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

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