Systolic geometry and regularization technique

TL;DR

This paper introduces the concept of systole in Riemannian manifolds and discusses the regularization technique that connects systolic volume to topological invariants like Betti numbers and minimal entropy.

Contribution

It presents the regularization technique as a new method to relate systolic geometry to homotopical invariants of manifolds.

Findings

Regularization technique links systolic volume to Betti numbers.

It connects systolic geometry with minimal entropy.

Provides an overview of systolic geometry concepts.

Abstract

The aim of this text is to present the concept of systole of a compact riemannian manifold and to give an overview of systolic geometry. I will also present the "regularization technique", which leads to major results in systolic geometry. I will detail how this technique allows to link the systolic volume of some closed riemannian manifolds to homotopical invariants of these manifolds, such as the Betti numbers (according to Gromov) and the minimal entropy (according to Sabourau).