∞}) ⊕ KSp_{∗}(RP^{∞})
KO_{∗}(BQ_{8}) ≅ ⨁_{3}KO_{∗}(RP^{∞}) ⊕ KSp_{∗}(RP^{∞})

3. The Bockstein differentials in the Adams spectral sequence

Recall that H^{∗}ko = A ⊗_{A(1)} Z/2 [16]. As a result, the Kunneth isomorphism

H^{∗}(ko ∧ X) ≅ A ⊗_{A(1)} Z/2 ⊗ H^{∗}X ≅ A ⊗_{A(1)} H^{∗}X

and the adjunction

Hom_{A}(A ⊗_{A(1)} M, N) ≅ Hom_{A(1)}(M, N)

give the ordinary mod 2 Adams spectral sequence the form

Ext_{A(1)}^{s,t}(H^{∗}X, Z/2) ⟹ ko_{t−s}X.

Recall also that for X = S^{0} the spectral sequence collapses at E_{2} to give

\begin{aligned}E_{2} & = & E_{∞} & = & Z/2[h_{0}, h_{1}, w, b]/(h_{0}h_{1}, h_{1}^{3}, h_{1}w, w^{2} + h_{0}^{2}b), \\ & & & & |h_{0}| = (0, 1), |h_{1}| = (1, 1), |w| = (4, 3), |b| = (8, 4), \end{aligned}

where |x| = (t − s, s) specifies the geometric degree t − s and the Adams filtration s. We will refer, ambiguously, to b and to a generator β ∈ ko_{8}(S^{0}) detected by b, as ‘the Bott element’, since multiplication by its image in KO_{8}(S^{0}) is the Bott periodicity isomorphism.

As an A(1)-module, H^{∗}(Σ^{−1}BS^{3}/BN) is a direct sum,

H^{∗}(Σ^{−1}BS^{3}/BN) = ⨁_{n≥1} H^{∗}M^{4n−2}(2),

where M^{n}(k) = S^{n−1} ∪_{k} e^{n} is the mod k Moore space with top cell in dimension n. It is well known, and easy to calculate, that, as an E_{2}(S^{0})-module,

E_{2}(M^{1}(2)) = E_{∞}(M^{1}(2))
= ⟨x, y | h_{0}x = 0, h_{1}^{2}x = h_{0}y, h_{1}^{2}y = wx, wy = 0⟩
⟹ ko_{∗}(M^{1}(2)),

with |x| = (t − s, s) = (0, 0) and |y| = (2, 1). It follows that E_{2}(Σ^{−1}BS^{3}/BN) is the direct sum of copies of E_{2}(M^{4n−2}(2)) generated by x_{n} and y_{n} for n ≥ 1 with |x_{n}| = (4n − 3, 0) and |y_{n}| = (4n − 1, 1). (See Figure 1.)

As an A(1)-module, H^{∗}(BSl_{2}(3)) is a direct sum of Z/2’s, one in each dimension congruent to 3 or 4 mod 4. Thus, E_{2}(BSl_{2}(3)) is the corresponding sum of copies of E_{2}(S^{0}), one starting in each dimension congruent to 3 or 4 mod 4. The theorem of [11] tells us that, for r ≥ 2, there is a 1-1 correspondence between infinite h_{0}- towers in E_{r} of the Adams spectral sequence, i.e. sets {h_{0}^{i}x|i ≥ 0} with h_{0}^{i}x ≠ 0 for all i, and the E_{r}-term of the homology Bockstein spectral sequence, and that, under this correspondence, the Adams d_{r} connecting these towers corresponds to the Bockstein β_{r}. We will compute the cohomology Bockstein spectral sequence and dualize. For ko, the Bockstein spectral sequence is well known ([12]) to collapse at E_{2}, with E_{2} = E_{∞} spanned by the equivalence classes of Sq^{4n}ι, where ι ∈ H^{0}ko is the generator. The Bockstein spectral sequence of BSl_{2}(3) has E_{1} = E_{3}, β_{3}(x̃w^{k}) = w^{k+1}, and E_{4} = E_{∞} = 0. Tensoring these, we find that the Bockstein spectral sequence for ko ∧ BSl_{2}(3) has

