CHAPTER I. LOO P THEOREM SPHERE THEOREM: The Tower Constructio n In this chapter I discuss the Loop Theorem, Dehn's Lemma, the Sphere Theorem and some of their more interesting generalizations. There is a very nice account of these theorems given in the book by John Hempel [He-]. In particular, one can find there complete proofs that are quite readable. There are a few reasons why I have chosen to begin these lectures at this point. The fundamental importance of these theorems to the methods of three-manifold topology, and particularly to the approach of this author, cannot be overestimated. I plan to give a modified version of the classical method of proof. The approach that I use here, to prove the Loop Theorem,can be used to give a unified proof of Dehn's Lemma and the Loop Theorem with their generalizations after Shapiro- Whitehead [S-W] and Waldhausen [W, J, respectively. I have here a forum to make a case for a new proof of the Sphere-Theorem using only the Loop-Theorem (or even better, a proof using equivariant surgery in a universal covering, see Chapter VI ) and to present some Interesting problems arising from the study of these theorems. However, certainly the main pursuasive for starting at this point is that many participants at these lectures are not familiar with the techniques of 3-manifold topology and are here to gain a working knowledge for the study of problems in this area. With this in mind, there is no better place to start. The results of this chapter can be considered as the first step in the program of studying singular mappings of surfaces into 3-manifolds. 1
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