Abstract view
Higher Dimensional Asymptotic Cycles


Published:20030601
Printed: Jun 2003
Abstract
Given a $p$dimensional oriented foliation of an $n$dimensional
compact manifold $M^n$ and a transversal invariant measure $\tau$,
Sullivan has defined an element of $H_p (M^n,R)$. 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 onedimensional case there was a natural 11
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
onedimensional 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.