Foliations and the topology of 3-manifolds. II
type: Item
instance of: scholarly article
title: Foliations and the topology of 3-manifolds. II
zbMATH Open document ID: 0627.57012
author: David Gabai
published in: Journal of Differential Geometry
publication date: 1987-01-01T00:00:00Z
review text: In this paper the author gives some new applications of the technique introduced in the first part of this paper [ibid. 18, 445-503 (1983; Zbl 0533.57013)]. So the basic result is again a theorem about the existence of nice foliations on a 3-manifold having a given Thurston norm minimizing surface as a leaf. This theorem is applied to a question on knots in 3-manifolds. In particular, three old problems of J. Martin [\textit{R. Kirby}, Algebr. geom. Topol., Stanford/Calif. 1976, Proc. Symp. Pure Math. 32, Pt. 2, 273-312 (1978; Zbl 0394.57002), Problems 1.18 A,B,C] are solved. The type of application is properly illustrated by the following result (the solution of one of Martin's problems; Corollary 2.5): Let k be a knot in \(D^ 2\times S^ 1\) of winding number 0 such that k does not lie in a 3-cell in \(D^ 2\times S^ 1\). If M is obtained by nontrivial surgery on k, then \(M\neq D^ 2\times S^ 1\).
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57R30
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 4020012
zbMATH Keywords: foliations on a 3-manifold
zbMATH Keywords: Thurston norm minimizing surface
zbMATH Keywords: knots in 3- manifolds
zbMATH Keywords: nontrivial surgery
Wikidata QID: Q115184351
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214441487
OpenAlex ID: W4246015810
DOI: 10.4310/JDG/1214441487
description: scientific article; zbMATH DE number 4020012