Differential Geometry

In recent years the graph of a foliation, an object which has been known for a long time, cf. 10], has known new interest. In fact there are two groupoids associated with a foliation, the homotopy groupoid and the holonomy groupoid, sometimes called the graph. It serves as a basis for the construction of the C-algebra associated to the foliation. Moreover, the homotopy groupoid of the characteristic foliation of a Poisson manifold is used in the symplectic integration of this Poisson manifold, cf. 22, 12, 15, 2]. For a general foliation we do not know whether its groupoids are Hausdorr manifolds, cf. 23, 21]. Recently P. Dazord and G. Hector proved that the homotopy groupoid is Hausdorr ii the foliation has no vanishing cycles, cf. 12]. In this note study the existence of vanishing cycles for some classes of foliations, including totally geodesic foliations and these which admit an Ehresmann connection. In this section, for the convenience of the reader, we will recall some basic deenitions and results. 1.1. The graph of a foliation The deenition and the basic properties of the graph of a foliation can be found in 23]. Let us recall the deenition. The graph GR(F) of the foliation F is the space of equivalence classes of triples (y; ; x) where x and y are points of the same leaf L of F and is a path in L linking x to y. Two triples (y; ; x) and (y 0 ; 0 ; x 0) are equivalent ii x = x 0 ; y = y 0 and the holonomy of the curve ?1 0 is trivial. In the same paper the author