Systolic inequalities for the number of vertices

Inspired by the classical Riemannian systolic inequality of Gromov we present a combinatorial analogue providing a lower bound on the number of vertices of a simplicial complex in terms of its edge-path systole. Similarly to the Riemannian case, where the inequality holds under a topological assumpt...

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Bibliographic Details
Published inarXiv.org
Main Authors Avvakumov, Sergey, Balitskiy, Alexey, Hubard, Alfredo, Karasev, Roman
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 14.07.2022
Subjects
Apexes
Combinatorial analysis
Graph theory
Homology
Lower bounds
Systole
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Summary:Inspired by the classical Riemannian systolic inequality of Gromov we present a combinatorial analogue providing a lower bound on the number of vertices of a simplicial complex in terms of its edge-path systole. Similarly to the Riemannian case, where the inequality holds under a topological assumption of "essentiality", our proofs rely on a combinatorial analogue of that assumption. Under a stronger assumption, expressed in terms of cohomology cup-length, we improve our results quantitatively. We also illustrate our methods in the continuous setting, generalizing and improving quantitatively the Minkowski principle of Balacheff and Karam; a corollary of this result is the extension of the Guth--Nakamura cup-length systolic bound from manifolds to complexes.
ISSN:2331-8422