 
Summary: ropoloy,v Vol. 29. No. 4. pp. 411I ~500. 19vo
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17 1990 Pergamon Press plc
PRODUCING REDUCIBLE 3MANIFOLDS BY SURGERY ON A
KNOT
MARTIN SCHARLEMANNt
(Received in revisedfirm 10 April 1989)
IT HASlong been conjectured that surgery on a knot in S3 yields a reducible 3manifold 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 3manifold A4
produces a reducible 3manifold M'. But this problem is too complex, since, dually, it asks
which knots in which manifolds arise from surgery on reducible 3manifolds. In this paper
we are able to show, approximately, that if M itself either contains a summand not a
rational homology sphere or is areducible, 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]:
