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
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