Systolic freedom and rigidity modulo 2

05.09.2022 14:15 – 15:15

The $k$-dimensional systole of an $n$-dimensional closed Riemannian manifold $M$ is the infimal $k$-volume of a non-trivial $k$-cycle (with some coefficients). In '90s, Gromov asked if the product of the $k$-systole and the $(n-k)$-systole is bounded from above by the volume of $M$ (up to a dimensional factor); this would manifest the \textit{systolic rigidity}. Freedman exhibited the first examples with $k=1$ and mod 2 coefficients where this fails; this manifests the \textit{systolic freedom}. I will give an overview of classical results and constructions, and will sketch the proof of our recent result with Hannah Alpert and Larry Guth. We show that Freedman's examples are almost as "free" as possible, and the systolic rigidity almost holds, with $k=1$ and mod 2 coefficients. Namely, on a manifold of bounded local geometry, $\mbox{systole}_1(M) \cdot \mbox{systole}_{n-1}(M) \le c_\epsilon \mbox{volume}(M)^{1+\epsilon}$, as long as the left-hand side is finite ($H_1(M; \mathbb{Z}/2)$ is non-trivial).

Lieu

Salle 1-15, Séminaire "Analysis seminar "

entrée libre