Cup-products for the polyhedral product functor

Davis-Januszkiewicz introduced manifolds which are now known as moment-angle manifolds over a polytope [6]. Buchstaber-Panov introduced and extensively studied moment-angle complexes defined for any abstract simplicial complex K [4]. They completely described the rational cohomology ring structure in terms of the Tor-algebra of the Stanley-Reisner algebra [4]. Subsequent developments were given in work of Denham-Suciu [7] and Franz [9] which were followed by [1, 2]. Namely, given a family of based CW-pairs (X, A) = {(Xi, Ai)}(i=1)(m) together with an abstract simplicial complex K with m vertices, there is a direct extension of the Buchstaber-Panov moment-angle complex. That extension denoted Z(K; (X, A)) is known as the polyhedral product functor, terminology due to Bill Browder, and agrees with the Buchstaber-Panov moment-angle complex in the special case (X, A) = (D-2, S-1) [1, 2]. A decomposition theorem was proven which splits the suspension of Z(K; (X, A)) into a bouquet of spaces determined by the full sub-complexes of K. This paper is a study of the cup-product structure for the cohomology ring of Z(K; (X, A)). The new result in the current paper is that the structure of the cohomology ring is given in terms of this geometric decomposition arising from the "stable" decomposition of Z(K; (X, A)) [1, 2]. The methods here give a determination of the cohomology ring structure for many new values of the polyhedral product functor as well as retrieve many known results. Explicit computations are made for families of suspension pairs and for the cases where Xi is the cone on Ai. These results complement and extend those of Davis-Januszkiewicz [6], Buchstaber-Panov [3, 4], Panov [13], Baskakov-Buchstaber-Panov, [3], Franz, [8, 9], as well as Hochster [12]. Furthermore, under the conditions stated below (essentially the strong form of the K "unneth theorem), these theorems also apply to any cohomology theory.