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

BΓ 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