P r e f a c e
The study of mapping class groups, the moduli space of Riemann surfaces,
Teichmuller geometry and related areas has seen a recent influx of young mathe-
maticians. Inspired by this, I had the idea to solicit from some of the senior people
in the area papers that would focus primarily on open problems and directions. I
proposed that these problems might range in scope from specific computations to
broad programs. The idea was then to bring these papers together into one source,
most likely a book. This book would then be a convenient location where younger
(and indeed all) researchers could go in order to find problems that might inspire
them to further work. I was especially interested in having problems formulated
explicitly and accessibly. The result is this book.
The appearance of mapping class groups in mathematics is ubiquitous; choosing
topics to cover seemed an overwhelming task. In the end I chose to solicit papers
which would likely focus on those aspects of the topic most deeply connected with
geometric topology, combinatorial group theory, and surrounding areas.
Content. For organizational purposes the papers here are divided into four groups.
This division is by necessity somewhat arbitrary, and a number of the papers could
just as easily have been grouped differently.
The problems discussed in Part I focus on the combinatorial and (co)homologi-
cal group-theoretic aspects of mapping class groups, and the way in which these
relate to problems in geometry and topology. The most remarkable recent success
in this direction has been the proof by Madsen and Weiss of the Morita-Mumford-
Miller Conjecture on the stable cohomology of mapping class groups. Further
problems arising from this work are described in Madsen's paper. Other coho-
mological aspects, including those related to various subgroups, most notably the
Torelli group, are discussed in the papers of Bestvina and Hain. The combinatorial
and geometric group theory of mapping class groups admits a rich and interesting
structure. Ideas and problems coming out of this point of view are discussed in the
papers of Farb, Ivanov, Korkmaz, Penner and Wajnryb.
Part II concentrates on connections between various classification problems in
topology and their combinatorial reduction to (still open) problems about map-
ping class groups. In dimension three this reduction is classical. It arises from
the fact that every 3-manifold is a union of two handlebodies glued along their
boundaries. This construction and many of the problems arising from it are de-
scribed in Birman's paper. The reduction of the classification of 4-dimensional
symplectic manifolds to purely combinatorial topological questions about surfaces
and mapping class groups is more recent. The general idea is that (by a theorem of
Donaldson) each closed symplectic 4-manifold admits a symplectic Lefschetz pencil
These are a kind of "fibration with singularities", and the main piece of data that