Essential laminations in 3-manifolds
type: Item
instance of: scholarly article
title: Essential laminations in 3-manifolds
zbMATH Open document ID: 0685.57007
DOI: 10.2307/1971476
author: David Gabai
author: Ulrich Oertel
published in: Annals of Mathematics. Second Series
publication date: 1989-01-01T00:00:00Z
review text: Haken 3-manifolds are those which contain a two-sided 2-manifold whose fundamental group is a subgroup of the fundamental group of the 3- manifold (i.e. the homomorphism of fundamental groups induced by inclusion is injective). The successful study of this class of 3- manifolds was initiated in the 60's by Waldhausen, and through work of many people their structure is well-understood. One of Waldhausen's early results was the fact that the universal cover of a Haken 3-manifold must be homeomorphic to a 3-cell from which some of the boundary has been removed. In particular, if the Haken 3-manifold is closed, then the universal cover is 3-dimensional Euclidean space, rather than one of the many other simply-connected open 3-manifolds which have a more complicated end structure. Non-Haken 3-manifolds (other than those whose universal cover is the 3- sphere), are still rather poorly understood. By the mid 70's, there were only a few examples known of non-Haken 3-manifolds, and it was felt that they were probably quite rare. This feeling turned out to be totally incorrect, because Thurston showed how to construct huge quantities of non-Haken 3-manifolds. Thurston's method shows that ``most'' ways of attaching solid tori to the torus boundaries of hyperbolic knot and link complements result in non-Haken 3-manifolds. Currently there is much effort to try to understand non-Haken 3-manifolds by using laminations to play the role that 2-dimensional submanifolds do in the Haken theory. A lamination is a foliation of a closed subset of a manifold, whose leaves have codimension 1 in the manifold. The systematic use of laminations in low-dimensional topology was initiated by Thurston, who introduced the device of ``train tracks'', or branched 1-dimensional submanifolds of 2-manifolds. By assigning numbers, called weights, to the tracks in a train track, one can construct the laminations ``carried'' by the train track. In this way, a train track parametrized a portion of the space…
reviewed by: Darryl McCullough
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57R30
zbMATH DE Number: 4122848
zbMATH Keywords: non-Haken 3-manifolds
zbMATH Keywords: Laminations in 3-manifolds
zbMATH Keywords: branched 2-manifolds
zbMATH Keywords: essential lamination
zbMATH Keywords: end-incompressible
zbMATH Keywords: essential branched 2-manifold
zbMATH Keywords: universal cover
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/1971476
OpenAlex ID: W2046756141
description: scientific article; zbMATH DE number 4122848