Covering instability for the existence of positive scalar curvature metrics

2025/06/16 by Li, Chao, Zhang, Boyu · 4 citations
#Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2506.13885

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≤ n≤ 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.

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