Covering instability for the existence of positive scalar curvature metrics

Abstract

We show that a closed non-orientable $3$-manifold admits a positive scalar curvature metric if and only if its orientation double cover does; however, for each $4\le n\le 7$, there exist infinitely many smooth non-orientable $n$-manifolds $M$ that are mutually non-homotopy equivalent, such that the orientation double cover of $M$ admits positive scalar curvature metrics, but every closed smooth manifold that is homotopy equivalent to $M$ cannot admit positive scalar curvature metrics. These examples were first introduced by Alpert-Balitskiy-Guth in the study of Urysohn widths. To prove the nonexistence result, we extend the Schoen-Yau inductive descent approach to non-orientable manifolds. We also discuss band width estimates and the notion of enlargeability for non-orientable PSC manifolds.