Symmetric Products and a Cartan-Type Formula for Polyhedral Products

We give a geometric method for determining the cohomology groups of a polyhedral product \(Z\big (K;( \underline {X}, \underline {A})\big )\) , under suitable freeness conditions or with coefficients taken in a field k. This is done by considering first the special case where the pair \((X_{i},A_{i }) = (B_{i}\vee C_{i}, B_{i}\vee E_{i})\) for all i and \(E_{i}\hookrightarrow C_{i} \) is a null-homotopic inclusion. We derive a decomposition for these polyhedral products, which resembles a Cartan formula. The theory of symmetric products is used then to generalize the result to arbitrary polyhedral products \(Z\big (K;( \underline {X}, \underline {A})\big )\) . This leads to a direct computation of the Hilbert-Poincaré series for \(Z\big (K;( \underline {X}, \underline {A})\big )\) and to other applications.