Ossa's Theorem via the Kunneth formula

Let $p$ be a prime. We calculate the connective unitary K-theory of the smash product of two copies of the classifying space for the cyclic group of order $p$, using a K\"{u}nneth formula short exact sequence. As a corollary, using the Bott exact sequence and the mod $2$ Hurewicz homomorphism we calculate the connective orthogonal K-theory of the smash product of two copies of the classifying space for the cyclic group of order two.


Introduction
This paper arose as a result of discussions during a graduate course at the University of Sheffield during 2008. In order to introduce Frank Adams' technique of constructing homology resolutions as realisations of iterated cofibrations of spectra a simpler example than the classical Adams spectral sequence was needed. We had the spectrum bu to hand but, in order to postpone the algebraic intricacies of spectral sequences, what was required was an example whose geometric resolution gave rise to a short exact sequence rather than a spectral sequence. As it happens the bu-resolution of BZ/2 yields such an example, which was simple enough for the purposes of the course. At that point, John Greenlees mentioned the existence of [9], which prompted the writing of §2. In §2 we use the bu-resolution of §2.1 to calculate bu * (BZ/p ∧ BZ/p) in terms of bu * (BZ/p) and mod p Eilenberg-MacLane spaces (Theorem 2.12).
Our method does not yield a homomorphismμ * induced on homotopy from a map of spectra, it is merely an algebraic homomorphism, as explained in §2.13. However, sinceμ * is closely related to the map induced by the multiplication on BZ/p it is virtually invariant under switching the BZ/p-factors, which may prove useful in calculations of bu * of other p-groups.
In [5] and [9] bo * -analogues of the bu-result are offered when p = 2. Our bo * calculations are consistent with the results proved in [5] and highlight the errors in the bo-analogue asserted in [9]. Consider the cofibration discovered by Raoul Bott during the proof of his famous Periodicity Theorem. Smashing this with X and taking homotopy groups yields the Bott sequence for X. In §3 we compute bo * (BZ/2 ∧ BZ/2) by comparing the Bott sequence for BZ/2 ∧ BZ/2 with that for BZ/2 and with mod 2 homology. Our calculations are relevant to [6] and [11], for example.
2. The connective unitary case 2.1. Let bu * denote connective unitary K-homology on the stable homotopy category of CW spectra [2] so that if X is a space without a basepoint its unreduced bu-homology is bu * (Σ ∞ X + ), the homology of the suspension spectrum of the disoint union of X with a base-point. In particular bu * (Σ ∞ S 0 ) = Z[u] where deg(u) = 2. Let p be a prime and consider the cofibration of pointed spaces From the Atiyah-Hirzebruch spectral sequence ( [2] p.47) we obtain the following result, which also follows from the Thom isomorphism bu * (W p ) ∼ = bu * (BS 1 ), since W p is Thom complex of the p-th tensor power of the canonical complex line bundle, by §2.1.

2.4.
The Atiyah-Hirzebruch spectral sequences for for bu * and KU * of Σ ∞ BZ/p both collapse for dimensional reasons and the map between them is injective so that bu * (Σ ∞ BZ/p) injects into KU * (Σ ∞ BZ/p) which, by the universal coefficient theorem for KU -theory [3] and the calculations of [4], is given by §2; see also [8] Chapter I, §2) and is zero in even dimensions. When p is odd it will be convenient to replace bu by buZ p , connective unitary K-theory with p-adic integers coefficients and similarly for KU Z p . These p-adic spectra possess Adams decompositions [1] (see also [8]) where deg(v) = 2p − 2 corresponds to u p−1 and multiplication by u translates the summand Σ 2i−2 lu to Σ 2i lu for 0 ≤ i ≤ p − 2 and Σ 2p−4 lu to lu. LU -theory is obtained from lu by localising to invert v. In addition there are canonical isomorphisms Corollary 2.5. Let p be a prime and let lu * be as in §2.4 when p is odd or .
Therefore this group must be cyclic and an ordercount in the collapsed Atiyah-Hirzebruch spectral sequence shows that the non- as well as similar resolutions for buZ p and lu.
. By a cell-by-cell induction, for all CW spectra of finite type X the external product gives isomorphisms Smashing the cofibration of §2.1 with X and applying the argument of [3] yields the following Künneth formula: There is a natural short exact sequence −→ 0 as well as similar exact sequences for buZ p and lu. Example 2.9. Let p be a prime. As in Corollary 2.5, let lu * be as in §2.4 when p is odd or lu = (buZ 2 ) * when p = 2. In Theorem 2.8 set X = Σ ∞ BZ/p. Then lu 2 * (Σ ∞ (BZ/p ∧ BZ/p)) comes entirely from the left-hand graded group which is generated by which is zero by induction and similarly if 2j−1 > 2(p−1). Therefore lu 2m (Σ ∞ (BZ/p∧ BZ/p)) is the graded F p -vector space spanned by v 1 ⊗ v 2m−1 , . . . , v 2m−1 ⊗ v 1 which are linearly independent, being detected by the canonical homomorphism to H 2m (Σ ∞ (BZ/p ∧ BZ/p); Z/p). Therefore for each m ≥ 1 Similarly lu 2 * +1 (Σ ∞ (BZ/p ∧ BZ/p)) comes entirely from the right-hand graded group This decomposition has a well-known geometric origin ([9] §2).

