Incompressibility of surfaces in surgered 3-manifolds
The problem we consider in this paper was raised in [\textit{M. Culler}, \textit{C. McA. Gordon}, \textit{J. Luecke} and \textit{P. B. Shalen}, Ann. Math., II. Ser. 125, 237-300 (1987; Zbl 0633.57006)]. Suppose \(T\) is a torus on the boundary of an orientable 3-manifold \(X\), and \(S\) is a surface on \(\partial X-T\) which is incompressible in \(X\). A slope \(\gamma\) is the isotopy class of a nontrivial simple closed curve on \(T\). Denote by \(X(\gamma)\) the manifold obtained by attaching a solid torus to \(X\) so that \(\gamma\) is the slope of the boundary of a meridian disc. Given two slopes \(\gamma_1\) and \(\gamma_2\), we denote their (minimal) geometric intersection number by \(\Delta (\gamma_1, \gamma_2)\). Theorem 1. Suppose \(X\) contains no incompressible annulus with one boundary component in \(S\) and the other in \(T\). If \(S\) compresses in \(X(\gamma_1)\) and \(X(\gamma_2)\), then \(\Delta (\gamma_1, \gamma_2) \leq 1\). Hence, there are at most three slopes \(\gamma\) such that \(S\) is compressible in \(X (\gamma)\). This proves Conjecture 2.4.1 of [Culler-Gordon-Luecke-Shalen, loc. cit.]. Besides its own interest, the theorem will also (at least conceptually) simplify part of the proof of the Cyclic Surgery Theorem. In that paper the authors proved some results on the above theorem: If the conditions are satisfied, then \(\Delta (\gamma_1, \gamma_2) \leq 2\). (They also proved that the theorem is true when \(S\) is a torus). Our task is to rule out the possibility of \(\Delta (\gamma_1, \gamma_2) =2\). Suppose \(S\) is compressible in \(X (\gamma_1)\), then the central curve \(K\) of the attached solid torus will be a knot in \(X (\gamma_1)\). Thus Theorem 1 follows immediately from the (equivalent) theorem below, which is more suitable to our method of proof: Theorem 2. Let \(S\) be a surface on the boundary of a 3-manifold \(M\). Let \(K\) be a knot in \(M\) which is not isotopic to a simple closed curve on \(S\). If \(S\) is compressible in \(M\) and is incompressible in \(M-K\), then \(S\) is incompressible in \((M,K; \gamma)\) unless \(\Delta (m, \gamma) \leq 1\), where \(m\) is the meridian slope of \(K\). In the Theorem, \((M,K; \gamma)\) denotes the manifold obtained by surgery on \(K\) with slope \(\gamma\). In the above notation, \((M,K; \gamma)= E(K) (\gamma)\), where \(E(K)= M-\text{Int} N(K)\) is the exterior of \(K\)> Note that in Theorem 2 we can not conclude that there are at most three surgeries to make \(S\) compressible: There may still be some annulus with one boundary on \(\partial M\) and the other a meridian curve on \(\partial N(K)\). In Theorem 1, if \(X\) contains some incompressible annulus with one boundary in \(S\) and the other in \(T\), then \(S\) may be compressible in \(X(\gamma)\) for infinitely many \(\gamma\). Examples are presented in [\textit{J. Berge}, Topology Appl. 38, 1-19 (1991; Zbl 0725.57001), \textit{D. Gabai}, Topology 28, 1-6 (1989; Zbl 0678.57004)] where \(\partial M\) is compressible in \((M,K; \gamma)\) for three different slopes, and yet there is no incompressible annulus between \(\partial M\) and \(\partial N(K)\). So the above theorems are the best possible. Nevertheless, we notice that the theorems are still true if \(S\) is a properly embedded surface: Cutting the manifold along \(S\), we will get back to the situation in the theorems. In section 1 we prove Theorem 2 under the additional assumption that \(K\) can be isotoped to \(\alpha \cup \beta\), where \(\alpha\) is an arc in \(\partial M\), and \(\beta\) is an arc properly embedded in \(M\) so that \(\partial M-\partial \beta\) is compressible in \(M-\beta\). Any compressing disc of \(\partial M\) in \(M\) will intersect \(K\). The above hypothesis means that \(K\) is not ``very knotted''. The intersection can be arranged to be on a boundary arc \(\alpha\). Examples of such knots are presented by the 1-bridge knots, in which case the arc \(\beta\) can be isotoped rel \(\partial \beta\) to an arc \(\beta' \) on \(\partial M\), but no such \(\beta'\) can be disjoint from \(\alpha\). The first proof of Theorem 2 was then completed by a result of Gordon and Luecke (unpublished). Using the ``representing all type'' techniques developed in [\textit{C. McA. Gordon} and \textit{J. Luecke}, J. Am. Math. Soc. 2, 371-415 (1989; Zbl 0678.57005)], they were able to prove that if \(\Delta (m, \gamma) =2\), then there exist compressing discs of \(S\) in \(M\) and \((M,K; \gamma)\) such that one of the intersection graphs \(\Gamma_1, \Gamma_2\) contains at least \(n\) parallel boundary edges. It is easy to see that in this case \(K\) can be isotoped to \(\alpha \cup \beta\) satisfying the hypothesis of Proposition 1, and the result follows. Sections 2 and 3 play the same role as the above result of Gordon and Luecke, but the proof is more elementary in the sense that we only use the results of [Culler-Gordon-Luecke-Shalen, loc. cit.], and the argument is simpler. In Section 2, we reduce the proof of Theorem 2 to the existence of some bands in \(M\) with certain ``nice'' properties. Then in section 3 we apply some results of [Culler-Gordon-Luecke-Shalen, loc. cit.] to show that such nice bands exist if \(\Delta (m, \gamma) \geq 2\). This completes the proof.
scientific article; zbMATH DE number 89833