Systolic freedom and rigidity modulo 2

Abstract: The $k$-dimensional systole of a closed Riemannian $n$-dimensional 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 \emph{systolic rigidity}. Freedman exhibited the first examples with $k=1$ and mod 2 coefficients where this fails; this manifests the \emph{systolic freedom}. In a joint work in progress 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). The proof, which I will explain, is based on the Schoen--Yau--Guth--Papasoglu minimal surface method.

differential geometrydynamical systemsgeometric topologysymplectic geometry

Audience: researchers in the topic

Geometry and Dynamics seminar

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