New Topological Restrictions For Spaces With Nonnegative Ricci Curvature

TL;DR

This paper establishes new topological restrictions for spaces with nonnegative Ricci curvature, including a Betti number rigidity theorem and a vanishing theorem for simplicial volume, extending classical results to synthetic spaces.

Contribution

It proves a Betti number rigidity theorem and a vanishing theorem for simplicial volume in the setting of RCD(0,n) spaces, answering longstanding open questions.

Findings

Proves a Betti number rigidity theorem for RCD(0,n) spaces.

Establishes a vanishing theorem for the simplicial volume.

Provides a new proof of the classification of noncompact 3-manifolds with nonnegative Ricci curvature.

Abstract

We obtain new topological restrictions for complete Riemannian manifolds with nonnegative Ricci curvature and RCD(0,n) spaces. Our main results are a Betti number rigidity theorem which answers a question open since work of M.-T. Anderson in 1990, and a vanishing theorem for the simplicial volume generalizing a theorem of M. Gromov from 1982. Combining such results we obtain a new proof of the classification of noncompact 3-manifolds with nonnegative Ricci curvature, originally due to G. Liu in 2011, which extends to the synthetic setting.