Positive scalar curvature on manifolds with fibered singularities

Boris Botvinnik; Jonathan Rosenberg

doi:10.1515/crelle-2023-0055

Published by De Gruyter September 6, 2023

Positive scalar curvature on manifolds with fibered singularities

Boris Botvinnik and Jonathan Rosenberg

From the journal Journal für die reine und angewandte Mathematik (Crelles Journal)

https://doi.org/10.1515/crelle-2023-0055

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Abstract

A (compact) manifold with fibered 𝑃-singularities is a (possibly) singular pseudomanifold M Σ with two strata: an open nonsingular stratum M ̊ (a smooth open manifold) and a closed stratum β ⁢ M (a closed manifold of positive codimension), such that a tubular neighborhood of β ⁢ M is a fiber bundle with fibers each looking like the cone on a fixed closed manifold 𝑃. We discuss what it means for such an M Σ with fibered 𝑃-singularities to admit an appropriate Riemannian metric of positive scalar curvature, and we give necessary and sufficient conditions (the necessary conditions based on suitable versions of index theory, the sufficient conditions based on surgery methods and homotopy theory) for this to happen when the singularity type 𝑃 is either Z / k or S 1 , and 𝑀 and the boundary of the tubular neighborhood of the singular stratum are simply connected and carry spin structures. Along the way, we prove some results of perhaps independent interest, concerning metrics on spin^𝑐 manifolds with positive “twisted scalar curvature,” where the twisting comes from the curvature of the spin^𝑐 line bundle.

Funding source: Simons Foundation

Award Identifier / Grant number: 708183

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-1607162

Funding statement: B. Botvinnik was partially supported by Simons collaboration grant 708183. J. Rosenberg partially supported by U.S. NSF grant number DMS-1607162.

Acknowledgements

We would like to thank Paolo Piazza for many useful suggestions on the subject of this paper. Paolo is a coauthor on a subsequent paper [10] dealing with other singularity types 𝑃. We would also like to thank Bernd Ammann, Bernhard Hanke, and André Neves for organizing an excellent workshop at the Mathematisches Forschungsinstitut Oberwolfach in August 2017 on Analysis, Geometry, and Topology of Positive Scalar Curvature Metrics, which led to the present work. We would also like to thank the referees for helpful suggestions and criticism regarding earlier versions of the paper.

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Received: 2022-01-07

Revised: 2023-08-09

Published Online: 2023-09-06

Published in Print: 2023-10-01

Positive scalar curvature on manifolds with fibered singularities

Abstract

Acknowledgements

References

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