Producing Reducible 3-manifolds by Surgery on a Knot

IT HAS long been conjectured that surgery on a knot in S3 yields a reducible 3-manifold if and only if the knot is cabled, with the cabling annulus part of the reducing sphere (cf. [7.8, 9, 10, 111). One may regard the Poenaru conjecture (solved in [S]) as a special case of the above. More generally, one can ask when surgery on a knot in an arbitary 3-manifold A4 produces a reducible 3-manifold M’. But this problem is too complex, since, dually, it asks which knots in which manifolds arise from surgery on reducible 3-manifolds. In this paper we are able to show, approximately, that if M itself either contains a summand not a rational homology sphere or is a-reducible, and M’ is reducible, then k must have been cabled and the surgery is via the slope of the cabling annulus. Thus the result stops short of proving the conjecture for M = S3, but (see below) does suffice to prove the conjecture for satellite knots. The results here are broader than this; for a context recall the main result of [3]: