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ropoloy,v Vol. 29. No. 4. pp. 411I ~500. 19vo Pnntcd I" Great Bntam.
 

Summary: ropoloy,v Vol. 29. No. 4. pp. 411I ~500. 19vo
Pnntcd I" Great Bntam.
lx4&93839oso?~-00
17 1990 Pergamon Press plc
PRODUCING REDUCIBLE 3-MANIFOLDS 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 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]:

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics