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.
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http://dx.doi.org/10.1090/cbms/043/01
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