Morse sequences on stacks and flooding sequences
Gilles Bertrand
TL;DR
This paper generalizes discrete Morse theory by introducing Morse sequences on stacks, where a monotone weight function $F$ on a simplicial complex $K$ governs allowable expansions and fillings. It shows that the gradient field of any Morse sequence on a stack can be recovered from a flooding sequence, a watershed-like refinement, and develops cosimplicial-based schemes to compute such sequences. Two main computational schemes are proposed: (i) left-to-right constructions of Morse sequences on cosimplicial sets with maximal, minimal, and min/max variants, and (ii) a per-level Scheme Sch2 that builds flooding sequences by concatenating level-wise subsequences, achieving $O(dN)$ time in the sequential setting and enabling parallelism. The work accommodates non-injective stacks, extends to weighted complexes, and points to future directions in persistent homology and TDA, with applicability to image segmentation and possible adaptation to cubical frameworks.
Abstract
This paper builds upon the framework of \emph{Morse sequences}, a simple and effective approach to discrete Morse theory. A Morse sequence on a simplicial complex consists of a sequence of nested subcomplexes generated by expansions and fillings-two operations originally introduced by Whitehead. Expansions preserve homotopy, while fillings introduce critical simplexes that capture essential topological features. We extend the notion of Morse sequences to \emph{stacks}, which are monotonic functions defined on simplicial complexes, and define \emph{Morse sequences on stacks} as those whose expansions preserve the homotopy of all sublevel sets. This extension leads to a generalization of the fundamental collapse theorem to weighted simplicial complexes. Within this framework, we focus on a refined class of sequences called \emph{flooding sequences}, which exhibit an ordering behavior similar to that of classical watershed algorithms. Although not every Morse sequence on a stack is a flooding sequence, we show that the gradient vector field associated with any Morse sequence can be recovered through a flooding sequence. Finally, we present algorithmic schemes for computing flooding sequences using cosimplicial complexes.