Morse frames
Gilles Bertrand, Laurent Najman
TL;DR
The paper addresses computing topological invariants of finite simplicial complexes by introducing Morse frames, a framework built on Morse sequences that attaches sets of critical faces to each simplex. The main construct, the Morse reference (and its co-reference), links to gradient flows and yields a Morse complex via $d_p(c)=\Upsilon(\partial_p(c))$, establishing an isomorphism $H_p(K) \cong H_p(\ddot{W})$ and enabling a new annotation-based view of persistent cohomology. It further provides a mod $2$ Betti-number computation approach leveraging the reference and discusses practical implementations, complexity, and memory advantages over existing methods. Overall, Morse frames unify gradient-flow intuition with discrete Morse theory, offering efficient, one-pass gradient-field construction and a two-pass route for Betti-number computation, with potential extensions to persistence and other coefficient fields $\mathbb{F}$.
Abstract
In the context of discrete Morse theory, we introduce Morse frames, which are maps that associate a set of critical simplexes to all simplexes. The main example of Morse frames are the Morse references. In particular, these Morse references allow computing Morse complexes, an important tool for homology. We highlight the link between Morse references and gradient flows. We also propose a novel presentation of the Annotation algorithm for persistent cohomology, as a variant of a Morse frame. Finally, we propose another construction, that takes advantage of the Morse reference for computing the Betti numbers in mod 2 arithmetic.