Order trees and laminations of the plane
type: Item
instance of: scholarly article
title: Order trees and laminations of the plane
zbMATH Open document ID: 0887.57031
author: David Gabai
author: William H. Kazez
published in: Mathematical Research Letters
publication date: 1998-05-10T00:00:00Z
review text: The authors define the concept of a lamination \(\lambda\) of \(\mathbb R^2\). The leaves of such a \(\lambda\) are smoothly embedded 1-manifolds which are never isolated. For an essential lamination, no leaves are circles and there are local charts whose intersection with every leaf is connected. They define also order trees and \(\mathbb R\)-order trees. One type of order tree is an \(\mathbb R\)-tree [\textit{J. W. Morgan} and \textit{P. B. Shalen}, Ann. Math., II. Ser. 120, 401-476 (1984; Zbl 0583.57005)]. For every lamination \(\lambda\) of \(\mathbb R^2\) there is an associated order tree \(T(\lambda)\) whose vertices are the closures of the complementary domains of \(\lambda\) along with the nonboundary leaves of \(\lambda\). For essential laminations, the natural map \(\nu:\mathbb R^2\rightarrow T(\lambda)\) is a quotient map and \(T(\lambda)\) is an \(\mathbb R\)-order tree. It is proved (Theorem 4.2) that every cyclically ordered \(\mathbb R\)-order tree \(T\) is isomorphic to the associated (order) tree of some essential lamination of \(\mathbb R^2\). There is a construction called ``splitting along leaves'' (section 5) which is used to replace non-boundary leaves by closed complementary regions. Whenever \(\lambda'\) is obtained from \(\lambda\) by such a process, then \(T(\lambda)\cong T(\lambda')\). In section 6, it is shown that each isomorphism of cyclically ordered \(\mathbb R\)-order trees is induced by a homeomorphism. Two later sections deal with the relation of essential laminations of \(\mathbb R^2\) to foliations of \(\mathbb R^2\) and essential laminations of \(\mathbb R^3\).
Mathematics Subject Classification ID: 57R30
zbMATH DE Number: 1082985
zbMATH Keywords: essential lamination
zbMATH Keywords: foliation
zbMATH Keywords: order tree
reviewed by: Leonard R. Rubin
MaRDI profile type: Publication
full work available at URL: mrl.1997.v4.n4.a14
OpenAlex ID: W2001704784
DOI: 10.4310/MRL.1997.V4.N4.A14
description: scientific article; zbMATH DE number 1082985