Discrete Morse Complexes

TL;DR

This paper explores the properties of discrete Morse complexes derived from simplicial complexes, revealing connections between combinatorial topology and enumerative combinatorics through the study of discrete Morse functions.

Contribution

It introduces the concept of discrete Morse complexes and demonstrates how existing results in topology and combinatorics can be reinterpreted within this framework.

Findings

Discrete Morse complexes form simplicial complexes from Morse functions.

Known combinatorial and topological results can be reinterpreted in this setting.

The study reveals new insights into the structure of discrete Morse functions.

Abstract

We investigate properties of the set of discrete Morse functions on a simplicial complex as defined by Forman. It is not difficult to see that the pairings of discrete Morse functions of a finite simplicial complex again form a simplicial complex, the discrete Morse complex. It turns out that several known results from combinatorial topology and enumerative combinatorics, which previously seemed to be unrelated, can be re-interpreted in the setting of these discrete Morse complexes.