E_{2} = E_{3} = {Sq^{4n}ι|n ≥ 0} ⊗ ∧(x̃) ⊗ Z/2[w]

with

β_{3}(Sq^{4n}ι ⊗ x̃w^{k}) = Sq^{4n}ι ⊗ w^{k+1}

and E_{4} = E_{∞} = 0. Thus, the towers in the Adams spectral sequence will be annihilated by Adams d_{3}’s. Examining E_{3} we find that this d_{3} must be

d_{3}(x_{4n}) = αh_{0}^{3}x_{4n−1} + βwx_{4n−5}

with α, β ∈ {0, 1} and x_{i} ∈ E_{3}^{0,i}. Comparison with the Adams spectral sequence for ko_{∗}M^{4n}(8) shows that α = 1. In order to determine β, it will help to examine the isomorphisms

Ext_{A}(H^{∗}(ko ∧ X), Z/2) ≅ Ext_{A}(H^{∗}(ko) ⊗ H^{∗}(X), Z/2)
≅ Ext_{A(1)}(H^{∗}(X), Z/2)

more carefully. The first isomorphism is induced by the usual Kunneth isomorphism H^{∗}(ko∧X) ≅ H^{∗}ko⊗H^{∗}X, in which the tensor product has the diagonal A action. The second isomorphism applies generally to extended modules, and requires that the tensor product be given the ‘extended’ A action in which A acts only on the left factor, H^{∗}ko ≅ A ⊗_{A(1)} Z/2. The two isomorphisms can thus be combined only because of a third isomorphism θ between the diagonal and extended actions. For convenience of notation, note that if we let { \atop H^{∗}X} denote H^{∗}X with trivial A action, i.e., A action pulled back along the augmentation A → Z/2, then the diagonal action on H^{∗}ko ⊗ { \atop H^{∗}X} is the extended action. The isomorphism

θ : H^{∗}ko ⊗ { \atop H^{∗}X} → H^{∗}ko ⊗ H^{∗}X

Figure 1. E_{2} for ko_{∗}Σ^{−1}BS^{3}/BN

Figure 2. E_{3} = E_{∞} = E^{0}(ko_{∗}Σ^{−1}BS^{3}/BN) is θ(aι ⊗ x) = ∑ a′ι ⊗ a′′x, with inverse θ^{−1}(aι ⊗ x) = ∑ a′ι ⊗ χ(a′′)x, where ψ(a) = ∑ a′ ⊗ a′′. These matters are discussed nicely in [1, §III.16] and [2, 3.1]. To apply the theorem of [11] to towers in Ext_{A(1)}(H^{∗}(X), Z/2), the convenient Bockstein is therefore β̂_{3} = θ^{−1}β_{3}θ. We find

β̂_{3}(ι ⊗ x̃w^{k}) = ι ⊗ w^{k+1}, and
β̂_{3}(Sq^{4}ι ⊗ x̃w^{k−1}) = Sq^{4}ι ⊗ w^{k} + ι ⊗ w^{k+1}.

Dualizing, we see that

d_{3}(x_{4n}) = h_{0}^{3}x_{4n−1} + wx_{4n−5}.

Thus, in E_{4}, we have generators x_{4n−1} and y_{4n+1} for n ≥ 1 with |x_{4n−1}| = (4n−1, 0) and |y_{4n+1}| = (4n + 1, 1), and with the relation wx_{4n−1} = h_{0}^{3}x_{4n+3} (and hence h_{0}^{2}bx_{4n−1} = h_{0}^{6}x_{4n+7}) for every n > 0. See Figure 3, in which we have replaced the class b^{i}x_{4n−1} by the sum b^{i}x_{4n−1} + h_{0}^{4i}x_{8i+4n−1} to simplify the diagram.

4. The ko-homology of Σ^{−1}BS^{3}/BN

Theorem 4. In the Adams spectral sequence for ko_{∗}Σ^{−1}BS^{3}/BN,

d_{2}(x_{n+1}) = h_{1}y_{n}

and E_{3} = E_{∞}. Furthermore, all possible extensions are nonzero. Therefore, for n ≥ 0,

ko_{8n+ϵ}Σ^{−1}BS^{3}/BN = \{ \begin{matrix}Z/2 & = & ⟨β^{n}x_{1}⟩ & ϵ = 1 \\ Z/2 & = & ⟨ηβ^{n}x_{1}⟩ & 2 \\ Z/2^{2n+2} & = & ⟨y_{2n+1}⟩ & 3 \\ Z/2^{2n+2} & = & ⟨y_{2n+2}⟩ & 7 \\ 0 & & & 4, 5, 6, 8 \end{matrix}

with wy_{n} = 4y_{n+1}, βy_{n} = 4y_{n+2}, and η^{2}β^{n}x_{1} = 2^{2n+1}y_{2n+1} for all n > 0.

