Discrete Morse Theory Is At Least As Perfect As Morse Theory

TL;DR

This paper demonstrates that Discrete Morse theory can match the precision of classical Morse theory in bounding manifold homology, with implications for triangulations and combinatorial properties.

Contribution

It proves that Discrete Morse theory can achieve the same bounds as classical Morse theory and extends its applications to triangulations and combinatorial topology.

Findings

Discrete Morse theory recovers classical Morse bounds on homology.

Every simply connected smooth d-manifold (except possibly d=4) admits a locally constructible triangulation.

Classical geometric connectivity can be interpreted via collapse depth in a combinatorial framework.

Abstract

In bounding the homology of a manifold, Forman's Discrete Morse theory recovers the full precision of classical Morse theory: Given a PL triangulation of a manifold that admits a Morse function with c_i critical points of index i, we show that some subdivision of the triangulation admits a boundary-critical discrete Morse function with c_i interior critical faces of dimension d-i. This dualizes and extends a recent result by Gallais. Further consequences of our work are: (1) Every simply connected smooth d-manifolds (except possibly when d=4) admits a locally constructible triangulation. (This solves a problem by Zivaljevic.) (2) Up to refining the subdivision, the classical notion of geometric connectivity can be translated combinatorially via the notion of collapse depth.