Discrete Stratified Morse Theory: Algorithms and A User's Guide

TL;DR

This paper introduces a discrete stratified Morse theory, providing foundational theorems, an algorithm for constructing Morse functions on simplicial complexes, and illustrating its utility with examples and classical theory relations.

Contribution

It develops a discrete analogue of stratified Morse theory, including algorithms and proofs connecting topology and critical simplices in simplicial complexes.

Findings

Proves fundamental theorems relating topology and critical simplices.

Provides an algorithm for constructing stratified Morse functions.

Demonstrates the utility of the theory with examples.

Abstract

Inspired by the works of Forman on discrete Morse theory, which is a combinatorial adaptation to cell complexes of classical Morse theory on manifolds, we introduce a discrete analogue of the stratified Morse theory of Goresky and MacPherson. We describe the basics of this theory and prove fundamental theorems relating the topology of a general simplicial complex with the critical simplices of a discrete stratified Morse function on the complex. We also provide an algorithm that constructs a discrete stratified Morse function out of an arbitrary function defined on a finite simplicial complex; this is different from simply constructing a discrete Morse function on such a complex. We then give simple examples to convey the utility of our theory. Finally, we relate our theory with the classical stratified Morse theory in terms of triangulated Whitney stratified spaces.