Morse sequences on stacks and flooding sequences
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918290807324672 |
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| author | Bertrand, Gilles |
| author_facet | Bertrand, Gilles |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_01384 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Morse sequences on stacks and flooding sequences Bertrand, Gilles Discrete Mathematics Algebraic Topology 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. |
| title | Morse sequences on stacks and flooding sequences |
| topic | Discrete Mathematics Algebraic Topology |
| url | https://arxiv.org/abs/2509.01384 |