Duality in Toric Topology

There are numerous notions of duality across mathematics. In this talk we explore the interplay between dualities in combinatorics, topology, and algebra, via a study of cup and cap products in moment-angle complexes and related polyhedral product spaces. We obtain both a description of the cap product and a characterisation of Poincaré duality in $\mathcal Z_K$, in terms of the combinatorics of the simplicial complex $K$. We then show how these results can be extended to a broader class of polyhedral products via simplicial substitution, and give a similar combinatorial characterisation of duality in such spaces.