Structure of metric cycles and normal one-dimensional currents

We prove that every one-dimensional real Ambrosio-Kirchheim normal current in a Polish (i.e. complete separable metric) space can be naturally represented as an integral of simpler currents associated to Lipschitz curves. As a consequence a representation of every such current with zero boundary (i.e. a cycle) as an integral of so-called elementary solenoids (which are, very roughly speaking, more or less the same as asymptotic cycles introduced by S. Schwartzman)is obtained. The latter result on cycles is in fact a generalization of the analogous result proven by S. Smirnov for classical Whitney currents in a Euclidean space. The same results are true for every complete metric space under suitable set-theoretic assumptions.

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