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The holonomy groupoid of a singular foliation. (English) Zbl 1161.53020
For a (regular) foliation $$\mathcal F$$, the holonomy groupoid (sometimes called the graph of a foliation) is defined as the space of equivalence classes of triples $$(y,\alpha,x)$$ where $$x$$ and $$y$$ are points of the same leaf $$L$$ of $$\mathcal F$$, $$\alpha$$ is a path in $$L$$ linking $$x$$ to $$y$$, and two triples $$(y,\alpha,x)$$, $$(y',\alpha',x')$$ are equivalent if and only if $$x=x'$$, $$y=y'$$ and the holonomy of the curve $$\alpha^{-1}\ast\alpha'$$ is trivial. This definition is due to H. E. Winkelnkemper [Ann. Global Anal. Geom. 1, No. 3, 51–75 (1983; Zbl 0526.53039)], who also proved that the holonomy groupoid is a, in general non-Hausdorff, manifold.
The analogous definition in the setting of singular foliations was given by various authors such as Pradines, Bigonnet and Debord [J. Pradines, Cah. Topol. Géom. Différ. Catégoriques 26, 339–380 (1985; Zbl 0576.57023); B. Bigonnet and J. Pradines, C. R. Acad. Sci., Paris, Sér. I 300, 439–442 (1985; Zbl 0581.57013); C. Debord, Banach Cent. Publ. 54, 21–33 (2001; Zbl 1007.22008); C. Debord, J. Differ. Geom. 58, No. 3, 467–500 (2001; Zbl 1034.58017)]. However, these authors dealt with singular foliations that can be defined by an almost injective Lie algebroid, i.e. a Lie algebroid whose anchor map is injective in a dense open subset of the base manifold.
The present article deals with the problem of defining a holonomy groupoid for a singular foliation without assuming the foliation to be defined by some algebroid. The properties of such groupoid may be summarized in the following theorem:
Let $$\mathcal F$$ be a (possibly singular) Stefan-Sussmann foliation on a manifold $$M$$. Then there exists a topological groupoid $${\mathcal H}({\mathcal F})\rightrightarrows M$$ such that: a) its orbits are the leaves of the given foliation $$\mathcal F$$, b) $${\mathcal H}({\mathcal F})$$ is minimal in the sense that if $$G\rightrightarrows M$$ is a Lie groupoid which defines the foliation $$\mathcal F$$ then there exists an open subgroupoid $$G_0$$ of $$G$$ and a morphism of groupoids $$G_0 \rightarrow {\mathcal H}({\mathcal F})$$ which is onto.
Moreover, it is shown that if $${\mathcal F}$$ is regular or is defined from an almost injective Lie groupoid then one gets the aforementioned definitions of Winkelnkemper and Pradines, Bigonnet, Debord.
In such construction, the key notion is that of a bi-submersion which plays the role of a local Lie groupoid defining the foliation. The holonomy groupoid is indeed given by the quotient of germs of these bi-submersions with respect to an appropriate equivalence relation.
Though, in the general case, such groupoid can be quite an ill behaved geometric object and its only possible topology is quite pathological, it often has a nice longitudinal smooth structure and it is used by the authors for the generalization in the singular setting the Connes’ construction of the $$C^{\ast}$$-algebra of a foliation.

##### MSC:
 53C12 Foliations (differential geometric aspects) 22A22 Topological groupoids (including differentiable and Lie groupoids) 53C29 Issues of holonomy in differential geometry
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