Greedy Morse matchings and discrete smoothness

TL;DR

This paper introduces the concept of discrete smoothness to ensure that greedy Morse matchings accurately reflect the dynamics of a sampled function on complexes, providing theoretical guarantees and applications in topology.

Contribution

It defines discrete smoothness as a minimal sampling condition and extends theoretical guarantees for greedy Morse matchings to this setting, including a combinatorial proof of CAT(0) cube complexes collapsibility.

Findings

Discrete smoothness guarantees the geometric faithfulness of discrete gradients.

Theoretical results are extended from general complexes to the smooth case.

A combinatorial proof shows all CAT(0) cube complexes are collapsible.

Abstract

Discrete Morse theory emerged as an essential tool for computational geometry and topology. Its core structures are discrete gradient fields, defined as acyclic matchings on a complex $C$, from which topological and geometrical informations of $C$ can be efficiently computed, in particular its homology or Morse-Smale decomposition. Given a function $f$ sampled on $C$, it is possible to derive a discrete gradient that mimics the dynamics of $f$. Many such constructions are based on some variant of a greedy pairing of adjacent cells, given an appropriate weighting. However, proving that the dynamics of $f$ is correctly captured by this process is usually intricate. This work introduces the notion of discrete smoothness of the pair $(f,C)$, as a minimal sampling condition to ensure that the discrete gradient is geometrically faithful to $f$. More precisely, a discrete gradient construction from a function $f$ on a polyhedron complex $C$ of any dimension is studied, leading to theoretical guarantees prior to the discrete smoothness assumption. Those results are then extended and completed for the smooth case. As an application, a purely combinatorial proof that all CAT(0) cube complexes are collapsible is given.