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

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.