Introduction
The papers contributed to the present volume commemorate my 70th birthday
on July 21, 2007. Most of them were presented as lectures in the Paulfest at the
Pontifical Catholic University of Rio de Janeiro (PUC-Rio) from August 6 to 10,
2007. I am deeply grateful to these friends and colleagues for this moving expression
of personal friendship and professional solidarity in our mathematical work.
This collection of papers on differential topology and geometry is concentrated
on the theory of foliations and related areas such as dynamical systems, group
actions on low dimensional manifolds, and geometry of hypersurfaces. There are
survey papers on classification of foliations and their dynamical properties, includ-
ing codimension one foliations with Bott-Morse singularities. Some of the papers
involve the relationship of foliations with characteristic classes, contact structures,
and Eliashberg-Mishachev wrinkled mappings.
Here is a brief description of each of the papers.
There are three surveys of aspects of foliation theory. Steve Hurder has written
a comprehensive and up-to-date survey of the dynamics, structure, and classifi-
cation of foliations, beginning with a short history and emphasizing transverse
dynamics. He describes the Haefliger classifying space and characteristic classes of
foliations with their relationship to dynamical properties such as entropy, amenabil-
ity, and invariant transverse measures. His recent result (Theorem 12.4) giving a
decomposition of a C1 foliation on a closed manifold into three saturated Borel
sets–the elliptic, parabolic, and (partially) hyperbolic leaves, according to the leaf
holonomy–leads to many results regarding the foliation dynamics and offers many
open problems. The lengthy bibliography is a comprehensive listing of papers in
these areas.
Takashi Tsuboi surveys the theory of classifying spaces for a groupoid Γ, in-
cluding a clear and detailed construction of the classification of the classifying space
and information about its topology for various interesting groupoids Γ. Special
attention is given to the Haefliger classifying space BΓq
r
for codimension q foliations
(where Γq r denotes the groupoid of germs of Cr diffeomorphisms of Rq) and related
classifying spaces for foliations with additional structures. He states the Gromov-
Phillips-Haefliger-Thurston existence theorem for foliations and gives a deep insight
into the Mather-Thurston isomorphism of the homology of the classifying space of
Diffq(Rq)
r
with that of
ΩqB¯q.
Γ r There are many exercises for the reader and a
number of unsolved problems at the frontier of current research.
Bruno Scardua and Jos´ e Seade give a survey of their study of foliations with
Bott-Morse singularities. These are codimension one singular foliations whose sin-
gular leaves are compact and each have a neighborhood diffeomorphic to the prod-
uct of the singular leaf by the level sets of a Morse function near a singular point.
vii
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