Flat currents modulo p in metric spaces and filling radius inequalities

TL;DR

This paper extends the theory of metric space currents to coefficients in Z_p, establishing isoperimetric inequalities mod(p) and applying them to prove Gromov's filling radius inequality for nonorientable manifolds.

Contribution

It develops a new framework for currents with Z_p coefficients in metric spaces and proves isoperimetric inequalities mod(p), enabling the extension of filling radius inequalities to nonorientable manifolds.

Findings

Established isoperimetric inequalities mod(p) in Banach spaces.

Proved Gromov's filling radius inequality for nonorientable manifolds.

Applied Ekeland's principle to find quasi-minimizers of mass mod(p).

Abstract

We adapt the theory of currents in metric spaces, as developed by the first-mentioned author in collaboration with B. Kirchheim, to currents with coefficients in Z_p. Building on S. Wenger's work in the orientable case, we obtain isoperimetric inequalities mod(p) in Banach spaces and we apply these inequalities to provide a proof of Gromov's filling radius inequality (and therefore also the systolic inequality) which applies to nonorientable manifolds, as well. With this goal in mind, we use the Ekeland principle to provide quasi-minimizers of the mass mod(p) in the homology class, and use the isoperimetric inequality to give lower bounds on the growth of their mass in balls.