Morse theory of Bestvina-Brady type for posets and matchings

TL;DR

This paper develops a Morse theory for posets of Bestvina-Brady type, generalizing Forman's discrete Morse theory to all finite posets and introducing a relative version for comparing topologies.

Contribution

It introduces a new Morse theory framework for posets of Bestvina-Brady type, extending previous theories to all finite posets and including a relative version for subposets.

Findings

Generalizes Forman's discrete Morse theory to all finite posets

Develops a relative Morse theory for posets and subposets

Provides tools for comparing topologies of posets and subposets

Abstract

We introduce a Morse theory for posets of Bestvina-Brady type combining matchings and height functions. This theory generalizes Forman's discrete Morse theory for regular CW-complexes and extends previous results on Morse theory for $h$-regular posets to all finite posets. We also develop a relative version of Morse theory which allows us to compare the topology of a poset with that of a given subposet.