viii INTRODUCTION had another motive for this approach namely, to prove that the universal covering space of an orientable, irreducible 3-manifold is itself irreduc- ible. I was only able to carry through equivarient surgery in the limited case of two-sheeted coverings (involutions) hence, I give an equivarient surgery proof of the Loop-Theorem and Dehn's Lemma but I give no new infor- mation on the Sphere Theorem. Since I presented these lectures, W. Meeks and S. T. Yau have given unified proofs of the Loop Theorem - Dehn's Lemma and the Sphere Theorem, using minimal surfaces to accomplish equivarient surgery. Their methods also show that the universal covering space of an orientable, irreducible 3-manifold is irreducible. In Chapter II the main result is the Prime Decomposition Theorem (II.A) for compact, orientable 3-manifolds. Here, the classical proof of existence is due to H. Kneser [Kn-] and is a very intriguing proof. However, a more natural approach is an argument based on reasoning by induction. I present such an argument using a theorem of W. Haken [Ha..] , which states that a closed, orientable 3-manifold admitting a connected sum decomposition, admits a connected sum decomposition by closed, orientable 3-manifolds having strictly smaller Heegaard genus (II.7). Even here I give a new proof of Hakenfs Theorem, which is surprisingly easy and, moreover, is really a 2- dimensional argument. In Chapter HL, I give the definition of an incompressible surface and of a Haken-manifold. I give some sufficient (and some necessary) condi- tions for the existence of an incompressible surface in a 3-manifold. I have included a number of examples, many of which I use later in the manu- script. Here, however, the main result is the so-called Haken-Finiteness Theorem (III.20). It gives a finiteness condition for collections of pair- wise disjoint, incompressible surfaces embedded in a Haken-manifold. It is
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