# A family of simplicial complexes and the cohomology of their associated polyhedral products

# A family of simplicial complexes and the cohomology of their associated polyhedral products

A polyhedral product space, $Z_K(X,A)$, is a subspace of a product of spaces. More specifically, it is the union of products of spaces indexed by a simplicial complex. The most well known examples are moment-angle complexes and their real analogue. The polyhedral join (a generalization of the J-construction) is an analogous construction for simplicial complexes. Most notably, these constructions provide a family of simplicial complexes for which moment-angle complexes and real moment-angle complexes are homeomorphic. In general, statements in one polyhedral product can be translated into statements in another polyhedral product of more complicated spaces $(X,A)$ but over a simpler simplicial complex, or vice versa. I will discuss the effect of the polyhedral join on the cohomology of the associated polyhedral product.