The Connectedness Homomorphism between Discrete Morse Complexes

TL;DR

This paper introduces the connectedness homomorphism in discrete Morse theory, providing a new framework for analyzing connectedness at the chain complex level and drawing parallels to smooth Morse theory.

Contribution

It presents a novel connectedness homomorphism for discrete Morse complexes and applies it to describe discrete cusp-degeneration phenomena.

Findings

Defines the connectedness homomorphism for discrete Morse complexes

Establishes a discrete analogue of cusp-degeneration in Morse theory

Provides a comparison between smooth and discrete Morse complexes

Abstract

Given two discrete Morse functions on a simplicial complex, we introduce the {\em connectedness homomorphism} between the corresponding discrete Morse complexes. This concept leads to a novel framework for studying the connectedness in discrete Morse theory at the chain complex level. In particular, we apply it to describe a discrete analogy to `cusp-degeneration' of Morse complexes. A precise comparison between smooth case and our discrete cases is also given.