Combinatorial systolic inequalities

TL;DR

This paper develops combinatorial analogues of classical systolic inequalities, linking geometric properties of smooth manifolds with combinatorial features of their triangulations, and establishes equivalences between geometric and combinatorial systolic inequalities.

Contribution

It introduces combinatorial systolic inequalities and proves their equivalence to classical inequalities for manifolds, bridging geometry and combinatorics in triangulations.

Findings

Establishes combinatorial systolic inequalities for triangulated manifolds.

Shows equivalence between geometric and combinatorial systolic inequalities.

Provides methods to relate Riemannian metrics to triangulation combinatorics.

Abstract

We establish combinatorial versions of various classical systolic inequalities. For a smooth triangulation of a closed smooth manifold, the minimal number of edges in a homotopically non-trivial loop contained in the $1$-skeleton gives an integer called the combinatorial systole. The number of top-dimensional simplices in the triangulation gives another integer called the combinatorial volume. We show that a class of smooth manifolds satisfies a systolic inequality for all Riemannian metrics if and only if it satisfies a corresponding combinatorial systolic inequality for all smooth triangulations. Along the way, we show that any closed Riemannian manifold has a smooth triangulation which "remembers" the geometry of the Riemannian metric, and conversely, that every smooth triangulation gives rise to Riemannian metrics which encode the combinatorics of the triangulation. We give a few applications of these results.