Surgery on links and double branched covers of S3.

Abstract

The author studies the relationship between 2-fold cyclic coverings of S3 branched over a link and closed, orientable 3-manifolds that are obtained by performing surgery on a link in S3. The links of central importance are the strongly invertible ones, namely, the links L in S3 for which there exists an orientation preserving involution of S3 that induces on each component of L an involution having exactly two fixed points. A key result is that a closed, orientable 3-manifold M can be obtained by performing surgery on a strongly invertible link L if and only if M is a 2-fold cyclic covering of S3 branched over some link L′. This result has several corollaries, among which is that every simply connected 2-fold cyclic branched covering of S3 is S3 if and only if every strongly invertible link has Property P. (A link has Property P if it is impossible to obtain a counterexample to the Poincaré conjecture by doing surgery on it.) The theorem is improved to yield the result that every 2-fold cyclic branched covering of S3 can be obtained by doing surbery on a member of a special family of strongly invertible links, and it yields a new proof of a result of O. Ja. Viro [Mat. Sb. (N.S.) 87 (129) (1972), 216–228;] and of J. S. Birman and H. M. Hilden [Trans. Amer. Math. Soc. 213 (1975), 315–352; #1662 above] that each closed, orientable 3-manifold of Heegaard genus ≤2 is a 2-fold cyclic branched covering of S3. In addition, the author generalizes the surgical modifications of H. Wendt [Math. Z. 42 (1937), 680–696; Zbl 16, 420] to produce a generalized surgery technique, in which n pairwise disjoint solid tori in S3 are replaced by special "graph-manifolds'' bounded by tori. The significant features developed here are the results that every manifold obtained by doing generalized surgery on a strongly invertible link is a 2-fold cyclic branched covering of S3 and that any simply connected 3-manifold obtained by doing generalized surgery on a link in S3 having Property P is S3. By way of application, there is no counter-example to the Poincaré conjecture among the 2-fold coverings of S3 branched over Kinoshita-Terasaka knots or over Conway's 11-crossing knot or over 3-braid knots.