Proof: Recall that

ko_{∗}(Σ^{−1}BS^{3}/BN) [b^{−1}] = KO_{∗}Σ^{−1}BS^{3}/BN ≅ KO_{∗}RP^{∞}

and is therefore 0 in dimensions congruent to 4,5, or 6, mod 8. We will show that this forces d_{2}(x_{2n}) = h_{1}y_{2n−1} for each n > 0. Similarly, the fact that KO_{∗}RP^{∞} has order 2 in dimensions 1 and 2 mod 8, and is 0 in dimensions congruent to 0 mod 8 will imply that d_{2}(x_{2n+1}) = h_{1}y_{2n}.

We proceed inductively. First, consider x_{2} ∈ E_{2}^{5,0}. If d_{2}(x_{2}) ≠ h_{1}y_{1}, then b^{n}x_{2} survives to E_{∞} for each n. Since KO_{5} = 0, b^{n}x_{2} must be a boundary for large enough n. Choose such an n, and let u_{i} = b^{n−i}x_{2i+2} and v_{i} = b^{n−i}h_{1}y_{2i+1} in order to simplify the notation. In order for u_{0} = b^{n}x_{2} to be a boundary, it must be d_{r_{1}}h_{1}u_{t_{1}} = u_{0} for some r_{1} and t_{1}. The fact that u_{0} is not a multiple of h_{1} implies that h_{1}u_{t_{1}} must not be an h_{1} multiple in E_{r_{1}}. Hence, u_{t_{1}} must have supported an earlier nonzero differential d_{r_{2}}u_{t_{1}} = v_{s_{2}}, r_{2} < r_{1}, and we must have had h_{1}v_{s_{2}} = 0 in E_{r_{2}} to allow h_{1}u_{t_{1}} to survive to E_{r_{1}}. This requires a still earlier differential d_{r_{3}}h_{1}u_{t_{2}} = h_{1}v_{s_{2}} with r_{3} < r_{2} < r_{1}. To allow v_{s_{2}} to survive to E_{r_{2}}, we find that h_{1}u_{t_{2}} must not be an h_{1} multiple in E_{r_{3}}. Continuing in this way, we find an infinite sequence of ever shorter differentials. This contradiction shows that b^{n}x_{2} cannot be the target of a differential, and we conclude that d_{2}(x_{2}) = h_{1}y_{1}.

Figure 3. E_{4} for ko_{∗}BSl_{2}(3)

Figure 4. E_{5} = E_{∞} = E^{0}(ko_{∗}(BSl_{2}(3)))

The remaining even differentials d_{2}(x_{2n}) = h_{1}y_{2n−1} follow by induction on n, since the inductive hypothesis implies that this is the only nontrivial differential x_{2n} can support. The possibility that x_{2n} could be the target of a differential is ruled out exactly as for x_{2}.

The same argument also shows that b^{n}x_{1} cannot be the target of any differential. Since KO_{∗}Σ^{−1}BS^{3}/BN ≅ KO_{∗}RP^{∞} as KO_{∗}-modules, it follows that b^{n}x_{1} and b^{n}h_{1}x_{1} survive to nonzero classes in E_{∞}. The remaining classes in stems congruent to 0,1 and 2, mod 8 must vanish upon inverting the Bott element, and, hence, the same argument as for the even differentials establishes the odd differentials, d_{2}(x_{2n+1}) = h_{1}y_{2n}, for n > 0.

This leaves no room for further differentials, so E_{∞} is as shown in Figure 2. The fact that each KO_{i}Σ^{−1}BS^{3}/BN is cyclic, together with the fact that the Bott element acts monomorphically on E_{∞}, implies that all possible extensions are nontrivial. Finally, the ko_{∗}-module structure is evident from the Adams spectral sequence. QED

