"Pradines-type groupoids over foliations; cohomology, connections and the Chern-Weil homomorphism"

Preprint Nr 2, August 1986, Institute of Mathematics, Technical University of Łódź.
publications

We briefly introduce our concept of a Pradines-type groupoid over a foliation, see [6] above. We define a cohomology module H(A, f) of the Lie algebroid A of a Pradines-type groupoid F   over a foliation, with values in some vector bundle f, with respect to a given representation of F  in f. It is shown that H(A, f) depends only on the derivative of this representaion. Afterwards, the theory of connections in A and in F  is built. The last part is devoted to defining the Chern-Weil homomorphism h_F  of F  and to proving its independence of the choice of connection. As an applications of the introduced characteristic classes we give some generalization of the Bott Vanishing Theorem.