On the homotopy and strong homotopy type of complexes of discrete Morse functions

TL;DR

This paper investigates the homotopy types of Morse complexes of simplicial complexes, establishing conditions under which they are collapsible and computing their types for various graph families, advancing understanding in topological combinatorics.

Contribution

It provides new results on the homotopy and strong homotopy types of Morse complexes, including collapsibility conditions and computations for specific graph classes.

Findings

Morse complex of complexes with shared leaf vertices is strongly collapsible.

Pure Morse complex of a tree is strongly collapsible.

Homotopy type of Morse complex of disjoint union equals that of the join.

Abstract

In this paper, we determine the homotopy type of the Morse complex of certain collections of simplicial complexes by studying dominating vertices or strong collapses. We show that if $K$ contains two leaves that share a common vertex, then the Morse complex is strongly collapsible and hence has the homotopy type of a point. We also show that the pure Morse complex of a tree is strongly collapsible, thereby recovering as a corollary a result of Ayala et al. In addition, we prove that the Morse complex of a disjoint union $K\sqcup L$ is the Morse complex of the join $K*L$. This result is used to compute the homotopy type of the Morse complex of some families of graphs, including Caterpillar graphs, as well as the automorphism group of a disjoint union for a large collection of disjoint complexes.