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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]: GABAI'S THEOREM. Let k be a knot in M = D2 x S' with nonzero wrapping number. If_ci' is a manifold obtained by non-trivial surgery on k then one of the following must hold: (1) M = D2 x S' = M' and both k and k' are 0 or l-bridge braids. (2) M' = WI # W,, where W2 is a closed 3-manifold and H, (W,) isjnite and non-tritial. (3) M' is irreducible and aM ' is incompressible.

Suppose M is a compact orientable irreducible 3-manifold with Heegaard splitting surfaces P and Q. Then either Q is isotopic to a possibly stabilized or boundarystabilized copy of P or the distance d.P / Ä 2genus.Q/.More generally, if P and Q are bicompressible but weakly incompressible connected closed separating surfaces in M then either P and Q can be well-separated or P and Q are isotopic or d.P / Ä 2genus.Q/. 57N10; 57M50

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If there are any 2-component counterexamples to the Generalized Property R Conjecture, a least genus component of all such counterexamples cannot be a fibered knot. Furthermore, the monodromy of a fibered component of any such counterexample has unexpected restrictions.The simplest plausible counterexample to the Generalized Property R Conjecture could be a 2-component link containing the square knot. We characterize all two-component links that contain the square knot and which surger to # 2 (S 1 × S 2 ). We exhibit a family of such links that are probably counterexamples to Generalized Property R. These links can be used to generate slice knots that are not known to be ribbon.