Groupoids and foliations (Q2785084)

The author considers foliations of codimension \(q\) on manifolds and associates to them (transverse) étale groupoids called their holonomy groupoid. After two sections where the basic facts are summarized, Section 3 treats homotopy and cohomology for étale groupoids whereas the use of cohomology in foliation theory is illustrated in Section 5. The problem of determining when a field of planes of codimension \(q\) on a manifold \(M\) is homotopic to the field of planes tangent to a codimension \(q\) foliation on \(M\) is considered by using suitable spaces according to Thurston's theorem. The relation between foliations and étale groupoids brings the author to the so-called realization problem: Given an étale groupoid \((G,T)\) such that \(T\) is a Hausdorff manifold, when is it equivalent to the fundamental or the holonomy groupoid of a foliation on a closed manifold? The paper discusses and gives the present situation of the problem. Finally, the interesting problem of classification of étale grupoids up to equivalence in connection to the study of foliations for the complete Riemannian case is reviewed in Section 7, where the most interesting results are stated.NEWLINENEWLINEFor the entire collection see [Zbl 0972.00029].