A homotopy theoretical generalisation of the Bestvina–Brady construction

Abstract

By using the notion of polyhedral products (X_,A_)K, we recognise the Bestvina–Brady construction [4] as the fundamental group of the homotopy fibre of (S1,⁎)L→S1, where L is a flag complex. We generalise their construction by studying the homotopy fibre F of (S1,⁎)L→(S1,⁎)K for an arbitrary simplicial complex L and K an (m−1)-dimensional simplex. For a particular class of simplicial complexes L, we describe the homology of F, its fixed points, and maximal invariant quotients for coordinate subgroups of Zm. This generalises the work of Leary and Saadetoğlu [13] who studied the case when m=1.