Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Only integral Dehn surgeries can yield reducible manifolds

C. McA. Gordona1 and J. Lueckea2

a1 University of Texas, Austin, TX 78712, U.S.A.

a2 Courant Institute, New York University, New York, NY 10012, U.S.A.

Let K be a knot in S3, and let K(a/b) denote the closed oriented 3-manifold obtained by a/b-Dehn surgery on K. (We parametrize the Dehn surgeries on K by xs211A xs2229 {1/0} as in [9]; in particular, K(1/0) = S3.) If K is a (p, q)-cable knot, then K(pq) is always reducible (see Section 3), and it is conjectured by González-Acuña and Short in [5] that these are the only examples where Dehn surgery on a knot in S3 yields a reducible manifold. One of the results in this direction proved in [5] is that if π1 (K(a/b)) is a non-trivial free product then |b| ≤ 5. We show that this may be sharpened to the assertion in the title.

(Received June 16 1986)