5. The ko-homology of BSl_{2}(3))

Theorem 5. In the Adams spectral sequence for ko_{∗}(BSl_{2}(3))

d_{4}(y_{4n+1}) = h_{1}bx_{4n−9}

and E_{5} = E_{∞}. As a ko_{∗}-module, ko_{∗}(BSl_{2}(3)) is generated by elements y_{5}, y_{9}, and x_{4n+3} for n ≥ 0. Furthermore, for n ≥ 0,

ko_{8n+ϵ}BSl_{2}(3) = \{ \begin{matrix}Z/2^{4n+3} ⊕ Z/2^{2n} & = & ⟨x_{8n+3}⟩ ⊕ ⟨x̄_{8n+3}⟩ & ϵ = 3 \\ Z/2 & = & ⟨ηx_{8n+3}⟩ & 4 \\ Z/2 ⊕ Z/2 & = & ⟨η^{2}x_{8n+3}⟩ ⊕ ⟨β^{n}y_{5}⟩ & 5 \\ Z/2 & = & ⟨ηβ^{n}y_{5}⟩ & 6 \\ Z/2^{4n+6} ⊕ Z/2^{2n} & = & ⟨x_{8n+7}⟩ ⊕ ⟨x̄_{8n+7}⟩ & 7 \\ Z/2 & = & ⟨ηx_{8n+7}⟩ & 8 \\ Z/2 ⊕ Z/2 & = & ⟨η^{2}x_{8n+7}⟩ ⊕ ⟨β^{n}y_{9}⟩ & 9 \\ Z/2 & = & ⟨ηβ^{n}y_{9}⟩ & 10 \end{matrix}

with relations η^{2}β^{n}y_{5} = 2^{4n+5}x_{8n+7}, η^{2}β^{n}y_{9} ≡ 2^{2n+1}x̄_{8n+11} mod ⟨2^{4n+6}⟩, and x̄_{n} ≡ βx_{n−8} − 16x_{n} mod ⟨32, β^{2}⟩ for all n ≥ 0.

Remark: Note that the elements ηx_{n} and η^{2}x_{n} are annihilated by β. Thus, the map ko_{∗}BSl_{2}(3) → KO_{∗}BSl_{2}(3) has a nontrivial kernel, unlike ko_{∗}BZ/2 → KO_{∗}BZ/2 or ko_{∗}Σ^{−1}BS^{3}/BN → KO_{∗}Σ^{−1}BS^{3}/BN.

Proof: The isomorphism of Corollary 3,

KO_{∗}BSl_{2}(3) ≅ KO_{∗}RP^{∞} ⊕ KSp_{∗}RP^{∞},

implies that KO_{4n} = 0, KO_{4n+1} ≅ KO_{4n+2} ≅ Z/2, and that η^{2} times KO_{4n+1} is nonzero.

We show first that this implies that b^{n}y_{5} and h_{1}b^{n}y_{5} survive to nonzero elements of E_{∞} for all n ≥ 0. Clearly, their filtration is too high to support nontrivial differentials. On the other hand, if b^{n}y_{5} is the target of a differential, then the same sort of infinite regress as in Theorem 4 leads to a contradiction. Therefore, b^{n}y_{5} maps to the nonzero class in KO_{8n+5}BSl_{2}(3), and hence, η^{2} times it is nonzero. It follows that h_{1}b^{n}y_{5} is also nonzero in E_{∞}.

Now, all other classes in dimensions 8n + 4, 8n + 5, and 8n + 6 must vanish upon inverting b. We show that this implies d_{4}(y_{8n+5}) = h_{1}bx_{8n−5} by induction. Suppose n = 1. If d_{4}(y_{13}) ≠ h_{1}bx_{3} then, for some n, we must have b^{n}y_{13} the target of a differential. Say d_{r}(h_{1}b^{n−i}y_{8i+13}) = b^{n}y_{13}. Since b^{n}y_{13} is not an h_{1} multiple, h_{1}b^{n−i}y_{8i+13} must be indecomposable under multiplication by h_{1} in E_{r}. Hence b^{n−i}y_{8i+13} supports an earlier differential than d_{r}, and we are led to a contradiction as before. The induction now proceeds as in Theorem 4: d_{4}(y_{8n+5}) = h_{1}bx_{8n−5} is the only possible nonzero differential on y_{8n+5} by the inductive hypothesis. The alternative possibility, that b^{k}y_{8n+5} is the target of a differential for some k, leads to a contradiction.

Now, the remaining survivors, b^{n}y_{9} and h_{1}b^{n}y_{9}, and the remaining differentials d_{4}(y_{8n+1}) = h_{1}bx_{8n−9}, follow by exactly the same arguments.

