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The holonomy of a singular leaf

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Abstract

We introduce the holonomy of a singular leaf L of a singular foliation as a sequence of group morphisms from \(\pi _n(L)\) to the \(\pi _{n-1}\) of the universal Lie \(\infty \)-algebroid of the transverse foliation of L. We include these morphisms in a long exact sequence, thus relating them to the holonomy groupoid of Androulidakis and Skandalis and to a similar construction by Brahic and Zhu for Lie algebroids.

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Notes

  1. More precisely, the universal Lie \(\infty \)-algebroid is shown in [24] to exist for any singular foliation that admits a geometric resolution. This happens in particular for locally real analytic singular foliations, that is singular foliations that have, in a neighborhood of every point, generators which are real analytic in some local coordinates. This class is quite large. The construction has been later extended by the first author and Ruben Louis (see [23]) to arbitrary Lie–Rinehart algebras (see also [16]), in particular to general singular foliations, at the cost that the Lie \(\infty \)-algebroid could have infinite dimension.

  2. The symbol \(\tilde{\otimes }\) stands for the completed tensor product, which is the right operation to deal with products of NQ-manifolds.

  3. As in Section 3.4.4 in [24], we implicitly consider elements of degree k in \({{\mathcal {O}}_\bullet ^{{\mathcal M}}} \tilde{\otimes } \Omega ^\bullet _I \) as being elements of the form \( F_t \otimes 1 + G_t \otimes dt \) with \(F_t,G_t \in {{\mathcal {O}}_\bullet ^{}}\) being elements of degree k and \(k-1\) respectively that depend smoothly on a parameter \(t \in I\).

  4. A regular point is a point around which all leaves have the same dimension (and are called regular leaves). Recall that if a geometric resolution of finite length exists, then all regular leaves have the same dimension.

  5. By universal cover, we mean that for each leaf \(m \in M\), \(\Pi ({\mathfrak F}) |_m=s^{-1}(m) \) is the universal cover of the fiber over m of Androulidakis and Skandalis’ holonomy groupoid.

  6. As usual for Lie groupoids, \( {\mathbf {F}} |_m = s^{-1}(m) \subset \mathbf{F}\) is the source fiber of m, and \(\widetilde{I_m({{\mathbf {F}}})}\) is the universal cover of the isotropy of \(I_m({\mathbf {F}}) \) of \( {\mathbf {F}}\) at m.

  7. A flat Ehresmann connection is a horizontal distribution whose sections are closed under the Lie bracket of vector fields.

  8. BZ stands for Olivier Brahic and Chenchang Zhu.

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Acknowledgements

We acknowledge crucial discussions with Iakovos Androulidakis (for \( {\mathfrak F}\)-connections) and Pavol Ševera (for fundamental groups of NQ-manifolds): We will mark them in the text in due places. We also thank the workshop and conference “Singular Foliations” in Paris Diderot and Leuven where the content of the article was presented. We also express special gratitude to Sylvain Lavau, Thomas Strobl, Marco Zambon and Chenchang Zhu. L. R. was supported by the Ruhr University Research School PLUS, funded by Germany’s Excellence Initiative [DFG GSC 98/3] and the PRIME programme of the German Academic Exchange Service (DAAD) with funds from the German Federal Ministry of Education and Research (BMBF). Both authors were supported by CNRS MITI 80 Prime projet “Granum”. Finally, we would like to thank the referee for his suggestions to clarify the article and improve the exposition.

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Correspondence to Camille Laurent-Gengoux.

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Laurent-Gengoux, C., Ryvkin, L. The holonomy of a singular leaf. Sel. Math. New Ser. 28, 45 (2022). https://doi.org/10.1007/s00029-021-00753-z

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