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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Asymptotic cycles for actions of Lie groups


Author: Sol Schwartzman
Journal: Proc. Amer. Math. Soc. 141 (2013), 1673-1677
MSC (2010): Primary 28D15, 54H20
Published electronically: November 6, 2012
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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$.


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Additional Information

Sol Schwartzman
Affiliation: Department of Mathematics, University of Rhode Island, Kingston, Rhode Island 02881
Email: solschwartzman@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11445-5
PII: S 0002-9939(2012)11445-5
Received by editor(s): February 3, 2011
Received by editor(s) in revised form: September 7, 2011
Published electronically: November 6, 2012
Communicated by: Kailash C. Misra
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.