Weakly reducible Heegaard splittings of Seifert fibered spaces
type: Item
instance of: scholarly article
title: Weakly reducible Heegaard splittings of Seifert fibered spaces
zbMATH Open document ID: 0939.57020
DOI: 10.1016/S0166-8641(98)00091-1
author: Jennifer Schultens
published in: Topology and its Applications
publication date: 2000-03-20T00:00:00Z
review text: The paper under review shows that a weakly reducible Heegaard splitting of an exceptional Seifert fibered space is reducible. This together with Theorem 2.6 in [\textit{Y. Moriah} and \textit{J. Schultens}, Topology 37, No. 5, 1089-1112 (1998; Zbl 0926.57016)] shows that an irreducible Heegaard splitting of an orientable Seifert fibered space over an orientable surface is either horizontal or vertical. A handlebody is a connected orientable \(3\)-(dimensional) manifold obtained by attaching \(1\)-handles to \(0\)-handles. A disc \(D\) in a handlebody \(V\) is called essential if its boundary loop \(\partial D\) is contained in \(\partial V\) and does not bound a disc in \(\partial V\). Every closed orientable \(3\)-manifold has a decomposition into two handlebodies \(W_1\), \(W_2\), called a Heegaard splitting. A Heegaard splitting is called reducible (weakly reducible respectively) if \(W_i\) contains an essential disc \(D_i\) for \(i=1\) and \(2\) such that \(\partial D_1 = \partial D_2\) (\(\partial D_1 \cap \partial D_2 = \emptyset\) respectively). A Seifert fibered space \(M\) is a \(3\)-manifold which admits a foliation of codimension \(2\) with all leaves being circles, called fibers. A fiber \(c\) is called regular if there is a regular neighbourhood \(N\) of \(c \cong S^1\) such that \(N\) is homeomorphic to a solid torus \(\cong D^2 \times S^1\) and that \(p \times S^1\) is a fiber for every point \(p\) in the \(2\)-disc \(D_2\). Otherwise, a fiber is called exceptional. When by identifying all the fibers to points we obtain a surface \(F\), we say that \(M\) is a Seifert fibered space over \(F\). An orientable Seifert fibered space is called exceptional if it is fibered over the \(2\)-sphere \(S^2\) with three exceptional fibers and rational Euler number \(0\). The proof of the theorem is based on the fact that every exceptional Seifert fibered space fibers as an \(S\)-bundle over \(S^1\) so that \(M \cong S \times [0,1] / \sim_\phi\), where \(\phi\) is a…
reviewed by: Chuichiro Hayashi
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 1395065
zbMATH Keywords: Heegaard splitting
zbMATH Keywords: exceptional Seifert fibered space
zbMATH Keywords: horizontal
zbMATH Keywords: vertical
MaRDI profile type: Publication
full work available at URL: s0166-8641(98)00091-1
OpenAlex ID: W2132802633
Wikidata QID: Q125552839
description: scientific article; zbMATH DE number 1395065