Star clusters in the Matching, Morse, and Generalized Morse complex

TL;DR

This paper determines the homotopy types of various Morse and matching complexes using star cluster collapses, providing new insights into their topological structures and relationships.

Contribution

It introduces a method using star cluster collapses to compute homotopy types of complexes, including new results for extended star graphs and Dutch windmill graphs.

Findings

Homotopy type of Morse complex of extended star graph computed

Homotopy type of matching complex of Dutch windmill graph determined

Relationships between matching complex and generalized Morse complex analyzed

Abstract

In this paper, we determine the homotopy type of the Morse complex and matching complex of multiple families of complexes by utilizing star cluster collapses and the Cluster Lemma. We compute the homotopy type of the Morse complex of an extended notion of a star graph, as well as the homotopy type of the matching complex of a Dutch windmill graph. Additionally, we provide alternate computations of the homotopy type of the Morse complex of paths, the homotopy type of the matching complex of paths, and the homotopy type of the matching complex of cycles. We then use this same method of computing homotopy types to investigate the relationship between the homotopy type of the matching complex and the generalized Morse complex.