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Holonomy transformations for singular foliations. (English) Zbl 1294.53025
The paper focuses on the study of the so called “holonomy transformations” for singular foliations. Recall that, for a regular foliation \((M, F)\) equipped with a transversal structure, the holonomy of \((M, F)\) at a point \(x\) of the foliated \(M\) is realized by a map \(h_{x}: \pi_{1}(L) \to \mathrm{GermDiff}(S)\), where \(L\) is the leaf containing \(x\), \(S\) is a transversal slice at \(x\) and \(\mathrm{GermDiff}(S)\) is the space of germs of local diffeomorphisms of \(S\). The paper extends the notion of holonomy from the regular case to the singular one.
Main tools of the authors are the construction of the holonomy groupoid for any singular foliation and the notion of bi-submersion which are given by the first author and G. Skandalis in [J. Reine Angew. Math. 626, 1–37 (2009; Zbl 1161.53020)]. Namely, instead of the familiar holonomy map, the authors introduce the notion of holonomy transformation and show how to associate holonomy transformations to a singular foliation. The paper also gives further applications for singular Riemannian foliations in Molino’s sense and some deformations of singular foliations.

MSC:
53C12 Foliations (differential geometric aspects)
53C29 Issues of holonomy in differential geometry
22A25 Representations of general topological groups and semigroups
93B18 Linearizations
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