Links, two-handles, and four-manifolds

Details Links, two-handles, and four-manifolds Links, two-handles, and four-manifolds

2004-12-28

arxiv.org/abs/math/0412511v4

Int. Math. Res. Not. 58 (2005), 3595-3624

Mathematics

Geometric Topology

23 pages, 9 figures. Final version

Scientific paper

We show that only finitely many links in a closed 3-manifold share the same complement, up to twists along discs and annuli. Using the same techniques, we prove that by adding 2-handles on the same link we get only finitely many smooth cobordisms between two given closed 3-manifolds. As a consequence, there are finitely many smooth closed 4-manifolds constructed from some Kirby diagram with bounded number of crossings, discs, and strands, or from some Turaev special shadow with bounded number of vertices. (These are the 4-dimensional analogues of Heegaard diagrams and special spines for 3-manifolds.) We therefore get two filtrations on the set of all smooth closed 4-manifolds with finite sets. The two filtrations are equivalent after linear rescalings, and their cardinality grows at least as n^{c*n}.

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