In the resulting E_{5}-term, the indecomposables which might support differentials lie in dimensions 4n − 1. The only remaining elements in dimensions 4n − 2 must be nonzero in E_{∞}. Hence, E_{5} = E_{∞}. (See Figure 4.)

The hidden η-extensions are propagated by multiplication by β once η^{2}y_{5} and η^{2}y_{9} are determined. The first, η^{2}y_{5}, is inherited from the Moore space M^{4}(8) → BSl_{2}(3). The second, η^{2}y_{9}, is detected by considering

BSl_{2}(3) ← B^{8} → B^{8}/B^{4} = M^{8}(8)

where B^{n} is the n-skeleton of BSl_{2}(3).

It remains to determine the extensions in dimensions 4n − 1. Multiplication by the Bott element is a monomorphism in dimensions 4n − 1 in E_{∞}, and hence in ko_{∗}. Since KO_{4n−1}BSl_{2}(3) = Z/2^{∞} ⊕ Z/2^{∞}, ko_{4n−1}BSl_{2}(3) has rank ≤ 2.

The E_{∞}-term shows that 2^{4n+3}ko_{8n+3} = 0 and 2^{4n+6}ko_{8n+7} = 0. An element detected in filtration 0 will generate a subgroup of this order, which is therefore a direct summand. Dividing by this subgroup must yield a cyclic group, forcing the extensions. QED

6. The ko-cohomology of BG

The preceding calculations of ko-homology are easily dualized to obtain the following.

Theorem 6. There are classes z_{n} ∈ ko^{4n}Σ^{−1}BS^{3}/BN, n ≥ 0, which generate ko^{∗}Σ^{−1}BS^{3}/BN as a ko_{∗}-module, with relations ηz_{n} = 0 if n ≠ 0, and βz_{n} = ωz_{n−1} = 4z_{n−2}. Thus

ko^{i}Σ^{−1}BS^{3}/BN = \{ \begin{matrix}Ẑ_{2} & i ≡ 0 mod 4 \\ Z/2 & i ≡ −1, −2 mod 4, i < 0 \\ 0 & otherwise \end{matrix}

Theorem 7. There are classes z_{n} ∈ ko^{4n}BSl_{2}(3), n > 0, which generate ko^{∗}BSl_{2}(3) as a ko_{∗}-module, with relations ηβz_{n} = 0 if n > 2, and ωz_{n+1} ≡ 8z_{n} mod (8, β). Further,

ko^{i}BSl_{2}(3) = \{ \begin{matrix}Ẑ_{2} ⊕ Ẑ_{2} & i ≡ 0 mod 4 \\ Z/2 & i ≡ −1, −2 mod 4 \\ 0 & otherwise \end{matrix}

Figure 5. E_{∞} = E^{0}(ko^{−∗} Σ^{−1}BS^{3}/BN)

Figure 6. E_{∞} = E^{0}(ko^{−∗}BSl_{2}(3))

Note: We will see that z_{1} and z_{2} each generate a copy of ko_{∗}, while the other generators generate a copy of ko_{∗}/(ηβ).

We will prove these two results in tandem as follows. We can filter B = BSl_{2}(3) and X = Σ^{−1}BS^{3}/BN so that

B^{0} = X^{0} = ∗
B^{n}/B^{n−1} = M^{4n}(8)
X^{n}/X^{n−1} = M^{4n−2}(2)

For convenience of indexing when we compute the coefficients of the Tate theory, we will henceforth index cohomology by the negative of the usual index. To calculate ko^{−∗}BSl_{2}(3) and ko^{−∗}Σ^{−1}BS^{3}/BN we use duality:

ko^{−q}B = ko^{−q} lim_{→} B^{n}
= lim_{←} ko^{−q} B^{n}
= lim_{←} ko_{q} DB^{n}

and similarly for X = Σ^{−1}BS^{3}/BN. It will emerge from the calculation of the inverse systems, that they are Mittag-Leffler and hence there are no lim_{←}^{1}-terms to consider. The finiteness of DX^{n} and DB^{n} implies the convergence of the inverse limit of the Adams spectral sequences.

The filtration of DX^{n} induced by that of X^{n} has filtration quotients

DX^{−n}/M^{4n+3}(2) = DX^{−n+1}

for n < 0. Thus, the inverse limit Adams spectral sequence for ko^{−∗}X has

E_{2} = ⨁_{n=−∞}^{−1} E_{2}(M^{4n+3}(2)).

