Right-angularity, flag complexes, asphericity

Abstract

The “polyhedral product functor” produces a space from a simplicial complex L and a collection of pairs of spaces, {(A(i), B(i))}, where i ranges over the vertex set of L. We give necessary and sufficient conditions for the resulting space to be aspherical. There are two similar constructions, each of which starts with a space X and a collection of subspaces, {Xi}, where \({i \in \{0,1. . . . , n\}}\) , and then produces a new space. We also give conditions for the results of these constructions to be aspherical. All three techniques can be used to produce examples of closed aspherical manifolds.