Producing reducible 3-manifolds by surgery on a knot. (English) Zbl 0727.57015

A long standing conjecture is that surgery on a knot k in \(S^ 3\) yields a reducible 3-manifold only if k is cabled. More generally, given a knot k in a compact orientable 3-manifold M, one would like to obtain a description of the 3-manifolds \(M'\) obtained from M by Dehn surgery on k, and of the knots \(k'\subset M'\) that are the core of the filling tori. This had been done by Gabai in the case that M is a solid torus. In the paper under review the author obtains the relation between \((M',k')\) and (M,k) for all 3-manifolds M such that \(\partial M\) compresses in M or M contains a sphere not bounding a rational homology ball. For example, \(M'\) is reducible only if k is cabled and the slope of the surgery is that of the cabling annulus, or if \(M=(S^ 1\times S^ 2)\#W\), \(M'=W_ 1\#W_ 2\), with W, \(W_ 1\), \(W_ 2\) rational homology spheres. The proofs use the theory and techniques of sutured manifolds, as developed by the author [J. Differ. Geom. 29, 557-614 (1989; Zbl 0673.57015)]. A corollary is a strengthening of Gabai’s result in the special case that M is a solid torus; another corollary is a proof of the above conjecture for satellite knots in \(S^ 3:\) in both cases, \(M'\) is reducible only if k is cabled.


57N10 Topology of general \(3\)-manifolds (MSC2010)


Zbl 0673.57015
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