## 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

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.

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

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 \(t \in I\).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.

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

BZ stands for Olivier Brahic and Chenchang Zhu.

## References

Androulidakis, I.: Personal Communication (2017)

Androulidakis, I., Skandalis, G.: The holonomy groupoid of a singular foliation. J. Reine Angew. Math.

**626**, 1–37 (2009)Androulidakis, I., Zambon, M.: Smoothness of holonomy covers for singular foliations and essential isotropy. Math. Z.

**275**(3–4), 921–951 (2013)Androulidakis, I., Zambon, M.: Holonomy transformations for singular foliations. Adv. Math.

**256**, 348–397 (2014)Berglund, A.: Rational homotopy theory of mapping spaces via Lie theory for \(L_\infty \)-algebras. Homol. Homotopy Appl.

**17**(2), 343–369 (2015)Bonavolontà, G., Poncin, N.: On the category of Lie \(n\)-algebroids. J. Geom. Phys.

**73**, 70–90 (2013)Brahic, O., Zhu, C.: Lie algebroid fibrations. Adv. Math.

**226**(4), 3105–3135 (2011)Brahic, O., Zambon, M.: L\(\infty \)-actions of lie algebroids. Commun. Contemp. Math. (2021). https://doi.org/10.1142/S0219199721500139

Cerveau, D.: Distributions involutives singulières. Ann. Inst. Fourier (Grenoble)

**29**(3), xii, 261–294 (1979)Cattaneo, A.S., Felder, G.: Poisson sigma models and symplectic groupoids. Quantization of Singular Symplectic Quotients. Based on a Research-in-Pairs Workshop, Oberwolfach, Germany, 2–6 Aug 1999, pp. 61–93. Birkhäuser, Basel (2001)

Crainic, M., Fernandes, R.L.: Integrability of Lie brackets. Ann. Math. (2)

**157**(2), 575–620 (2003)Dazord, P.: Feuilletages à singularités. Indag. Math.

**47**, 21–39 (1985)Debord, C.: Holonomy groupoids of singular foliations. J. Differ. Geom.

**58**(3), 467–500 (2001)Debord, C.: Longitudinal smoothness of the holonomy groupoid. C. R. Math. Acad. Sci. Paris

**351**(15–16), 613–616 (2013)Ehresmann, C.: variétés feuilletées, Sur la théorie des. Univ. Roma Ist. Naz. Alta Mat. Rend. Mat. e Appl. (5)

**10**, 64–82 (1951)Frégier, Y., Juarez-Ojeda, R.A.: Homotopy theory of singular foliations. arXiv:1811.03078 (2018)

Frejlich, P.: Submersions by Lie algebroids. J. Geom. Phys.

**137**, 237–246 (2019)Haefliger, A.: feuilletées, Variétés. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3)

**16**, 367–397 (1962)Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)

Hermann, R.: The differential geometry of foliations. II. J. Math. Mech.

**11**, 303–315 (1962)Hermann, R.: On the accessibility problem in control theory. International Symposium Nonlinear Differential Equations and Nonlinear Mechanics, pp. 325–332. Academic Press, New York (1963)

Khudaverdian, H.M., Voronov, T.T.: Higher Poisson brackets and differential forms. Geometric Methods in Physics. Proceedings of the XXVII Workshop on Geometric Methods in Physics, Białowieża, Poland, 29 June–5 July 2008, pp. 203–215. American Institute of Physics (AIP), Melville, NY (2008)

Laurent-Gengoux, C., Louis, R.: Lie–Rinehart algebra \(\simeq \) acyclic Lie \(\infty \)-algebroid. arXiv:2106.13458 (2021)

Laurent-Gengoux, C., Lavau, S., Strobl, T.: The universal Lie \(\infty \)-algebroid of a singular foliation. Doc. Math.

**25**, 1571–1652 (2020)Loday, J.-L., Vallette, B.: Algebraic Operads, vol. 346. Springer, Berlin (2012)

Mackenzie, K.C.H.: Lie Groupoids and Lie Algebroids in Differential Geometry, volume 124 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1987)

Mackenzie, K.C.H.: General Theory of Lie Groupoids and Lie Algebroids, volume 213 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (2005)

Moerdijk, I., Mrčun, J.: Introduction to Foliations and Lie Groupoids, vol. 91. Cambridge University Press, Cambridge (2003)

Reeb, G.: Sur certaines propriétés topologiques des variétés feuilletées. Actualités Sci. Ind., no. 1183. Hermann & Cie., Paris (1952) (Publ. Inst. Math. Univ. Strasbourg 11, pp. 5–89 155–156)

Ševera, P.: Letters to Alan Weinstein about Courant algebroids. arXiv:1707.00265 (2017)

Ševera, P., Širaň, M.: Integration of differential graded manifolds. Int. Math. Res. Not.

**2020**(20), 6769–6814 (2020)Spanier, E.H.: Algebraic topology. Springer, New York (1995) (

**Corrected reprint of the 1966 original**)Tuynman, G.M.: Supermanifolds and Supergroups, volume 570 of Mathematics and Its Applications. Kluwer Academic Publishers, Dordrecht (2004) (

**Basic theory**)Villatoro, J., Garmendia, A.: Integration of singular foliations via paths (2019). https://doi.org/10.1093/IMRN/RNAB177

Voronov, T.: \(Q\)-manifolds and higher analogs of Lie algebroids. XXIX Workshop on Geometric Methods in Physics, volume 1307 of AIP Conference Proceedings, pp. 191–202. American Institute of Physics, Melville, NY (2010)

## 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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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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DOI: https://doi.org/10.1007/s00029-021-00753-z