Classifying sufficiently connected PSC manifolds in 4 and 5 dimensions

Chodosh, Otis; Li, Chao; Liokumovich, Yevgeny

doi:10.2140/gt.2023.27.1635

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Classifying sufficiently connected PSC manifolds in $4$ and $5$ dimensions

Otis Chodosh, Chao Li and Yevgeny Liokumovich

Geometry & Topology 27 (2023) 1635–1655

DOI: 10.2140/gt.2023.27.1635

Abstract

We show that if $N$ is a closed manifold of dimension $n = 4$ (resp. $n = 5$ ) with $π_{2} (N) = 0$ (resp. $π_{2} (N) = π_{3} (N) = 0$ ) that admits a metric of positive scalar curvature, then a finite cover $\hat{N}$ of $N$ is homotopy equivalent to $S^{n}$ or connected sums of $S^{n - 1} \times S^{1}$ . Our approach combines recent advances in the study of positive scalar curvature with a novel argument of Alpert, Balitskiy and Guth.

Additionally, we prove a more general mapping version of this result. In particular, this implies that if $N$ is a closed manifold of dimensions $4$ or $5$ , and $N$ admits a map of nonzero degree to a closed aspherical manifold, then $N$ does not admit any Riemannian metric with positive scalar curvature.

Keywords

positive scalar curvature, Urysohn width, minimal surfaces, $\mu$–bubbles

Mathematical Subject Classification

Primary: 53C21

References

Publication

Received: 21 June 2021

Revised: 13 September 2021

Accepted: 15 October 2021

Published: 15 June 2023

Proposed: John Lott

Seconded: Urs Lang, Gang Tian

Authors

Otis Chodosh
Department of Mathematics Stanford University Stanford, CA United States

Chao Li
Courant Institute of Mathematical Sciences New York University New York, NY United States

Yevgeny Liokumovich
Department of Mathematics University of Toronto Toronto ON Canada

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Volume 27, issue 4 (2023)