Criteria for asphericity of polyhedral products: corrigenda to “right-angularity, flag complexes, asphericity”

Abstract

Given a simplicial complex \(L\) with vertex set \(I\) and a family \({\mathbf A}=\{(A(i),B(i))\}_{i\in I}\) of pairs of spaces with base points \(*_i\in B(i)\), there is a definition of the “polyhedral product” \({{\mathbf A}^L}\) of \({\mathbf A}\) with respect to \(L\). Sometimes this is called a “generalized moment angle complex”. This note concerns two refinements to earlier work of the first author. First, when \(L\) is infinite, the definition of polyhedral product needs clarification. Second, the earlier paper omitted some subtle parts of the necessary and sufficient conditions for polyhedral products to be aspherical. Correct versions of these necessary and sufficient conditions are given in the present paper.