Systolic inequalities for the number of vertices

TL;DR

This paper establishes combinatorial systolic inequalities relating the number of vertices in a simplicial complex to its edge-path systole, extending classical Riemannian results to a discrete setting with topological and cohomological assumptions.

Contribution

It introduces a combinatorial analogue of Gromov's systolic inequality, providing new bounds on vertices based on edge-path systole and cohomology cup-length, and extends these ideas to continuous complexes.

Findings

Lower bounds on vertices in terms of edge-path systole

Quantitative improvements under cohomology cup-length assumptions

Extension of systolic bounds from manifolds to simplicial complexes

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.