Obtaining 3-Manifolds by Surgery on S ³

Abstract

The aim of this chapter is to show, in Theorem 12.14, that every closed connected orientable 3-manifold can be obtained by “surgery” on S ³. The method used is a version of that of [77]. An elementary r-surgery on a general n-manifold M is the operation of removing from M an embedded copy of S ^r × D ^n-r and replacing it with a copy of D ^r+1 × S ^n-r-1 the replacement being effected by means of the obvious homeomorphism between the boundaries of the removed set and its replacement. Surgery in general is a sequence of elementary surgeries. In the case of surfaces, instances of 1-surgery and 0-surgery have already been employed in earlier chapters, usually when the surface was contained in S ³. The only surgeries needed in this chapter are 1-surgeries on a 3-manifold, and it is easy to see they can be performed “simultaneously”. The surgery process will consist of the removal from S ³ of disjoint copies of S ¹ × D ² and their replacement by copies of D ² × S ¹ Of course, the set removed and its replacement are homeomorphic, but the parametrisation of the removed set as disjoint copies of S ¹ × D ², and the canonical method of replacement with respect to that, ensure that the new manifold is usually not S ³. A collection of disjoint solid tori in S ³ is just a regular neighbourhood of a link, and a parametrisation of a neighbourhood of each component by S ¹ × D ² is called a framing of the link. Thus it will be shown that 3-manifolds can be interpreted by means of framed links in S ³.