Surgery on a knot in.surface I/ MARTIN

Suppose F is a compact orientable surface, K is a knot in F I, and.F I/surg is the 3–manifold obtained by some nontrivial surgery on K. If F f0g compresses in.F I/surg, then there is an annulus in F I with one end K and the other end an essential simple closed curve in F f0g. Moreover, the end of the annulus at K determines the surgery slope. An application: Suppose M is a compact orientable 3–manifold that fibers over the circle. If surgery on K M yields a reducible manifold, then either the projection K M! S1 has nontrivial winding number, K lies in a ball, K lies in a fiber, or K is cabled. 57M27 The study of Dehn surgery on knots in 3–manifolds has a long and rich history, interacting in a deep way with sophisticated combinatorics (in Gordon and Luecke [6] and Culler, Gordon, Luecke and Shalen [2]), the theory of character varieties (in [2] and Boyer, Gordon and Zhang [1]) and sutured manifold theory (in Gabai [3] and Scharlemann [8]). It is pleasing then to find a result that is simple to state, easy to understand and yet has so far escaped explicit notice. Yi Ni has pointed out that there is at least implicit overlap of our results with his paper [7, Theorem 1.4 and Section 3]. Theorem 0.1 Suppose F is a compact orientable surface, K is a knot in F I, and.F I/surg is the 3–manifold obtained by some nontrivial surgery on K. If F f0g compresses in.F I/surg, then K is parallel to an essential simple closed curve in F f0g. Moreover, the annulus that describes the parallelism determines the slope of the surgery