Orientability of metric manifolds and metric spaces with curvature bounds

We explore orientability of metric spaces that are homeomorphic to a manifold without boundary (so-called metric manifolds), without assuming any smooth structure. For metric manifolds that satisfy certain weak assumptions, we show that orientability is equivalent to the existence of a locally integral cycle of multiplicity one. This current can be viewed as an analytical analogue of the topological fundamental class of the manifold. We then extend the construction to spaces which are manifolds up to a small singular set. As applications, we obtain a version of Stokes’ theorem for non-collapsed RCD spaces and local Poincaré inequalities for a class of spaces with curvature bounded above. Based on joint work with N. Cavallucci, A. Mondino and R. Perales.