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arXiv · 2106.10429

Systolic inequalities for the number of vertices

Sergey Avvakumov·Alexey Balitskiy·Alfredo Hubard·Roman Karasev
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Abstract

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.

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Sergey Avvakumov, Alexey Balitskiy, Alfredo Hubard, Roman Karasev. 2021-06-19. Systolic inequalities for the number of vertices. https://arxiv.org/abs/2106.10429

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