On Kunneth suspensions

0. Introduction. In (2) we denned the Kunneth suspension of a cohomology operation —the Kunneth suspension involves an arbitrary ess-complex Y rather than the 1-sphere S 1 , as with the usual suspension of a cohomology operation. Now the suspension homomorphism is well known to be related to the operation of forming loop spaces (cf. (4)). The main object of this paper is to prove a similar result for the Kunneth suspension. Our results fall under the following general scheme. There is a natural function where square brackets denote homotopy classes of maps, and X r , Y r are function complexes. Although this function is perfectly explicit, it is not obvious how to compute /? in general, part of the difficulty being that the spaces X r , Z Y have to be computed before ft can be. However, in the case when Z = A, a css-Abelian group, the homotopy type of A T is very simply related to the cohomology of Y and homo-topy of A. Hence in this case, and when X also is a css-Abelian group, we can hope for more convenient expressions for /?; for example, when X and Z are both css-Abelian groups, we show how to express /? in terms of the Kunneth suspension. In section 3 we show how the methods given here may be used to determine the homotopy type of X T by induction on the Postnikov system of X. The problem from which the present work arose was pointed out to me by Dr M. G. Barratt; similar problems are considered by Thorn in (5). The results of this paper, and of (2), formed part of an Oxford doctoral thesis written under the supervision of Dr Barratt, to whom I am deeply indebted for advice and criticism. 1. Preliminaries. We refer the reader to (2) for any notations and definitions not discussed here. The category of ess-complexes with base point is written 2£. The correct product in 2E is the collapsed or smash product X%Y = Yx 7/(Xx*u*x 7). The standard ^-simplex A 9 has no base point, and so we define the complex with base point A«#X = A«xZ/A«x*.