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Enlargeability, foliations, and positive scalar curvature

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Abstract

We extend the deep and important results of Lichnerowicz, Connes, and Gromov–Lawson which relate geometry and characteristic numbers to the existence and non-existence of metrics of positive scalar curvature (PSC). In particular, we show: that a spin foliation with Hausdorff homotopy groupoid of an enlargeable manifold admits no PSC metric; that any metric of PSC on such a foliation is bounded by a multiple of the reciprocal of the foliation K-area of the ambient manifold; and that Connes’ vanishing theorem for characteristic numbers of PSC foliations extends to a vanishing theorem for Haefliger cohomology classes.

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Acknowledgements

It is a pleasure to thank Fernando Alcalde Cuesta, Mikhael Gromov, Gilbert Hector, Steven Hurder, Paul Schweitzer, SJ, and Shing-Tung Yau for helpful information, and the referee for cogent comments which helped improve the presentation. MB wishes to thank the french National Research Agency for support via the project ANR-14-CE25-0012-01 (SINGSTAR).

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Authors and Affiliations

  1. Institut Montpellierain Alexander Grothendieck, UMR 5149 du CNRS, Université de Montpellier, Montpellier, France

    Moulay-Tahar Benameur

  2. Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, USA

    James L. Heitsch

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  1. Moulay-Tahar Benameur
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  2. James L. Heitsch
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Correspondence to James L. Heitsch.

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Benameur, MT., Heitsch, J.L. Enlargeability, foliations, and positive scalar curvature. Invent. math. 215, 367–382 (2019). https://doi.org/10.1007/s00222-018-0829-6

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  • DOI: https://doi.org/10.1007/s00222-018-0829-6

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