Proof
For simplicity we prove this only for p = 2. The proof, which uses KU , may be modified for odd primes but requires a more careful analysis of the splittings of is injective, because it is localisation by inverting u. We shall use this observation to show that is injective, which suffices to prove the result when p = 2. Since BZ/2 = RP ∞ a skeletal approximation to the multiplication gives Consider the effect on reduced, periodic complex K-theory If L is the Hopf line bundle then µ * (L − 1) = (L − 1) ⊗ (L − 1) so that µ * is onto and, by the universal coefficient formula for KU * , KU * , Letting r, s tend to infinity yields the result.
Similarly at odd primes, using the multiplication µ together with the stable homotopy splittings of Σ ∞ BZ/p [9], yields an isomorphism . By Example 2.9, the F p -vector space bu 2 * (Σ ∞ (BZ/p ∧ BZ/p)) is detected in mod p homology and there is a map of spectra a=0 ∨ i,j>0 Σ 2a+2i+2j−2 HZ/p) which induces an isomorphism on even dimensional homotopy. Therefore we obtain the following result: Theorem 2.12. ( [9]; see also [5] and [7]) There is an isomorphism π * (Σ 2a+2i+2j−2 HZ/p). Remark 2.13. The composition of maps of spectra k(1 ∧ µ) used in §2.11 is not nullhomotopic, although is it zero on homotopy groups. It is for this reason that our method does not yield a homomorphismμ * induced by a map of spectra.
3. The connective orthogonal case 3.1. In this section we shall concentrate on p = 2 and connective orthogonal Ktheory bo. Consider the following commutative diagram of spectra of horizontal and vertical cofibrations in which c is complexification and η is multiplication by the generator of π 1 (bo). The notation for bo 1 is taken from [5].
We have the following table of (reduced) orthogonal connective K-theory groups: The graded group bo * (RP ∞ ) is a module over and multiplication by η is nontrivial from dimension 8n + 1 to 8n + 2 and from 8n + 2 to 8n + 3. Multiplication by α has kernel of order 4 from dimension 8n + 3 to 8n + 7 and is one-one from dimension 8n + 7 to 8n + 11. Multiplication by β is always one-one.
The central horizontal cofibration yields a long exact sequence of reduced homology theories and there is a factorisation bo −→ bu −→ HZ. Using the fact that H i (RP ∞ ; Z) ∼ = Z/2 for odd i > 0 and is zero otherwise we may calculate bo 1 * (RP ∞ ). In addition we may double-check the results from the long exact homotopy sequence of the lefthand vertical fibration in the diagram of §3.1 Diagram chasing yields the following table: We can now state the main result of this section (see also [5]), whose proof will be sketched in §3.6.

Theorem 3.2.
There are homomorphisms of graded groups characterised in §3.6, and is an isomorphism.
Consider the Bott sequence which is isomorphic to the homotopy sequence of the cofibration where the middle spectrum is identified with bu ∧ X via an equivalence due to Anderson-Wood [10].