Incompressibility of surfaces in surgered 3-manifolds
type: Item
instance of: scholarly article
title: Incompressibility of surfaces in surgered 3-manifolds
zbMATH Open document ID: 0872.57022
DOI: 10.1016/0040-9383(92)90020-I
author: Ying-Qing Wu
published in: Topology
publication date: 1993-01-16T00:00:00Z
review text: The problem we consider in this paper was raised in [\textit{M. Culler}, \textit{C. McA. Gordon}, \textit{J. Luecke} and \textit{P. B. Shalen}, Ann. Math., II. Ser. 125, 237-300 (1987; Zbl 0633.57006)]. Suppose \(T\) is a torus on the boundary of an orientable 3-manifold \(X\), and \(S\) is a surface on \(\partial X-T\) which is incompressible in \(X\). A slope \(\gamma\) is the isotopy class of a nontrivial simple closed curve on \(T\). Denote by \(X(\gamma)\) the manifold obtained by attaching a solid torus to \(X\) so that \(\gamma\) is the slope of the boundary of a meridian disc. Given two slopes \(\gamma_1\) and \(\gamma_2\), we denote their (minimal) geometric intersection number by \(\Delta (\gamma_1, \gamma_2)\). Theorem 1. Suppose \(X\) contains no incompressible annulus with one boundary component in \(S\) and the other in \(T\). If \(S\) compresses in \(X(\gamma_1)\) and \(X(\gamma_2)\), then \(\Delta (\gamma_1, \gamma_2) \leq 1\). Hence, there are at most three slopes \(\gamma\) such that \(S\) is compressible in \(X (\gamma)\). This proves Conjecture 2.4.1 of [Culler-Gordon-Luecke-Shalen, loc. cit.]. Besides its own interest, the theorem will also (at least conceptually) simplify part of the proof of the Cyclic Surgery Theorem. In that paper the authors proved some results on the above theorem: If the conditions are satisfied, then \(\Delta (\gamma_1, \gamma_2) \leq 2\). (They also proved that the theorem is true when \(S\) is a torus). Our task is to rule out the possibility of \(\Delta (\gamma_1, \gamma_2) =2\). Suppose \(S\) is compressible in \(X (\gamma_1)\), then the central curve \(K\) of the attached solid torus will be a knot in \(X (\gamma_1)\). Thus Theorem 1 follows immediately from the (equivalent) theorem below, which is more suitable to our method of proof: Theorem 2. Let \(S\) be a surface on the boundary of a 3-manifold \(M\). Let \(K\) be a knot in \(M\) which is not isotopic to a simple closed curve on \(S\). If \(S\) is compressible…
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 89833
zbMATH Keywords: cyclic surgery theorem
zbMATH Keywords: 3-manifold
zbMATH Keywords: surface
zbMATH Keywords: incompressible
zbMATH Keywords: intersection number
zbMATH Keywords: knots
MaRDI profile type: Publication
full work available at URL: 0040-9383(92)90020-i
OpenAlex ID: W2036976496
description: scientific article; zbMATH DE number 89833