Let x_{n} and y_{n} generate the n^{th} summand here, as in §3, so that |x_{n}| = (4n + 2, 0) and |y_{n}| = (4n+4, 1) for each n < 0. The geometric boundary map for this filtration of DX^{n}, namely, the map

M^{−4n−1}(2) ⟶ DX^{n} ⟶ M^{−4n−4}(2)

is dual to the geometric boundary map

M^{4n+2}(2) ⟵ X^{n} ⟵ M^{4n+5}(2).

Since the latter map induces the differential d_{2}(x_{n+1}) = h_{1}y_{n}, we see that the diagram

S^{4n+4}  ⟶^{η^{2}} S^{4n+2}
↓  ↑
M^{4n+5}(2) ⟶ M^{4n+2}(2)

must commute after smashing with ko. Hence the dual diagram

S^{−4n−4}  ⟵^{η^{2}} S^{−4n−2}
↑  ↓
M^{−4n−4}(2) ⟵ M^{−4n−1}(2)

must also. This implies that the geometric boundary map for DX^{n} also induces the differential

d_{2}(x_{n+1}) = h_{1}y_{n}.

There are no further differentials possible. The resulting E_{∞}-term is shown in Figure 5. Note that no elements of filtration zero survive to E_{∞}.

The inverse limit Adams spectral sequence for ko^{−∗}B has

E_{2} = ⨁_{n=−∞}^{−1} E_{2}(S^{4n}) ⊕ E_{2}(S^{4n+1})

with the first differential again implied by the Bockstein, leaving an E_{4}-term generated by x_{4n} and y_{4n+2}, −∞ < n < 0, with |x_{4n}| = (4n, 0) and |y_{4n+2}| = (4n + 2, 1), and with the relation wx_{4n} = h_{0}^{3}x_{4n+4}. Applying duality as we did for X, we conclude that

d_{4}(y_{4n+2}) = h_{1}bx_{4n−8}

and that there are no further differentials possible. The extensions are resolved in the same way as they were for ko_{∗}BSl_{2}(3). The resulting E_{∞}-term is shown in Figure 6.

Additively, ko^{−∗}B = ⊕Σ^{4n}A summed over all integers n, where A is the graded module Ẑ_{2} ⊕ Ẑ_{2}, Z/2, Z/2, 0 with generators x, y, ηx, and η^{2}x, in degrees 0,0,1, and 2 respectively. The action of the Bott generator β ∈ ko^{−8} is more interesting. In negative degrees, i.e. ko^{n}B with n < 0, multiplication by β is an isomorphism, as suggested by the right hand side of Figure 6. In positive degrees however, multiplication by β annihilates the η multiples and introduces complicated relations among the torsion-free summands, as can be deduced from the hidden extensions in Figure 6. If we let x_{n} be the filtration 0 generator of Σ^{4n}A in degree 4n, then

4βx_{n} ≡ 64x_{n+2} mod ⟨128, β^{2}⟩.

We shall see in the next section that the norm sequence relating ko_{∗}, ko^{−∗} and the Tate theory t(ko)_{∗} works to insert ko_{∗}BG and the ko-homology of a mod 8 Moore space coming from the disjoint basepoints in ko_{q}BG_{+} and ko^{−q}BG_{+}, into this picture in such a way that it becomes perfectly uniform.

7. Coefficients of the Tate theory t(ko)^{G}

Greenlees and May [9] have recently given a definition of a spectrum t(k), which they call the Tate spectrum of k, for any G-spectrum k. Their definition agrees with previous, ad hoc, constructions of candidates for general ‘Tate theories’ and enjoys many good properties. Since generalized Tate theories are not yet widely known, we will discuss a few of these properties, then explain the conjecture which we disprove and the context which made it a reasonable conjecture. Finally, we suggest a minimal modification of the conjecture consistent with the results here. For a comprehensive treatment of this material, see [9]. Note that the construction takes place in the category of G-spectra, though some, but not all, of its good properties can be seen on the non-equivariant level by taking G-fixed points.

