Klein bottle and toroidal Dehn fillings at distance 5
Let \(M\) be a compact, connected, orientable \(3\)-manifold with a torus boundary component \(T\). A slope on \(T\) is the isotopy class of an essential simple closed curve. Given a slope \(\gamma\), \(M(\gamma)\) denotes the \(3\)-manifold obtained by \(\gamma\)-Dehn filling \(M\) along \(T\). For two slopes \(\gamma _1\) and \(\gamma _2\) on \(T\), \(\Delta(\gamma _1, \gamma _2)\) denotes the distance between them, which is their geometric intersection number. \textit{C. McA. Gordon} and \textit{Y.-Q. Wu} [Toroidal Dehn fillings on hyperbolic 3-manifolds. Mem. Am. Math. Soc. 909, 140 p. (2008; Zbl 1166.57014)] determined all hyperbolic \(3\)-manifolds admitting two toroidal Dehn fillings at distance \(4\) or \(5\). In the paper under review it is proved that if \(M\) is a hyperbolic \(3\)-manifold with a torus boundary component and there are two slopes \(\pi\), \(\tau\) such that \(M(\pi)\) contains a Klein Bottle, \(M(\tau)\) contains an essential torus and \(\Delta(\pi, \tau)=5\), then \(M(\pi)\) is toroidal and \(M\) is one of several \(3\)-manifolds obtained by certain Dehn fillings along two boundary components of the magic manifold \(N\); this is the exterior of the 3-chain link, which was studied by \textit{B. Martelli} and \textit{C. Petronio} [Commun. Anal. Geom. 14, No.~5, 969--1026 (2006; Zbl 1118.57018)]. As a corollary it is obtained that if \(M\) is a hyperbolic \(3\)-manifold with a single torus boundary component and we suppose that there are two slopes \(\pi\), \(\tau\) such that \(M(\pi)\) is a lens space containing a Klein bottle and \(M(\tau)\) contains an essential torus then \(\Delta(\pi , \tau) \leq 4\). The proofs are made using graph intersection theory and a deep analysis of Scharlemann cycles.
scientific article; zbMATH DE number 5809046