Integration of vector fields on cell complexes and Morse theory

TL;DR

This paper develops a theory of gradient vector fields on polyhedral complexes, extending Morse theory to non-manifold spaces using discrete Morse functions and metrics, enabling topological analysis similar to smooth cases.

Contribution

It introduces a novel class of gradient vector fields on polyhedral complexes derived from functions and metrics, improving upon Forman's discrete Morse functions.

Findings

Existence of gradient vector fields with desired properties on polyhedral complexes

A new class of functions improves discrete Morse theory

Framework captures topology of complexes similar to classical Morse theory

Abstract

In this paper, we investigate vector fields on polyhedral complexes and their associated trajectories. We study vector fields which are analogue of the gradient vector field of a function in the smooth case. Our goal is to define a nice theory of trajectories of such vector fields, so that the set of them captures the topology of the polyhedral complex, as in classical Morse theory. Since we do not assume the polyhedral complex to be a manifold, the definition of vector fields on it is very different from the smooth case. Nevertheless, we will show that there exist nice classes of functions and metrics which give gradient vector fields with desired properties. Our construction relies on Forman's discrete Morse theory. In particular, the class of functions we use is an improvement of Forman's discrete Morse functions. A notable feature of our theory is that our gradient vector fields are defined purely from functions and metrics as in the smooth case, contrary to the case of discrete Morse theory where we need the data of dimension of cells. This allows us to implement several useful constructions which were not available in the discrete case.