Moduli Spaces of Metrics of Positive Scalar Curvature on Topological Spherical Space Forms

Philipp Reiser

doi:10.4153/s0008439520000132

Abstract

AbstractLet $M$ be a topological spherical space form, i.e., a smooth manifold whose universal cover is a homotopy sphere. We determine the number of path components of the space and moduli space of Riemannian metrics with positive scalar curvature on $M$ if the dimension of $M$ is at least 5 and $M$ is not simply-connected.

Highlights

Introduction and Main ResultLet M be a closed smooth manifold
We denote by R+(M) the space of Riemannian metrics with positive scalar curvature on M and by M+(M) the corresponding moduli space, i.e., the quotient of R+(M) by the pull-back action of the di eomorphism group
Is article addresses the problem of determining the number of path components of both R+(M) and M+(M) if M is a non--connected topological spherical space form, i.e., a non-trivial quotient of a homotopy sphere by a free action of a nite group

Summary

Introduction and Main Result

Let M be a closed smooth manifold. We denote by R+(M) the space of Riemannian metrics with positive scalar curvature on M and by M+(M) the corresponding moduli space, i.e., the quotient of R+(M) by the pull-back action of the di eomorphism group. Is article addresses the problem of determining the number of path components of both R+(M) and M+(M) if M is a non--connected topological spherical space form, i.e., a non-trivial quotient of a homotopy sphere by a free action of a nite group. Main eorem Let M be a topological spherical space form of dimension at least that is not -connected and admits a metric with positive scalar curvature. ], where the authors determined under which conditions a topological spherical space form admits a metric with positive scalar curvature. Eorem B ([ , Main eorem]) Let M be a topological spherical space form of dimension n ≥. For the last case in eorem B, note that by [ ] the universal cover of the topological space form M admits a metric with positive scalar curvature if and only if its alpha invariant vanishes. In Section , we use these tools to prove the Main eorem

Preliminaries

Proof of the Main Theorem