In this paper, we define the homological Morse numbers of a filtered cell complex in terms of the 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 phonological Morse numbers. Furthermore, we prove a sharp upper bound for homological Morse numbers, expressed again in terms of the Betti tables.

Morse inequalities for the Koszul complex of multi-persistence / Guidolin, Andrea; Landi, Claudia. - In: JOURNAL OF PURE AND APPLIED ALGEBRA. - ISSN 0022-4049. - 227:7(2023), pp. 107319---. [10.1016/j.jpaa.2023.107319]

Morse inequalities for the Koszul complex of multi-persistence

Landi, Claudia
2023-01-01

Abstract

In this paper, we define the homological Morse numbers of a filtered cell complex in terms of the 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 phonological Morse numbers. Furthermore, we prove a sharp upper bound for homological Morse numbers, expressed again in terms of the Betti tables.
227
7
107319
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Morse inequalities for the Koszul complex of multi-persistence / Guidolin, Andrea; Landi, Claudia. - In: JOURNAL OF PURE AND APPLIED ALGEBRA. - ISSN 0022-4049. - 227:7(2023), pp. 107319---. [10.1016/j.jpaa.2023.107319]
Guidolin, Andrea; Landi, Claudia
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11380/1295215
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