The disjoint curve property and genus 2 manifolds
A closed orientable \(2\)-(dimensional) manifold \(R\), embedded in an orientable \(3\)-manifold \(M\), is called compressible if there is an embedded disc \(D\) (called a compressing disc) in \(M\) such that \(R \cap D = \partial D\) and \(\partial D\) is essential in \(R\), i.e., \(\partial D\) bounds no discs in \(R\). A \(3\)-manifold is called toroidal if it contains an incompressible torus. A handlebody of genus \(n\) is an orientable \(3\)-manifold obtained from a ball by attaching \(n\) \(1\)-handles. Suppose that a closed orientable \(2\)-manifold \(F\) is embedded in a closed orientable \(3\)-manifold \(M\) and separates \(M\) into two handlebodies \(H_1\) and \(H_2\) of genus \(n\). Then we call \((H_1, H_2, F)\) a genus \(n\) Heegaard splitting. The important notion of weak reducibility is introduced by \textit{A. J. Casson} and \textit{C. McA. Gordon} in [Topology Appl. 27, 275-283 (1987; Zbl 0632.57010)]. In the paper under review, a condition weaker than weak reducibility is introduced. A Heegaard splitting \((H_1, H_2, F)\) is said to have the disjoint curve property if there exists a compressing disc \(D_i\) of \(F\) in \(H_i\) for \(i=1\) and \(2\) such that \(F\) contains an essential loop \(c\) disjoint from \(\partial D_1\) and \(\partial D_2\). It is shown that any Heegaard splitting of a toroidal \(3\)-manifold has the disjoint curve property. If a genus \(2\) Heegaard splitting \((H_1, H_2, F)\) of a \(3\)-manifold \(M\) has the disjoint curve property, then either \(M\) is not hyperbolic or \(M\) is obtained by a Dehn surgery on a \(1\)-bridge braid. Here, a Dehn surgery is an operation removing a neighbourhood of a knot and attaching a solid torus. A knot \(K\) in a closed \(3\)-manifold \(X\) is called a \(1\)-bridge braid if there is a Heegaard splitting torus \(T\) of \(X\) such that \(K\) intersects each solid torus bounded by \(T\) in a single boundary parallel arc. If the curve \(c\) intersects a compressing disc of \(F\) in a single point neither in \(H_1\) nor in \(H_2\), then \(M\) is not hyperbolic. Similar results as above are obtained by \textit{J. Hempel} in [Three manifolds as viewed from the curve complex, preprint]. If a genus \(2\) Heegaard splitting surface admits compressing discs on both sides such that their boundary loops intersect each other in three or less points, then the splitting has the disjoint curve property and the \(3\)-manifold is not hyperbolic.
scientific article; zbMATH DE number 1367549