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Morse inequalities for the Koszul complex of multi-persistence

Morse inequalities for the Koszul complex of multi-persistence
Morse inequalities for the Koszul complex of multi-persistence
In this paper, we define the homological Morse numbers of a filtered cell complex in terms of relative homology of nested filtration pieces, and derive inequalities relating these numbers to the Betti tables of the multi-parameter persistence modules of the considered filtration. Using the Mayer-Vietoris spectral sequence we first obtain strong and weak Morse inequalities involving the above quantities, and then we improve the weak inequalities achieving a sharp lower bound for homological Morse numbers. Furthermore, we prove a sharp upper bound for homological Morse numbers, expressed again in terms of the Betti tables.
0022-4049
Guidolin, Andrea
40011dc4-77ce-4d11-90bd-02e76c0b375a
Landi, Claudia
b4b4ed74-f626-4733-900f-96f93af92f99
Guidolin, Andrea
40011dc4-77ce-4d11-90bd-02e76c0b375a
Landi, Claudia
b4b4ed74-f626-4733-900f-96f93af92f99

Guidolin, Andrea and Landi, Claudia (2023) Morse inequalities for the Koszul complex of multi-persistence. Journal of Pure and Applied Algebra, 227 (7), [107319]. (doi:10.1016/j.jpaa.2023.107319).

Record type: Article

Abstract

In this paper, we define the homological Morse numbers of a filtered cell complex in terms of relative homology of nested filtration pieces, and derive inequalities relating these numbers to the Betti tables of the multi-parameter persistence modules of the considered filtration. Using the Mayer-Vietoris spectral sequence we first obtain strong and weak Morse inequalities involving the above quantities, and then we improve the weak inequalities achieving a sharp lower bound for homological Morse numbers. Furthermore, we prove a sharp upper bound for homological Morse numbers, expressed again in terms of the Betti tables.

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e-pub ahead of print date: 9 January 2023
Published date: 19 January 2023

Identifiers

Local EPrints ID: 500360
URI: http://eprints.soton.ac.uk/id/eprint/500360
DOI: doi:10.1016/j.jpaa.2023.107319
ISSN: 0022-4049
PURE UUID: 5e7d54de-3626-4b8f-9201-fb3cbdbc6390
ORCID for Andrea Guidolin: ORCID iD orcid.org/0000-0002-7397-475X

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Date deposited: 28 Apr 2025 16:36
Last modified: 22 Aug 2025 02:47

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Author: Andrea Guidolin ORCID iD
Author: Claudia Landi

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