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The construction of a simplicial complex given by polyhedral joins (introduced by Anton Ayzenberg), generalizes Bahri, Bendersky, Cohen and Gitlers $J$-construction and simplicial wedge construction. This article gives a cohomological decomposition of a polyhedral product over a polyhedral join for certain families of pairs of simplicial complexes. A formula for the Hilbert-Poincar{e} series is given, which generalizes Ayzenbergs formula for the moment-angle complex.
Polyhedral products were defined by Bahri, Bendersky, Cohen and Gitler, to be spaces obtained as unions of certain product spaces indexed by the simplices of an abstract simplicial complex. In this paper we give a very general homotopy theoretic cons
A generalised Postnikov tower for a space $X$ is a tower of principal fibrations with fibres generalised Eilenberg-MacLane spaces, whose inverse limit is weakly homotopy equivalent to $X$. In this paper we give a characterisation of a polyhedral prod
A panel structure on a topological space is just a locally finite family of closed subspaces. A space together with a panel structure is called a space with faces. In this paper, we define the notion of polyhedral product over a space with faces. Thi
For a d-dimensional polyhedral complex P, the dimension of the space of piecewise polynomial functions (splines) on P of smoothness r and degree k is given, for k sufficiently large, by a polynomial f(P,r,k) of degree d. When d=2 and P is simplicial,
For every knot K with stick number k there is a knotted polyhedral torus of knot type K with 3k vertices. We prove that at least 3k-2 vertices are necessary.