Morse inequalities for the Koszul complex of multi-persistence

Guidolin, Andrea; Landi, Claudia

doi:10.1016/j.jpaa.2023.107319

Cited by 3 publications

(3 citation statements)

References 26 publications

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“…In Section 3.1, upon briefly recalling general definitions and results, we provide a more detailed description of Koszul complexes of multiparameter persistent homology modules. We claim no original results in this subsection, as the Koszul complex is a standard tool, and the explicit description of its chain modules and differentials in the case of persistent homology modules is included, for example, in [GL23,Sect. 3].…”

Section: The Koszul Complex Of a Persistence Modulementioning

confidence: 99%

“…and call it the set of q-homological critical grades (see [GL23]). For any fixed u ∈ N 2 and any q ∈ N, let us recall the following known inequalities (see [LS22, Corollary 1], and [GL23] for a generalization to the case n ≥ 2):…”

Section: Homological Critical Grades and Support Of Betti Tables For ...mentioning

confidence: 99%

“…Starting from the observation that for a 1-parameter persistent homology module the support of the Betti tables coincides with the set of entrance grades of critical cells in the filtration under consideration, our goal is to understand whether and to what extent this fact can be generalized to multiparameter persistence. An indication that this may be the case comes from the results of [GL23], which establish Morse inequalities involving, on the one hand, the values of the Betti tables of a multiparameter persistent homology module, and, on the other hand, the so-called homological critical numbers of the same filtration. The latter numbers can be viewed as theoretical lower bounds of the numbers of critical cells entering the filtration at each filtration grade for any choice of a discrete gradient vector field consistent with the filtration.…”

Section: Introductionmentioning

confidence: 99%

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On the support of Betti tables of multiparameter persistent homology modules

Guidolin¹,

Landi²

2023

Preprint

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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 a relation between Betti tables, a standard invariant for multigraded modules commonly used in multiparameter persistence, and the critical cells determined by discrete Morse theory on the filtered cell complex originating the module. In particular, we show that for a discrete gradient vector field consistent with a given multiparameter sublevel set filtration, the grades at which its critical cells appear in the filtration reveal all positions in which the Betti tables are possibly nonzero. This result is refined in the case of bifiltrations by considering homological critical grades of a filtered chain complex instead of entrance grades of critical cells.

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Section: The Koszul Complex Of a Persistence Modulementioning

confidence: 99%

Section: Homological Critical Grades and Support Of Betti Tables For ...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the support of Betti tables of multiparameter persistent homology modules

Guidolin¹,

Landi²

2023

Preprint

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Koszul Complexes and Relative Homological Algebra of Functors Over Posets

Chachólski,

Guidolin,

Ren

et al. 2024

Found Comput Math

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Under certain conditions, Koszul complexes can be used to calculate relative Betti diagrams of vector space-valued functors indexed by a poset, without the explicit computation of global minimal relative resolutions. In relative homological algebra of such functors, free functors are replaced by an arbitrary family of functors. Relative Betti diagrams encode the multiplicities of these functors in minimal relative resolutions. In this article we provide conditions under which grading the chosen family of functors leads to explicit Koszul complexes whose homology dimensions are the relative Betti diagrams, thus giving a scheme for the computation of these numerical descriptors.

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From the Mayer–Vietoris spectral sequence to überhomology

Caputi,

Celoria,

Collari

2023

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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We prove that the second page of the Mayer–Vietoris spectral sequence, with respect to anti-star covers, can be identified with another homological invariant of simplicial complexes: the $0$ -degree überhomology. Consequently, we obtain a combinatorial interpretation of the second page of the Mayer–Vietoris spectral sequence in this context. This interpretation is then used to extend the computations of bold homology, which categorifies the connected domination polynomial at $-1$ .

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Morse inequalities for the Koszul complex of multi-persistence

Cited by 3 publications

References 26 publications

On the support of Betti tables of multiparameter persistent homology modules

On the support of Betti tables of multiparameter persistent homology modules

Koszul Complexes and Relative Homological Algebra of Functors Over Posets

From the Mayer–Vietoris spectral sequence to überhomology

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