Classifying foliations

Steven Hurder

Abstract. We give a survey of the approaches to classifying foliations, start-

ing with the Haefliger classifying spaces and the various results and examples

about the secondary classes of foliations. Various dynamical properties of fo-

liations are introduced and discussed, including expansion rate, local entropy,

and orbit growth rates. This leads to a decomposition of the foliated space

into Borel or measurable components with these various dynamical types. The

dynamical structure is compared with the classiﬁcation via secondary classes.

1. Introduction

A basic problem of foliation theory is how to “classify all the foliations” of ﬁxed

codimension-q on a given closed manifold M, assuming that at least one such

foliation exists on M. This survey concerns this classiﬁcation problem for foliations.

Kaplan [192] proved the ﬁrst complete classiﬁcation result in the subject in 1941.

For a foliation F of the plane by lines (no closed orbits) the leaf space T =

R2/F

is a (possibly non-Hausdorﬀ) 1-manifold, and F is characterized up to orientation-

preserving diﬀeomorphism by the oriented leaf space T . (See [47, Appendix D]; also

[126, 127, 324].) Palmeira [252] proved that in the case of a simply connected

manifold of dimension at least three, the leaf space is a complete invariant for

foliations by hyperplanes.

Research on classiﬁcation advanced dramatically in 1970, with three seminal works:

Bott’s Vanishing Theorem [28], Haefliger’s construction of a “classifying space” for

foliations [119, 120], and Thurston’s profound results on existence and classiﬁ-

cation of foliations [297, 298, 299, 300], in terms of the homotopy theory of

Haefliger’s classifying spaces BΓq. The rapid progress during this period can be

seen in the two survey works by H. Blaine Lawson: ﬁrst was his article “Foliations”

[205], which gave a survey of the ﬁeld up to approximately 1972; second was the

CBMS Lecture Notes [206] which included developments up to 1975, including

the Haefliger-Thurston Classiﬁcation results. The work of many researchers in the

1970’s ﬁlled in more details of this classiﬁcation scheme, as we discuss below.

1991 Mathematics Subject Classiﬁcation. Primary 22F05, 37C85, 57R20, 57R32, 58H05,

58H10; Secondary 37A35, 37A55, 37C35 .

Key words and phrases. Foliations, diﬀerentiable groupoids, smooth dynamical systems, er-

godic theory, classifying spaces, secondary characteristic classes .

The author was supported in part by NSF Grant #0406254.

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Contemporary Mathematics

Volume 498, 2009

c 2009 American Mathematical Society

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http://dx.doi.org/10.1090/conm/498/09741