Actions of Lie groups on closed manifolds

Abstract

We associate to each non-singular action A of a connected Lie group G on a closed manifold M and each A-invariant probability measure \mu on M, a linear mapping P_A from the cohomology H^*(\mathcal{F}) of the foliation \mathcal{F} given by the orbits of A onto the cohomology H^*(\mathcal{G}) of the Lie algebra \mathcal{G} of G. This extends to actions the asymptotic cycles of S. Schwartzman and D. Sullivan and gives a strong connection between the ergodic properties of A and the characteristic mapping \chi of \mathcal{F} as defined by dos Santos (Contemp. Math.161 (1994), 41–57). The ergodic notion of E-action introduced by Arraut and dos Santos (Topology31 (1992), 545–555) extends naturally to actions of Lie groups and we prove the main result of this article which shows that a non-singular E-action A of a nilpotent Lie group G completely determines the characteristic mapping \chi of the foliation \mathcal{F} given by A. This shows the strength of the above connection and gives as a corollary the vanishing theorem for foliated bundles proved by dos Santos.