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Higher Dimensional Asymptotic Cycles

Published online by Cambridge University Press:  20 November 2018

Sol Schwartzman
Sol Schwartzman*
Affiliation:
Department of Mathematics, University of Rhode Island, Kingston, Rhode Island 02881, USA
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Abstract

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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 ${{H}_{p}}\left( {{M}^{n}},\,R \right)$. 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.

Keywords

57R3057S20
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 55 , Issue 3 , 01 June 2003 , pp. 636 - 648
Copyright
Copyright © Canadian Mathematical Society 2003

References

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