Birth and death in discrete Morse theory

TL;DR

This paper investigates the birth and death of critical cells in discrete Morse functions over a finite simplicial complex across different parameter slices, providing an algorithm for cell pairing with applications in data imaging.

Contribution

It introduces an algorithm for pairing critical cells across slices in discrete Morse theory, including cases with varying triangulations, with potential applications in data imaging.

Findings

Algorithm for pairing critical cells in fixed triangulations

Generalization to varying triangulations

Potential applications in data imaging

Abstract

Suppose $M$ is a finite simplicial complex and that for $0=t_0,t_1,...,t_r=1$ we have a discrete Morse function $F_{t_i}:M\to \zr$. In this paper, we study the births and deaths of critical cells for the functions $F_{t_i}$ and present an algorithm for pairing the cells that occur in adjacent slices. We first study the case where the triangulation of $M$ is the same for each $t_i$, and then generalize to the case where the triangulations may differ. This has potential applications in data imaging, where one has function values at a sample of points in some region in space at several different times or at different levels in an object.