Reducible manifolds and Dehn surgery.

*(English)*Zbl 0859.57016Let \(X\) be an orientable irreducible 3-manifold whose boundary \(\partial X\) is a torus. As noted in the introduction of the paper, it is an observed fact that the topological and geometric properties of the manifold \(X\) tend to persist in most of the manifolds obtained by Dehn surgery (Dehn filling) on \(\partial X\). In the present paper the question is considered of how many different Dehn surgeries on \(X\) can give reducible manifolds. The main result says that if Dehn surgery on two distinct isotopy classes of simple closed curves (“slopes”) on \(\partial X\) gives reducible manifolds then the geometric intersection number of the two slopes is one (it was known before that it is less than or equal to two). In particular, at most three different slopes can give reducible manifolds, and this result is best possible.

For the proof (outlined in the introduction of the paper) it is shown that two different slopes which are the boundaries of two essential planar surfaces in \((X, \partial X)\) have geometric intersection number one. This is obtained by a careful combinatorial analysis of the intersection pattern of the two planar surfaces. Such methods have been also at the basis of the author’s proof of the long-standing conjecture that knots in the 3-sphere are determined by their complements.

For the proof (outlined in the introduction of the paper) it is shown that two different slopes which are the boundaries of two essential planar surfaces in \((X, \partial X)\) have geometric intersection number one. This is obtained by a careful combinatorial analysis of the intersection pattern of the two planar surfaces. Such methods have been also at the basis of the author’s proof of the long-standing conjecture that knots in the 3-sphere are determined by their complements.

Reviewer: B.Zimmermann (Trieste)

##### MSC:

57N10 | Topology of general \(3\)-manifolds (MSC2010) |

57M99 | General low-dimensional topology |