Stable splitting of $K(G,\,1)$

Abstract

the splitting of the cohomology (ordinary and generalized) of K(G, 1), for finite abelian G, is realized topologically by taking suspensions. The cohomology of K(ZF,, 1), both ordinary and K-theoretic, splits over all operations into a direct sum of p 1 components. This suggests asking whether the space itself splits, at least stably. The answer, in a somewhat more general situation, turns out to be affirmative. The trick used to obtain this splitting is probably at least as interesting as the result. THEOREM. There are simply connected spaces X1, * *, X,1 and a homotopy equivalence from the suspension SK(Zpn, 1) to the one-point union Xl v * * v X,_1 such that Xi has homology only in dimensions of the form 2k(p 1) + 2i. PROOF. Let r be an integer representing a generator of the multiplicative group of units in Z,. Multiplication by r defines an isomorphism of Zin , and hence a homotopy equivalence f :K K (K = K(Zin, 1)). This map induces multiplication by ri on H2i(K; Z) = Z,,,. Now, for any integer s, let gs: SK -SK be s times (with respect to the suspension structure) the identity map. The induced map on cohomology is multiplication by s in every dimension. Then for s = ri, hi = Sf g, induces multiplication by ri ri on H2i+1(SK), thus also on H2i(SK). This multiplication is an isomorphism when i and j represent different classes in Z,, and has nontrivial kernel when i -j (modp 1). For 1 _ i <p 1, let Mi = hlo .. o hii ? ... o h.,1. This composition induces isomorphisms on homology in dimensions of the form 2k(p 1) 2i, but not on any other nontrivial positive dimensional homology. Now, using mapping cylinders, we can form the direct limit Xi of the sequence SK -* SK SK *... (all maps being mi) in such a way that the limit space is a CW-complex with homology isomorphic to the direct limit of the homology of the individual spaces, i.e., Hj(X) = 0 unless j has the form Received by the editors May 21, 1971. AMS 1970 subject classifications. Primary 55D20; Secondary 55D40.