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

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