Systolic almost-rigidity modulo 2

TL;DR

This paper proves a form of almost-rigidity for the product of mod 2 systoles in Riemannian manifolds, showing they are bounded by a volume-dependent inequality, indicating limited systolic freedom.

Contribution

It establishes a new systolic inequality for manifolds with non-trivial first homology mod 2, demonstrating a form of almost-rigidity in systolic geometry.

Findings

No power law systolic freedom for mod 2 systoles of dimension 1 and codimension 1.

Systolic inequalities hold for manifolds with bounded local geometry.

Product of mod 2 systoles is bounded by volume to a power close to 1.

Abstract

No power law systolic freedom is possible for the product of mod $2$ systoles of dimension $1$ and codimension $1$. This means that any closed $n$-dimensional Riemannian manifold $M$ of bounded local geometry obeys the following systolic inequality: the product of its mod $2$ systoles of dimensions $1$ and $n-1$ is bounded from above by $c(n,\varepsilon) \mbox{Vol}(M)^{1+\varepsilon}$, if finite (if $H_1(M; \mathbb{Z}/2)$ is non-trivial).