The holonomic imperative and the homotopy groupoid of a foliated manifold.(English)Zbl 0641.57011

Let F be a codimension p $$C^{\infty}$$-foliation of a $$p+q$$-dimensional connected manifold V. As is well-known, there is associated a holonomy groupoid Hol(V,F) which is a basic element in the construction of the $$C^*$$-algebra of F. There is also the fundamental groupoid of F, which the author calls the homotopy groupoid. It consists of homotopy classes of paths in leaves of F. Let Hom(V,F) denote it. Both groupoids are shown to be $$C^{\infty}$$-groupoids of dimension $$p+2q$$ and there exists a natural groupoid morphism Hom(V,F)$$\twoheadrightarrow Hol(V,F)$$. In the main theorem of the paper, the author shows that under certain conditions, Hol(V,F) is the smallest $$C^{\infty}$$-groupoid associated with F. Namely, he proves the following: Let F be as above and $${\mathcal R}$$ the equivalence relation determined by leaves of F. Let G be a $$C^{\infty}$$-groupoid of dimension $$p+2q$$ such that $$G^{(0)}$$ $$(=the$$ space of units) $$=V$$ and $$s\times r: G\to {\mathcal R}\subset V\times V$$ (source $$\times$$ range map) is surjective and $$C^{\infty}$$. If there is a $$C^{\infty}$$-groupoid morphism Hom(V,F)$$\twoheadrightarrow G$$ compatible with the projections onto $${\mathcal R}$$, then there is a unique morphism $$G\to Hol(V,F)$$, compatible with the projections onto $${\mathcal R}$$, such that the composition Hom(V,F)$$\to G\to Hol(V,F)$$ is the natural map. Some remarks on these groupoids are found in the case of specific foliations.
Reviewer: T.Mizutani

MSC:

 57R30 Foliations in differential topology; geometric theory 46L05 General theory of $$C^*$$-algebras
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