INTRODUCTION ix indespensible to the approach of this author and I have presented it in both detail and generality. I also present a convenient generalization due to Peter Shalen and myself. I believe the reader will find that Chapter IV is a fun chapter. It is a very important chapter since the existence of a hierarchy for a Haken-manifold provides an inductive method of proof, which has been a major tool employed in the study of this important class of 3-manifolds. But, in this chapter I introduce the notion of a partial hierarchy and give some fun examples of infinite partial hierarchies for compact 3-manifolds. I also define the (closed) Haken number of a compact 3-manifold and the length of a Haken-manifold and I discuss different inductive methods of proof (advan- tages and perils). I end Chapter IV with Theorem IV.19, where I prove that any Haken-manifold has a hierarchy of length no more than four— a result that I have never been able to use, Chapter V is an abbreviated version of what might have been a revision of my Princeton Lecture Notes on the structure of three-manifold groups, I first indicate the restrictive nature of three-manifold groups by classifying the abelian three-manifold groups. I present the Scott- Shalen Theorem (V.16) that any finitely generated three-manifold group is finitely presented. This is done using the idea of indecomposably covering a group, which I like very much. However, a large part of Chapter V is devoted to open questions about three-manifold groups and properties for three-manifold groups e.g,, the finitely generated intersection property for groups, which is important in later chapters (particularly, Chapter VII). Chapter VI is in some sense the beginning of the new material. Here I present new results about Seifert fibered manifolds. Namely, I prove that if a finite sheeted covering space of a Haken manifold M is
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