A Cartan formula for the cohomology of polyhedral products and its application to the ring structure

Abstract

We give a geometric method for determining the cohomology groups and the product structure of a polyhedral product Z K; (X, A) , under suitable freeness conditions or with coefficients taken in a field k. This is done by considering first the special caseand then deriving a decomposition for these polyhedral products which resembles a Cartan formula. The result is then generalized to arbitrary Z K; (X, A) . This leads to a direct computation of the Hilbert-Poincaré series for Z K; (X, A) . Other applications are includedThe product structure on H * Z K; (X, A) is described in terms of the additive generators, labelled via the Cartan decomposition. This is done by combining with the Wedge Lemma, the description in [3] of the * -product structure of a polyhedral product which is induced from the stable splitting The description given suffices to enable explicit calculations.