Abstract
It has been known for over 30 years that every closed connected orientable 3manifold is obtained by surgery on a link in S3 [8]. However, a classification of such 3-manifolds in terms of this surgery construction has remained elusive. This is due primarily to the lack of uniqueness of the surgery description. In [5], Kirby gave us a calculus of surgery diagrams. However, the lack of a 'canonical' surgical method of getting from one 3-manifold to another has hampered further progress. In this paper, we show that if one restricts attention to the case where a surgery curve is homotopically trivial in a 3-manifold, then one has the following uniqueness theorem.
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