INTRODUCTION

This manuscript is intended to present the development of three-

manifold topology evolving from the study of incompressible surfaces em-

bedded in three-manifolds. This, of course, is quite a restriction to

come under the broad title of the manuscript. But even here, the reader

will find many important aspects in the theory of incompressible surfaces

missing. I am not trying to make the manuscript all inclusive (I do not

believe that a possible task) and I have not tried to make the bibliography

complete. The manuscript is exactly what I would do in ten lectures with

the above intention.

The reader will find the subject, through the first six chapters,

overlappling with the book by John Hempel, [He ] . I have very high regard

for Hempel's book and debated a bit about presuming its contents. However,

I decided that this manuscript would better serve if the development began

more at the foundations. And in the end, I believe that the reader will

find the overlap mostly in spirit and terminology. I have developed the

material from my point of view and I give a number of new proofs to the

classical results. If I needed material that appears in Hempel*s book, and

if I felt that the development there was consistent with my development,

then I refer the reader to the appropriate result. While this manuscript

certainly can be considered independent of Hempelfs book, it can also be

considered as a sequel to it.

In Chapter I, I wanted togive a unified proof of the Loop Theorem -

Dehn's Lemma and the Sphere Theorem, using equivarient surgery. I also