# On the structure of flat chains modulo $p$

created by marchese on 18 Jul 2016
modified by stuvard on 27 Jun 2019

[BibTeX]

Published Paper

Inserted: 18 jul 2016
Last Updated: 27 jun 2019

Volume: 11
Number: 3
Pages: 309-323
Year: 2018
Doi: 10.1515/acv-2016-0040

ArXiv: 1607.05138 PDF

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 $1$.