Connecting Discrete Morse Functions via Birth-Death Transitions

TL;DR

This paper investigates how discrete Morse functions on finite complexes can be transformed into each other through elementary birth-death transitions, providing new proofs of key results and insights into their topological space.

Contribution

It introduces a framework for connecting discrete Morse functions via birth-death transitions and proves that any two such functions are linked by a finite sequence of these moves.

Findings

Any two discrete Morse functions are connected by a finite sequence of birth-death transitions.

Provides alternative proofs of fundamental results in discrete Morse theory.

Studies the topology of the space of discrete Morse functions.

Abstract

We study transformations between discrete Morse functions on a finite simplicial complex via birth-death transitions--elementary chain maps between discrete Morse complexes that either create or cancel pairs of critical simplices. We prove that any two discrete Morse functions $f_1$, $f_2$ on a finite simplicial complex $K$ are linked by a finite sequence of such transitions.As applications, we present alternative proofs of several of Forman's fundamental results in discrete Morse theory and study the topology of the space of discrete Morse functions.