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

ix