Figure 7. E_{∞} = E^{0}(t(ko)_{∗}^{Sl_{2}(3)})

Figure 8. E_{∞} for the part of E^{0}(t(ko)_{∗}^{Q_{8}}) coming ‘from X’

When G acts freely on the unit sphere of a representation V, there is a particularly simple description of the G-fixed points t(k)^{G},

t(k)^{G} = lim_{←} BG^{Ad(G)−nV} ∧ Σk,

where Ad(G) is the adjoint representation of G on the tangent space at the identity and BG^{α} is the Thom spectrum of a virtual representation α. When G is Z/2, Z/p for an odd prime p, or S^{1} this becomes

t(k)^{Z/2} = lim_{←} ℝP_{−n}^{∞} ∧ Σk

t(k)^{Z/p} = lim_{←} 𝕃_{−n}^{∞} ∧ Σk

t(k)^{S^{1}} = lim_{←} ℂP_{−n}^{∞} ∧ Σ^{2}k

where the extra suspension in the last case comes from Ad(S^{1}) = ℝ. (Here ℝP_{−n}^{∞}, 𝕃_{−n}^{∞} and ℂP_{−n}^{∞} are the usual stunted real projective space, lens space and complex projective space, respectively.) Thus, Tate theories provide a vast generalization of Mahowald’s remarkable root invariant inverse system and spectral sequence, which led to Lin’s theorem and the proof of the Segal conjecture. We shall exploit this isomorphism for G = Q_{8}. Let QP_{−n}^{∞} be the Thom space of the virtual representation −nℍ, where ℍ is the quaternions with the usual Q_{8} action. Then

t(k)^{Q_{8}} = lim_{←} QP_{−n}^{∞} ∧ Σk.

James periodicity applied to the finite stunted projective spaces QP_{m}^{n} will allow us to determine t(k)_{∗}^{Q_{8}} immediately from the calculations already done.

Perhaps the most remarkable property of Greenlees and May’s construction of Tate spectra is the fact that t(k) is a ring spectrum when k is, and that the natural map k → t(k) is a ring map. In particular, this shows that the root invariant spectral sequence, properly graded, is multiplicative, a fact which is not apparent from ad hoc constructions of Tate spectra.

The properties which one would expect of a Tate theory are, first, that it vanishes on free G-spaces (or spectra), second, that it represents classical Swan-Tate homology if k is ordinary homology and G is finite, and, third, that there is a long exact norm sequence

⋯ → t(k)_{n+1}^{G} → k_{n}BG_{+} → k^{−n}BG_{+} → t(k)_{n}^{G} → ⋯

in which the middle map is a generalized norm map which is simply multiplication by the Euler characteristic of G on the components coming from the disjoint basepoint of BG_{+}.

Roughly speaking, for K-theory, the norm sequence exhibits k-cohomology as a lattice in the Tate theory t(k), with quotient the suspension of the k-homology. For periodic K-theory, and 2-groups, this is a sum of copies of the sequence ℤ̂_{2} → ℚ_{2} → ℚ_{2}/ℤ̂_{2}, in which the connective K-theory is imbedded as a sum of copies of ℤ̂_{2} → ℤ̂_{2} → ℤ/2^{n} for various n. More precisely, Greenlees and May have shown that t(K) and t(KO) are rational [9]. (In general, Greenlees and Sadofsky have shown that if k is a v_{n}-spectrum, the t(k)^{G} is a v_{n−1}-spectrum [8].) The norm sequence for periodic K-theory then takes the form shown in the following table, in which the maps of the norm sequence are the obvious natural maps. Let RO, RU, and RSp be the real and complex representation rings and the quaternionic representation module. Let JO and JSp be the kernels of the augmentation, and let ˆ( ) denote JO-adic completion.

n mod 8 KO^{−n}BG_{+} →  t(KO)_{n}^{G}  →  KO_{n−1}BG_{+}
0  R̂O  ĴO ⊗ ℚ  ĴO ⊗ ℚ/ℤ
1  { \atop RO/RU}  0  ℤ
2  { \atop RU/RSp}  { \atop RU/RSp} ⊗ ℚ  ({ \atop RU/RSp} ⊗ ℚ/ℤ) ⊕ { \atop RO/RU}
3  0  0  Torsion({ \atop RU/RSp})
4  R̂Sp  ĴSp ⊗ ℚ  ĴSp ⊗ ℚ/ℤ
5  { \atop RSp/RU}  0  ℤ
6  { \atop RU/RO}  { \atop RU/RO} ⊗ ℚ  ({ \atop RU/RO} ⊗ ℚ/ℤ) ⊕ { \atop RSp/RU}
7  0  0  Torsion({ \atop RU/RO})

Since rational spectra split into a wedge of Eilenberg-MacLane spectra, it follows that t(KO)^{G} is the wedge of rational Eilenberg-MacLane spectra determined by the right hand column above. Greenlees and May conjectured that the following analog of this fact holds for t(ko)^{G}.

