Some remarks on Morse theory for posets, homological Morse theory and finite manifolds

TL;DR

This paper develops a discrete Morse theory for posets, linking the topology of order complexes to critical points of matchings, and extends it to homological variants for studying triangulable manifolds.

Contribution

It introduces a novel discrete Morse theory for posets and a homological version applicable to triangulable homology manifolds, generalizing previous results.

Findings

Established a Morse theory framework for posets.

Connected critical points to the topology of order complexes.

Extended the theory to homological manifolds.

Abstract

We introduce a version of discrete Morse theory for posets. This theory studies the topology of the order complexes K(X) of h-regular posets X from the critical points of admissible matchings on X. Our approach is related to R. Forman's discrete Morse theory for CW-complexes and generalizes Forman and Chari's results on the face posets of regular CW-complexes. We also introduce a homological variant of the theory that can be used to study the topology of triangulable homology manifolds by means of their order triangulations.