Decomposition of acyclic normal currents in a metric space

TL;DR

This paper generalizes Smirnov's classical Euclidean space result by proving that acyclic normal currents in Polish metric spaces can be decomposed into curves, extending the theory to more general metric spaces.

Contribution

It extends the decomposition theorem of acyclic normal currents from Euclidean spaces to general Polish metric spaces, under certain set-theoretic assumptions.

Findings

Acyclic normal currents in Polish spaces can be decomposed into curves.

The decomposition holds in complete metric spaces with set-theoretic assumptions.

Generalization of classical Euclidean results to broader metric space contexts.

Abstract

We prove that every acyclic normal one-dimensional real Ambrosio-Kirchheim current in a Polish (i.e. complete separable metric) space can be decomposed in curves, thus generalizing the analogous classical result proven by S. Smirnov in Euclidean space setting. The same assertion is true for every complete metric space under a suitable set-theoretic assumption.