Conjecture 8 ([9], Conj. 13.4). With the wedge taken over all integers i,

t(ko) ≃ ⋁ K(ĴO, 8i) ∨ K(({ \atop RU/RSp})/torsion, 8i + 2)

∨K(ĴSp, 8i + 4) ∨ K(({ \atop RU/RO})/torsion, 8i + 6)

The Conjecture holds for G of prime order by results of Davis and Mahowald [4]. However, the conjecture fails for G = Q_{8} or Sl_{2}(3) for the following reason. It would imply that t(ko)_{∗}^{G} ⟶ t(KO)_{∗}^{G} is a monomorphism. However, the Bott element acts invertibly on t(KO)_{∗}^{G}, but annihilates some elements of t(ko)_{∗}^{G}, so this is impossible. Also, if t(ko)^{G} is to split into Eilenberg-MacLane spectra, then η will act trivially on t(ko)_{∗}^{G}. Our calculation shows that there are nontrivial ηmultiples in t(ko)_{∗}^{G}. In fact, these are the elements which are annihilated by the Bott element. Thus, eliminating the elements annihilated by the Bott element leads to the following minimal modification to the conjecture consistent with our calculations here.

Conjecture 9. There exist a spectrum t(ko)/(ker β) and maps

t(ko) ⟶ t(ko)/(ker β) ⟶ t(KO)

factoring the natural map. Further, t(ko)/(ker β) splits into Eilenberg-MacLane spectra as in Conjecture 8.

Our calculation which disproves Conjecture 8 and suggests our conjecture is as follows.

Theorem 10. If G = Q_{8} or Sl_{2}(3) then, 2-locally, the coefficients of the Tate G-spectrum of ko are

t(ko)_{∗}^{Q_{8}} = t(ko)_{∗}^{Sl_{2}(3)} ⊕ ⨁_{n=−∞}^{∞} Σ^{4n}(Ẑ_{2} ⊕ Ẑ_{2})

and

t(ko)_{∗}^{Sl_{2}(3)} = ⨁_{n=−∞}^{∞} Σ^{4n}A

where A is the graded module (Ẑ_{2} ⊕ Ẑ_{2}, Z/2, Z/2, 0), with generators x, y, ηx, and η^{2}x .

Proof: We may assemble the Adams spectral sequences converging to

\begin{aligned}ko^{−∗} BG_{+} & = & π_{∗} F (BG_{+}, ko), and \\ ko_{∗} ΣBG_{+} & = & π_{∗} ko ∧ ΣBG_{+} \end{aligned}

in standard fashion to get a spectral sequence converging to π_{∗}t(ko)^{G} with differentials derived from the differentials in the two halves and from the boundary map

ko ∧ BG_{+} ^{\overline{}τ} F (BG_{+}, ko).

However, the differentials implied by τ are difficult to determine in general. When G = Q_{8}, periodicity implies them as follows. We use the description of the Tate spectrum as

t(ko)^{Q_{8}} = QP_{−∞}^{∞} ∧ Σko = lim_{←} QP_{−n}^{∞} ∧ Σko

as in [9, Thm 16.1], where QP_{−n}^{∞} is the Thom complex of −n times the quaternionic line bundle over BQ_{8}. The differentials in the resulting Adams-Atiyah-Hirzebruch spectral sequence are then periodic by James periodicity applied to the finite stunted quaternionic projective spaces. As a consequence, the differentials which hold in the Adams spectral sequence for ko^{−n}BG, G = Q_{8} or BSl_{2}(3), for n < 0, hold for all n in the spectral sequence for t(ko)_{∗}^{G}. A similar periodicity applies to the extension questions. The resulting E_{∞}-terms for t(ko)_{∗}^{BSl_{2}(3)} and each of the two other identical pieces of t(ko)_{∗}^{Q_{8}} are displayed in Figures 7 and 8. QED

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Mathematics Department, Wayne State University, Detroit, Michigan, 48202 E-mail address: dbayen@math.wayne.edu

Mathematics Department, Wayne State University, Detroit, Michigan, 48202 E-mail address: rrb@math.wayne.edu