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Equivariant basic cohomology under deformations

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Abstract

By methods of Ghys and Haefliger–Salem it is possible to deform a Riemannian foliation on a simply connected compact manifold, or more generally a Killing foliation, into a closed foliation, i.e., a foliation whose leaves are all closed. Certain transverse geometric and topological properties are preserved under these deformations, as previously shown by the authors. For instance the Euler characteristic of basic cohomology is invariant, whereas its Betti numbers are not. In this article we show that the equivariant basic cohomology ring structure is invariant. This leads to a sufficient algebraic condition, namely equivariant formality, for the Betti numbers to be preserved as well. In particular, this is true for the deformation of the Reeb orbit foliation of a K-contact manifold. Another consequence is that there is a universal bound on the sum of basic Betti numbers of any equivariantly formal, positively curved Killing foliation of a given codimension. We also show that a Killing foliation with negative transverse Ricci curvature is closed. If the transverse sectional curvature is negative we show, furthermore, that its fundamental group has exponential growth. Finally, we obtain a transverse generalization of Synge’s theorem to Killing foliations.

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Acknowledgements

We are grateful to Professors Marcos M. Alexandrino, Oliver Goertsches and Hiraku Nozawa, for the insightful discussions, and to the anonymous referee for the valuable comments and suggestions. Francisco C. Caramello Jr. also thanks the Department of Mathematics of the University of São Paulo for the welcoming environment where a substantial part of this work was developed, and FAPESP for the research funding.

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  1. Francisco C. Caramello Jr.

    Present address: Departamento de Matemática, Universidade Federal de Santa Catarina, R. Eng. Agr. Andrei Cristian Ferreira, Florianópolis, SC, 88040-900, Brazil

Authors and Affiliations

  1. Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, São Paulo, SP, 05508-090, Brazil

    Francisco C. Caramello Jr.

  2. Departamento de Matemática, Universidade Federal de São Carlos, Rod. Washington Luís Km 235, São Carlos, SP, 13565-905, Brazil

    Dirk Töben

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  1. Francisco C. Caramello Jr.
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  2. Dirk Töben
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Correspondence to Francisco C. Caramello Jr..

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F.C. was supported by grant #2018/14980-0, São Paulo Research Foundation (FAPESP).

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Caramello, F.C., Töben, D. Equivariant basic cohomology under deformations. Math. Z. 299, 2461–2482 (2021). https://doi.org/10.1007/s00209-021-02768-w

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