The cohomology ring of product complexes

Abstract

Introduction. T he KIiiiieth formula enables one to determine the integral cohomology group of the product XX Y of the spaces X and Y in terms of the integral cohomology groups H(X) anid H(Y). However, this formula does not enable one to determine the multiplicative structure of the cohomology rinig II(XX Y) in terms of the initegral cohomology rings(2) 1l(X) and II(Y). It is niatural to ask the question: Is the integral cohomology ring H1(XX Y) determinied by the integral cohomology rings 11(X) and H(Y)? This questioni is aniswered in the niegative by the following example: Let X1 = Y1 be the union of the real projective plane aind a one-sphere (circle) with onie point in commoni. Let X2 Y2 be a Klein bottle. It is easy to check that the rings 11(X1) and 1I(X2) are isomorphic. However, the rings II(X1 X Y1) and II(X2 X Y2) are not isomorphic. In particular, there is a onedimenisionial cohomology class aiid a three-dimensional cohomology class in Ii(X2X Y2) whose product is a nionizero four-dimenisionial cohomology class. Oii the other hanid, all products of onie-dimenisionial aiid three-dimensional cohomology classes of II(X1 X Y1) are zero. Hence these two cohomology rings caninot be isomorplhic. These assertions will follow readily from the theorems provecl below. Sinice the aniswerto the above questioni is niegative, it is niatural to inquire: What iniformation about the colhomology rinigs of X aiid Y is needed to determinie the integral cohomology rinig of XX Y? Let II(X, n) denote the cohomology rinig of X with the integers modulo n as coefficienits. Following J. H. C. Whitehlead [8], and Bocksteini [2], we define the spectrum of coho. mology rings of X, or simiiply the cohomology spectrum of X, to be the set of cohomology rinigs IH(X, n), n_()0, together with the coefficient homomorphisms: II(X, n)-*II(X, m), m>O and the Bocksteini homomorphisms of degree +1: II(X, m)--IH(X, 0), m>O. (These homomorphisms are defined in sectioni (1).) Bocksteiii [2] stated but did not prove that the cohomology spectra of X aiid Y determinie the cohomnology rinig IH(XX Y). From these spectra he constructed a group isomorphic to the group HI(X X Y), however, he did not iintroduce aniy products in this construction; thus the question of