Higher Dimensional Asymptotic Cycles

Given a p-dimensional oriented foliation of an n-dimensional compact manifold Mn and a transversal invariant measure (tau), Sullivan has defined an element of Hp (M^n,R). This generalized the notion of a (mu)-asymptotic cycle, which was originally defined for actions of the real line on compact spaces preserving an invariant measure (mu). In this one-dimensional case there was a natural 1-1 correspondence between transversal invariant measures (tau) and invariant measures (mu) when one had a smooth flow without stationary points. For what we call an oriented action of a connected Lie group on a compact manifold we again get in this paper such a correspondence, provided we have what we call a positive quantifier. (In the one-dimensional case such a quantifier is provided by the vector field defining the flow.) Sufficient conditions for the existence of such a quantifier are given, together with some applications