A finite set of local moves for Kirby calculus

ABSTRACT. We exhibit a finite set of local moves that connect any two surgery presen-tations of the same 3-manifold via framed links in S3. The moves are handle-slides and blow-downs/ups of a particular simple kind. A framed link L ⊂ S3 in the three-sphere is a link equipped with a section (the fram-ing) of the unitary normal bundle. The framing is considered only up to isotopy and is notoriously determined by assigning an integer to each component of L, which counts the algebraic intersection of the framing with the standard longitude. Let L ⊂ S3 be a framed link in the three-sphere. A surgery along L is a standard cut-and-paste operation in three-dimensional topology that consists of removing from S3 a solid torus neighborhood of each component of L and gluing it back via a different map, hence producing a new closed 3-manifold N. The map sends a meridian of the solid torus to a curve parallel to the framing of L, and this requirement is enough to determine the new 3-manifold N up to diffeomorphism. This cut-and-paste operation has a natural four-dimensional interpretation: we may see surgery as the result of attaching a four-dimensional 2-handle to S3, thus constructing a four-dimensional cobordism from S3 to N. A celebrated theorem of Lickorish-Wallace [3, 4] states that every closed orientable 3-manifold N can be obtained by surgerying along some framed link L in the three-sphere. It is easy to construct many different framed links giving rise to the same 3-manifold N: there are in fact various types of moves that transform L into a new framed link L ′ withouth affecting N. The most important ones are handles-slides and blow-downs. A handle-slide consists of sliding one 2-handle onto another; this move does not affect the four-dimensional cobordism and hence N is also preserved. The link is modified as follows: pick two components of L, select a handlebody containing them as in Fig. 1, and then modify one component as shown in the picture. If a component K of L is unknotted and has framing±1, it determines a±CP2-summand in the four-dimensional cobordism: the removal of this factor is a topological blow-down which also does not affect N. The corresponding move is shown in Fig. 2. The inverse of a blow-down is of course called a blow-up. FIGURE 1. A handle-slide. The framing of all components is horizontal. Note that the handlebody may be kotted in S3. 1 a