INTRODUCTION x i results of the paper [J-S«]. I use covering space arguments rather than the formal homotopy language used there. I believe that the reader should find the material here much more intuitive. I have not, however, covered many of the surprising results obtained in that manuscript. Time and space would not allow this. I am disappointed in not doing so but if the reader has interest after completing Chapter VII, I believe revisiting [J-S^] may be a more pleasant experience. In Chapter VHI^ I present a major part of the joint work of Shalen and myself [J-S..]. My approach here follows very much the lines of our original approach. While this approach is similar to the presentation in [J-S ], I do not go into the generalities of that manuscript (thereby, I considerably reduce the notation and length of the presentation) and I have set up a different foundation with the material of Chapter VII. I give proofs of the Essential Homotopy Theorem (VIII.4), the Homotopy Annulus Theorem (VIII.10) and the Homotopy Torus Theorem (VIII.11) and the Annulus- Torus Theorems (VIII.13 and VIII.14). I believe that the presentation of Chapter IX will introduce the reader to an understanding of the characteristic Seifert pair of a Haken- manifold with a minimum amount of work. In fact, this chapter is really very short, By introducing the idea of a perfectly-embedded Seifert pair in a 3-manifold, I am able to show (under the partial ordering that one perfectly-embedded Seifert pair (Z',^1) is less than or equal to another perfectly-embedded Seifert pair (£,$) if there is an ambient isotopy taking ST into Int £ and $T into Int $) that a Haken-manifold with incom- pressible boundary admits a unique (up to ambient isotopy) maximal, perfectly- embedded Seifert pair. This unique maximal, perfectly-embedded Seifert pair
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