# zbMATH — the first resource for mathematics

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.

##### MSC:
 57N10 Topology of general $$3$$-manifolds (MSC2010)
##### Keywords:
surgery; knot; reducible 3-manifold; Dehn surgery; sutured manifolds
Full Text: