Persistence of Heegaard structures under Dehn filling
type: Item
instance of: scholarly article
title: Persistence of Heegaard structures under Dehn filling
zbMATH Open document ID: 0968.57018
DOI: 10.1016/S0166-8641(99)00147-9
author: Yo'av Rieck
author: Eric Sedgwick
published in: Topology and its Applications
publication date: 2001-09-19T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57R65
Mathematics Subject Classification ID: 57M99
zbMATH DE Number: 1566033
zbMATH Keywords: Heegaard genus
zbMATH Keywords: Dehn surgery
zbMATH Keywords: 3-manifold
zbMATH Keywords: Heegaard surface
reviewed by: Alberto Cavicchioli
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1566033