Shellability and Regularity of Chain Complexes over a Principal Ring

TL;DR

This paper extends combinatorial algebraic topology tools to study the homology of abstract chain complexes, introducing shellability and regularity concepts that generalize classical notions and reveal new homological properties.

Contribution

It defines shellability and regularity for abstract chain complexes, broadening the scope of algebraic topology methods beyond classical topological complexes.

Findings

Shellability is a weaker invariant in the abstract setting.

Complete homological information is obtained for regular chain complexes.

Special chain complexes like cones are studied in detail.

Abstract

The goal of this paper is to generalize some of the existing toolkit of combinatorial algebraic topology in order to study the homology of abstract chain complexes. We define shellability of chain complexes in a similar way as for cell complexes and introduce the notion of regular chain complexes. In the case of chain complexes coming from simplicial complexes we recover the classical notions but, in contrast to the topological case, in the abstract setting shellings turn out to be a weaker homological invariant. In particular, we study special chain complexes, which are cones, and a class of regular chain complexes, for which we can obtain complete homological information.