Systolic almost-rigidity modulo 2

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 $c(n,\epsilon )\mathrm{Vol}(M{)}^{1+\epsilon }$, if finite (if $H_1(M;\mathbb{Z}/2)$ is non-trivial).