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A characteristic map for the holonomy groupoid of a foliation

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Abstract

We prove a generalisation of Bott’s vanishing theorem for the full transverse frame holonomy groupoid of any transversely orientable foliated manifold. As a consequence we obtain a characteristic map encoding both primary and secondary characteristic classes. Previous descriptions of this characteristic map are formulated for the Morita equivalent étale groupoid obtained via a choice of complete transversal. By working with the full holonomy groupoid we obtain novel geometric representatives of characteristic classes. In particular we give a geometric, non-étale analogue of the codimension 1 Godbillon–Vey cyclic cocycle of Connes and Moscovici in terms of line integrals of the curvature form of a Bott connection.

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Notes

  1. i.e. locally leafwise

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Acknowledgements

I wish to thank the Australian Federal Government for a Research Training Program scholarship, which supported this research. I also thank Moulay Benameur for supporting a visit to Montpellier in late 2018, and Magnus Goffeng for supporting a visit to Gothenburg in early 2019, where parts of this research were conducted. I also thank Magnus Goffeng, James Stasheff and the anonymous referee for helpful comments on the paper. Finally, I would like to extend deep thanks to Adam Rennie, whose consistent (but never overbearing) guidance and support have greatly benefited my growth as a mathematician.

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  1. School of Mathematical Sciences, University of Adelaide, Adelaide, SA, 5005, Australia

    Lachlan E. MacDonald

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MacDonald, L.E. A characteristic map for the holonomy groupoid of a foliation. Math. Z. 300, 1093–1115 (2022). https://doi.org/10.1007/s00209-021-02832-5

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