Asymptotic cycles for actions of Lie groups

Abstract:

Let $M^k$ be a compact $C^\infty$ manifold and suppose we are given a $C^\infty$ action of $\mathbb {R}^n$ on $M^k$. If $p$ is a quasiregular point for this action and $v$ is an $r$-vector over the Lie algebra of $\mathbb {R}^n$, we show how to associate with $p$ and $v$ an element $A_p^v$ in $H_r(M^k;\mathbb {R})$. When $n=1$ and $v$ is the usual generator for the Lie algebra of $\mathbb {R}$, $A_p^v$ coincides with the asymptotic cycle associated with $p$ by our flow. Just as in the one dimensional case, with any invariant probability measure we can associate an element $A_\mu ^v$ in $H_r(M^k;\mathbb {R}).$

Several results known in the one dimensional case generalize to our present situation. The results we have stated for actions of $\mathbb {R}^n$ are obtained from a discussion of what we can say when we have a smooth action of an arbitrary connected Lie group on $M^k$.