One-dimensional and codimension one homology of metric manifolds

TL;DR

This paper compares singular homology and integral current homology in metric manifolds, establishing conditions for their equivalence and the surjectivity of certain homomorphisms, with implications for isoperimetric inequalities.

Contribution

It provides new conditions under which integral current homology matches singular homology in metric manifolds, and links isoperimetric inequalities to homology equivalence.

Findings

Surjective homomorphism from integral current homology to singular homology under certain conditions

Equivalence of one-dimensional homology groups via isoperimetric inequality

Conditions ensuring homology groups coincide in metric manifolds

Abstract

We compare singular homology and homology via integral currents in metric spaces that are homeomorphic to smooth manifolds. For such spaces, we provide sufficient conditions that guarantee the existence of a surjective homomorphism from the codimension one homology group via integral currents to the codimension one singular homology group. Moreover, we show that a one-dimensional isoperimetric inequality for integral currents implies that the one-dimensional homology groups coincide.