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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Global invariants for measured foliations

Author: Steven Hurder
Journal: Trans. Amer. Math. Soc. 280 (1983), 367-391
MSC: Primary 57R30; Secondary 28D20, 57R20
MathSciNet review: 712266
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Abstract: New exotic invariants for measured foliations are constructed, which we call the $ \mu $-classes of a pair $ (\mathcal{F},\mu)$. The dependence of the $ \mu $-classes on the geometry of the foliation $ \mathcal{F}$ is examined, and the dynamics of a foliation is shown to determine the $ \mu $-classes in many cases. We use the $ \mu $-classes to study the classifying space $ B{\Gamma_{S{L_q}}}$ of foliations with a transverse invariant volume form, and we show the homotopy groups of $ B{\Gamma _{S{L_q}}}$ are uncountably generated starting in degrees $ q + 3$. New invariants for groups of volume preserving diffeomorphisms also arise from the $ \mu $-classes; these invariants are nontrivial and related to the geometric aspects of the group action. Relations between the $ \mu $-classes and the secondary classes of a foliation are exhibited.

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Keywords: Foliations, invariant measures, characteristic classes, group homology, geometry of foliations
Article copyright: © Copyright 1983 American Mathematical Society