The metric fundamental class of non-orientable manifolds and manifolds with boundary

TL;DR

This paper introduces a new analytic tool called the metric fundamental class for non-orientable manifolds with boundary, extending previous results and enabling new rectifiability insights.

Contribution

It defines the metric fundamental class for non-orientable manifolds with boundary and proves its existence under weak geometric conditions, extending prior work on orientable manifolds.

Findings

Existence of the metric fundamental class under weak conditions

Extension of rectifiability results to non-orientable manifolds

Analytic representation of the topological fundamental class

Abstract

We introduce the metric fundamental class for metric spaces that are homeomorphic to compact, non-orientable, smooth manifolds with (possibly empty) boundary. This is an integer rectifiable current that provides an analytic representation of the topological fundamental class of the space. Under certain weak geometric conditions, we show the existence of such a current, extending earlier results for orientable, closed manifolds obtained in collaboration with Basso and Wenger. As an application, we present new rectifiability results.