Regular geometric cycles

TL;DR

This paper provides a detailed explanation and complete proof of Gromov's theorem on the existence of regular geometric cycles representing homology classes with controlled volume and systolic properties.

Contribution

It offers a comprehensive proof of Gromov's result on regular geometric cycles, clarifying the construction and properties of these cycles in relation to systolic volume.

Findings

Existence of regular geometric cycles representing homology classes.

Control over volume of small-radius balls within cycles.

Approximation of systolic volume by these cycles.

Abstract

Let $\pi$ be a finitely presented group. If h is a non trivial homology class in Hn($\pi$; Z), a theorem of Gromov (see [Gro83], 6) asserts the existence of regular geometric cycles which represent h, whose relative systolic volume is as close as desired to the systolic volume of h, in which we can control the volume of balls of radius less than half of the cycle's relative systol. The aim of this note is to explain and provide a complete proof of this result.