Global invariants for measured foliations

Hurder, Steven

doi:10.1090/S0002-9947-1983-0712266-3

Global invariants for measured foliations
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Trans. Amer. Math. Soc. 280 (1983), 367-391 Request permission

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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