Homology by metric currents

TL;DR

This paper demonstrates that homology defined via metric normal currents aligns with traditional singular homology on suitable metric spaces, linking geometric and topological perspectives.

Contribution

It provides evidence that homology from metric normal currents coincides with singular homology on well-behaved metric spaces, bridging geometry and topology.

Findings

Homology from metric normal currents matches singular homology on nice metric spaces

Supports the geometric interpretation of homology via currents

Establishes a connection between metric currents and classical topological invariants

Abstract

Metric currents are, in a certain sense, a metric analogous of flat currents, therefore are related to the geometry of the space and of their support. In this short note, we try to give some evidence for the previous statement, by showing that the homology which can be defined by means of metric normal currents coincides, on nice enough metric spaces, with the usual singular homology.