Higher connectivity of the Morse complex

TL;DR

This paper investigates the connectivity properties of the Morse complex of a finite simplicial complex, revealing that higher connectivity is achieved as the maximum degree or number of edges increases, with classifications for connectivity levels.

Contribution

It introduces new results on the high connectivity of Morse complexes and classifies when these complexes are connected or simply connected, using advanced Morse theory techniques.

Findings

Morse complex becomes highly connected as maximum degree increases

Connectivity classification for Morse complex in various cases

Application of Bestvina-Brady Morse theory to generalized Morse complexes

Abstract

The Morse complex $\mathcal{M}(\Delta)$ of a finite simplicial complex $\Delta$ is the complex of all gradient vector fields on $\Delta$. In this paper we study higher connectivity properties of $\mathcal{M}(\Delta)$. For example, we prove that $\mathcal{M}(\Delta)$ gets arbitrarily highly connected as the maximum degree of a vertex of $\Delta$ goes to $\infty$, and for $\Delta$ a graph additionally as the number of edges goes to $\infty$. We also classify precisely when $\mathcal{M}(\Delta)$ is connected or simply connected. Our main tool is Bestvina-Brady Morse theory, applied to a "generalized Morse complex."