Right angularity, flag complexes, asphericity

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 ∈ {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. AMS classification numbers. Primary: 20E40, 20F65, 57M07, 57M10. Secondary: 20F36, 20E42, 20F55.