Abstract
The main objective of this paper is to extend Morse–Forman theory to vector-valued functions. This is mostly motivated by the need to develop new tools and methods to compute multiparameter persistence. To generalize the theory, in addition to adapting the main definitions and results of Forman to this vectorial setting, we use concepts of combinatorial topological dynamics studied in recent years. This approach proves to be successful in the following ways. First, we establish a result which is more general than that of Forman regarding the sublevel sets of a multidimensional discrete Morse function. Second, we find a way to induce a Morse decomposition in critical components from the critical points of such a function. Finally, we deduce a set of Morse equation and inequalities specific to the multiparameter setting.
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Funding
Guillaume Brouillette was supported by the Natural Sciences and Engineering Council of Canada (NSERC) under grant number 569623 and the Fonds de recherche du Québec – Nature et technologies (FRQNT) under grant number 289126. Tomasz Kaczynski was supported by a Discovery Grant from the NSERC.
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Brouillette, G., Allili, M. & Kaczynski, T. Multiparameter discrete Morse theory. J Appl. and Comput. Topology (2024). https://doi.org/10.1007/s41468-024-00176-7
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DOI: https://doi.org/10.1007/s41468-024-00176-7
Keywords
- Discrete Morse theory
- Multiparameter persistent homology
- Discrete gradient field
- Combinatorial dynamics
- Morse decomposition
- Singularity theory