On the structure of flat chains modulo $p$

TL;DR

This paper establishes that every equivalence class of integral 1-currents modulo p contains an integral current with controlled mass, and explores the implications for the structure and closedness of flat chains modulo p.

Contribution

It proves the existence of integral representatives in each class modulo p and links this to the closedness of certain flat chain families, advancing understanding of flat chains modulo p.

Findings

Every class in the quotient of integral 1-currents modulo p has an integral representative.

The closedness of flat chains of the form pT is established for specific dimensions.

Implications for flat chains of codimension 1 and 0 are derived.

Abstract

In this paper, we prove that every equivalence class in the quotient group of integral $1$-currents modulo $p$ in Euclidean space contains an integral current, with quantitative estimates on its mass and the mass of its boundary. Moreover, we show that the validity of this statement for $m$-dimensional integral currents modulo $p$ implies that the family of $(m-1)$-dimensional flat chains of the form $pT$, with $T$ a flat chain, is closed with respect to the flat norm. In particular, we deduce that such closedness property holds for $0$-dimensional flat chains, and, using a proposition from "The structure of minimizing hypersurfaces mod $4$" by Brian White, also for flat chains of codimension $1$.