On the support of Betti tables of multiparameter persistent homology modules
On the support of Betti tables of multiparameter persistent homology modules
Persistent homology encodes the evolution of homological features of a multifiltered cell complex in the form of a multigraded module over a polynomial ring, called a multiparameter persistence module, and quantifies it through invariants suitable for topological data analysis. In this paper, we establish relations between the Betti tables, a standard invariant for multigraded modules commonly used in multiparameter persistence, and the multifiltered cell complex. In particular, we show that the grades at which cells of specific dimensions first appear in the filtration reveal all positions in which the Betti tables are possibly non-zero. This result can be used in combination with discrete Morse theory on the multifiltered cell complex originating the module to obtain a better approximation of the support of the Betti tables. In the case of bifiltrations, we refine our results by considering homological critical grades of a filtered chain complex instead of entrance grades of cells.
Guidolin, Andrea
40011dc4-77ce-4d11-90bd-02e76c0b375a
Landi, Claudia
eb305c65-24f4-4cea-b3c0-1492a2496dd2
5 August 2025
Guidolin, Andrea
40011dc4-77ce-4d11-90bd-02e76c0b375a
Landi, Claudia
eb305c65-24f4-4cea-b3c0-1492a2496dd2
Guidolin, Andrea and Landi, Claudia
(2025)
On the support of Betti tables of multiparameter persistent homology modules.
The Quarterly Journal of Mathematics.
(doi:10.1093/qmath/haaf021).
Abstract
Persistent homology encodes the evolution of homological features of a multifiltered cell complex in the form of a multigraded module over a polynomial ring, called a multiparameter persistence module, and quantifies it through invariants suitable for topological data analysis. In this paper, we establish relations between the Betti tables, a standard invariant for multigraded modules commonly used in multiparameter persistence, and the multifiltered cell complex. In particular, we show that the grades at which cells of specific dimensions first appear in the filtration reveal all positions in which the Betti tables are possibly non-zero. This result can be used in combination with discrete Morse theory on the multifiltered cell complex originating the module to obtain a better approximation of the support of the Betti tables. In the case of bifiltrations, we refine our results by considering homological critical grades of a filtered chain complex instead of entrance grades of cells.
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GL-Pure-support-Betti
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haaf021
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Accepted/In Press date: 3 July 2025
Published date: 5 August 2025
Identifiers
Local EPrints ID: 504162
URI: http://eprints.soton.ac.uk/id/eprint/504162
ISSN: 0033-5606
PURE UUID: b9fd95d7-1319-4fdd-83fd-240318e2601e
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Date deposited: 28 Aug 2025 16:39
Last modified: 29 Aug 2025 02:19
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Author:
Andrea Guidolin
Author:
Claudia Landi
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