On Kirby's calculus of links

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(Receiued in revised form 21 July 1978) 4 FRAMED OR labelled link in S3 is a finite collection L of embedded circles in S’, each one of which is labelled by an integer. In [3] and [7] Lickorish and Wallace showed that any orientable 3-manifold can be obtained by surgery on S3 using such a link. Furthermore in [21 Kirby shows that two such manifolds are homeomorphic if and only if the links are related by a series of combinatorial moves. Thus there is a classification of orientable 3-manifolds in terms of equivalence classes of links. In this paper we present an exposition of Kirby’s theorem in a form which applies to links in a general 3-manifold and we also give a classification of non-orientable 3-manifolds by equivalence classes of links in the non-orientable S* bundle over S’ denoted S’ 5 S2. Our exposition reduces the dependence on ‘Cerf Theory’ which plays a central role in Kirby’s paper[2], and clarifies the connection between the various allowable moves. This clarification allows us to state the main classification theorem in the following simpler form: THEOREM. Orientation preserving homeomorphism classes of compact closed oriented 3-manifolds corresponds bijectively to equivalence class of labelled links in S’ where the equivalence is generated by a single move-the “Kirby move” (see §l