This manuscript is a detailed presentation of the ten lectures
given by the author at the Conference Board of Mathematical Sciences
Regional Conference on Three-Manifold Topology, held October 8-12, 1977,
at Virginia Polytechnic Institute and State University. The purpose of
the conference was to present the current state of affairs in three-manifold
topology and to integrate the classical results with the many recent ad-
vances and new directions.
The ten principal lectures presented here deal with the study of
three-manifolds via incompressible surfaces. At the conference, these lec-
tures were supplemented by eight special lectures: two lectures each by
Professor Joan Birman of Columbia University (Heegaard Theory), Professor
Sylvain Cappell of the Courant Institute (branched coverings and applica-
tions to 4-dimensional topology), Professor Robion Kirby of The University
of California/Berkeley (three-dimensional knot theory) and Professor
William Thurston of Princeton University (three-dimensional hyperbolic geo-
I wish to thank the host university, the members of the Department
of Mathematics at V.P.I. & S.U., and the Conference Director, Professor
Charles Feustel. A special thanks to the Conference Coordinator, Professor
Ezra fBudT Brown.
I believe that the conference was a success and that much of the
success was due to the special lectures given by the previously named parti-
cipants. I wish to acknowledge my thanks to Professors Birman, Cappell,
Kirby and Thurston.