The disjoint curve property and genus 2 manifolds
type: Item
instance of: scholarly article
title: The disjoint curve property and genus 2 manifolds
zbMATH Open document ID: 0935.57022
DOI: 10.1016/S0166-8641(98)00063-7
published in: Topology and its Applications
publication date: 2000-05-03T00:00:00Z
review text: 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 …
reviewed by: Chuichiro Hayashi
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M50
zbMATH DE Number: 1367549
zbMATH Keywords: Heegaard splitting
zbMATH Keywords: curve complexity
zbMATH Keywords: weakly reducible
zbMATH Keywords: genus \(2\)
zbMATH Keywords: atoroidal \(3\)-manifold
zbMATH Keywords: small Seifert fibered space
zbMATH Keywords: hyperbolic
zbMATH Keywords: \(1\)-bridge braid
author: Abigail Thompson
MaRDI profile type: Publication
Wikidata QID: Q126582987
full work available at URL: s0166-8641(98)00063-7
OpenAlex ID: W2058849533
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description: scientific article; zbMATH DE number 1367549