Spheres and balls as independence complexes
Susan M. Cooper, Sara Faridi, Thiago Holleben, Lisa Nicklasson, Adam Van Tuyl
TL;DR
The paper investigates when independence complexes arising from grafting/whiskering constructions are topologically balls or spheres, linking combinatorial, topological, and algebraic properties. By identifying grafted complexes with generalized Bier balls via polarization, it provides a unifying framework that yields ball/sphere dichotomies and CM characterizations, and extends these insights to independence complexes of very well-covered graphs. Key contributions include a precise characterization of independence complexes of grafted complexes as generalized Bier balls (hence balls or spheres), a thorough analysis of generalized whiskering and colorings, and a complete classification showing that, for very well-covered graphs without isolated vertices, Ind$(G)$ is a pseudomanifold and a ball or sphere exactly when $I(G)$ is CM; it also connects Gorenstein properties to complete intersections through polarization. Together, these results deepen the understanding of the interplay between polarization, Cohen–Macaulayness, and topological types in square-free monomial ideals and their associated independence complexes, with implications for Gorenstein rings and edge ideals.
Abstract
The terms "whiskering", and more generally "grafting", refer to adding generators to any monomial ideal to make the resulting ideal Cohen-Macaulay. We investigate the independence complexes of simplicial complexes that are constructed through a whiskering or grafting process, and we show that these independence complexes are (generalized) Bier balls. More specifically, the independence complexes are either homeomorphic to a ball or a sphere. In a related direction, we classify when the independence complexes of very well-covered graphs are homeomorphic to balls or spheres.