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Higher Dimensional Asymptotic Cycles

Open Access article
 Printed: Jun 2003
  • Sol Schwartzman
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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 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.
MSC Classifications: 57R30, 57S20 show english descriptions Foliations; geometric theory
Noncompact Lie groups of transformations
57R30 - Foliations; geometric theory
57S20 - Noncompact Lie groups of transformations

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