GRAPHS, EHRESMANN CONNECTIONS AND VANISHING CYCLES

Robert A. Wolak

Key words: 57R30, 53C05, 53C15.

1991 Math. Subject Classification: Foliation, Ehresmann
connection, vanishing cycle, homotopy groupoid, holonomy groupoid.

In recent years the graph of a foliation, an object which has
been known for a long time, cf. \cite{EH}, 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 $C^{\ast}$-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.
\cite{WE,DH,HE,AH}. For a general foliation we do not know whether its
groupoids are Hausdorff manifolds, cf. \cite{WI,PH}.
Recently P. Dazord and G. Hector proved that the homotopy groupoid is
Hausdorff iff the foliation has no vanishing cycles, cf.
\cite{DH}. 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.