Discrete Morse theory for the barycentric subdivision

TL;DR

This paper develops a method to transfer discrete Morse functions from a simplicial complex to its barycentric subdivision, preserving critical points and gradient paths, thus facilitating analysis of complex topological structures.

Contribution

It introduces a construction of a discrete Morse function on the barycentric subdivision that retains the properties of the original function on the simplicial complex.

Findings

The constructed Morse function on the barycentric subdivision has the same number of critical simplexes as the original.

The gradient path structure is preserved in the subdivision.

The method enables better analysis of topological features through subdivision.

Abstract

Let $F$ be a discrete Morse function on a simplicial complex $L$. We construct a discrete Morse function $\Delta(F)$ on the barycentric subdivision $\Delta(L)$. The constructed function $\Delta(F)$ "behaves the same way" as $F$, i. e. has the same number of critical simplexes and the same gradient path structure.