Persistence of Heegaard structures under Dehn filling
A classical way for constructing 3-manifolds is Dehn surgery, where a solid torus is attached to an incompressible torus \(T\) contained in the boundary of a compact connected 3-manifold \(X\). The infinite collection of 3-manifolds, obtained by this technique, is called the Dehn filling space \(D(X)\) of \(X\). The authors study the connection between the Heegaard structure of \(X\) and that of manifolds which belong to \(D(X)\). It is known that any Heegaard surface for \(X\) is a Heegaard surface for any \(M\in D(X)\) [\textit{Y. Rieck}, Topology 39, No. 3, 619-641 (2000; Zbl 0944.57013)]. The converse is not always true: after the filling, Heegaard surfaces for \(X\) may destabilize yielding Heegaard surfaces of arbitrarily lower genus for manifolds in \(D(X)\). This phenomenon is classified according to whether or not the core of the attached solid torus is isotopic into the destabilized surface. When this occurs, the destabilized surface will be a Heegaard surface for infinitely many fillings, arranged along a destabilization line in \(D(X)\). The authors prove that a destabilization line corresponds to a slope bounding an essential surface. Applications are obtained about the Heegaard genus of Dehn fillings on a a-cylindrical manifold.
scientific article; zbMATH DE number 1566033