Gromov's Simplicial Volume Vanishing Conjecture for Positive Scalar Curvature

Abstract

In this paper, we prove Gromov's simplicial volume vanishing conjecture for closed manifolds with spin universal cover. More precisely, we show that if a closed oriented manifold admits a metric of nonnegative scalar curvature and its universal cover is spin, then its simplicial volume vanishes. In particular, a closed oriented aspherical manifold with nonzero simplicial volume admits no metric of nonnegative scalar curvature.

23 pages. v4: Removing the extra group decay condition from an early version, we prove Gromov's simplicial volume vanishing conjecture unconditionally for closed manifolds whose universal cover is spin. Comments are welcome!