x INTRODUCTION a Seifert fibered manifold, then M itself is a Seifert fibered manifold (VI.29) and I develop the topological study of Seifert fibered manifolds needed in the later chapters to present the recent work of Shalen and my- self, Waldhausen and Johannson. Also, I present a proof of the Gordon- Heil prediction that a Haken-manifold, having an infinite cyclic, normal subgroup of its fundamental group, is a Seifert fibered manifold (VI.24) and a description of compact, incompressible surfaces in Seifert fibered manifolds (VI.34). I hope that this chapter on Seifert fibered manifolds will serve as an introduction to this important class of 3-manifolds for the beginners in the subject and provide some enjoyable reading for the more advanced. I begin Chapter VII by giving general conditions that are suffi- cient for a noncompact 3-manifold to admit a manifold compactification. The basic result here (VII.1) is after T. Tucker's work [Tu ]. I give a new proof of J. Simon's Theorem [Si ] that the covering space of a Haken- manifold corresponding to the conjugacy class of a subgroup of the funda- mental group, which has the finitely generated intersection property, admits a manifold-compactification to a Haken-manifold (VII.4). This result is then applied to covering spaces corresponding to finitely generated peri- pheral subgoups (finitely generated, peripheral subgroups have the finitely generated intersection property (V.20), [J-M] and [J-S~]) and to covering spaces corresponding to the fundamental groups of well-embedded submani- folds. This latter material is based on work of B. Evans and myself and allows certain isotopes that are used in all the later chapters. While this work on compactifications is itself very important to me, its use here pro- vides the foundation for the existence and uniqueness of the characteristic pair factor of a Haken-manifold pair. This is a new presentation of the main

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