Lipschitz and quasiconformal approximation of homeomorphism pairs
type: Item
instance of: scholarly article
title: Lipschitz and quasiconformal approximation of homeomorphism pairs
zbMATH Open document ID: 0964.57023
DOI: 10.1016/S0166-8641(99)00145-5
author: Jouni Luukkainen
published in: Topology and its Applications
publication date: 2001-07-23T00:00:00Z
Mathematics Subject Classification ID: 57N35
Mathematics Subject Classification ID: 30C65
Mathematics Subject Classification ID: 57R12
zbMATH DE Number: 1566038
zbMATH Keywords: quasisymmetric
zbMATH Keywords: locally flat
zbMATH Keywords: diffeomorphism
reviewed by: Jouni Luukkainen
MaRDI profile type: Publication
cites work: Locally tame sets are tame
cites work: Q4403472
cites work: The Hauptvermutung for 3-Complexes
cites work: LOCALLY NICE EMBEDDINGS IN CODIMENSION THREE
cites work: Counting topological manifolds
cites work: Homeomorphisms of 𝑅ⁿ are 𝑘-stable for 𝑘≤𝑛-3
cites work: Q4766258
cites work: Small ambient isotopies of a 3-manifold which transform one embedding of a polyhedron into another
cites work: Quasiconformal 4-manifolds
cites work: The equivalence of close piecewise linear embeddings
cites work: Topological Orbifolds
cites work: Lipschitz and quasiconformal flattening of spheres and cells
cites work: Boundary correspondence under quasiconformal mappings
cites work: Q3854329
cites work: On the triangulation of manifolds and the Hauptvermutung
cites work: Q4096151
cites work: Foundational Essays on Topological Manifolds, Smoothings, and Triangulations. (AM-88)
cites work: Q3744437
cites work: Q5685358
cites work: Topologically, quasiconformally or Lipschitz locally flat embeddings in codimension one
cites work: Respectful deformation of bi-Lipschitz and quasisymmetric embeddings
cites work: Q3800340
cites work: Bi-Lipschitz concordance implies bi-Lipschitz isotopy
cites work: Q4272418
cites work: Respectful quasiconformal extension from dimension \(n-1\) to \(n\)
cites work: Topologically, quasiconformally or Lipschitz locally flat approximation of embeddings
cites work: Quasisymmetric and Lipschitz approximation of embeddings
cites work: Elements of Lipschitz topology
cites work: Close Isotopies on Piecewise-Linear Manifolds
cites work: Approximating codimension 3 embeddings
cites work: Obstructions to the smoothing of piecewise-differentiable homeomorphisms
cites work: Q3293867
cites work: Higher \(G\)-signatures for Lipschitz manifolds
cites work: Group actions and equivariant Lipschitz analysis
cites work: Q4048264
cites work: A ''piecewise-linear'' method for triangulating 3-manifolds
cites work: Q3935870
cites work: Q3935869
cites work: Lipschitz approximation of homeomorphisms
cites work: Quasisymmetric embeddings of metric spaces
cites work: Lipschitz and quasiconformal approximation and extension
cites work: Quasiconformal extension from dimension n to n+1
cites work: Bilipschitz extensions of maps having quasiconformal extensions
cites work: Lectures on \(n\)-dimensional quasiconformal mappings
cites work: Piecewise linear approximation of lipeomorphisms
cites work: Quasi-Symmetric Embeddings in Euclidean Spaces
cites work: Quasiconformal concordance
Wikidata QID: Q126470553
Recommended article: Q4548121
Recommended article: Q5485272
Recommended article: Quasiconformal and Bi-Lipschitz Homeomorphisms, Uniform Domains and the Quasihyperbolic Metric
Recommended article: Quotients of \(c_0\) and Lipschitz-homeomorphisms
Recommended article: Lipschitz classes and quasiconformla mappings
Recommended article: Homeomorphisms Lipschitzian in the mean
Recommended article: Diffeomorphic approximation of quasiconformal and quasisymmetric homeomorphisms
Recommended article: On Lipschitz continuity of quasiconformal mappings in space
Recommended article: Q3113587
Recommended article: Quasiconformal homeomorphisms and the convex hull boundary
review text: This paper deals with manifolds in the categories LIP of locally bi-Lipschitz homeomorphisms and LQC of locally quasiconformal (or equivalently locally quasisymmetric) homeomorphisms. The following theorem consists of representative special cases of the main results of the paper. Let CAT be LIP or LQC, let \(0\leq q\leq n\) with \(n,q\neq 4\), let \(M\) be a topological \(n\)-manifold without boundary, and let \(Q\) be a closed (as a subset) locally flat \(q\)-submanifold of \(M\) without boundary. Then the pair \((M,Q)\) can be given an essentially unique locally CAT flat CAT manifold pair structure as follows.NEWLINENEWLINENEWLINE(1) There is a CAT structure on \(M\) for which \(Q\) is a locally CAT flat submanifold of \(M\).NEWLINENEWLINENEWLINE(2) If \({\mathcal A}\) and \({\mathcal B}\) are two CAT structures on \(M\) satisfying (1), then there is an arbitrarily small isotopy of \((M,Q)\) from the identity to a CAT homeomorphism \(f:(M_{\mathcal A},Q)\to(M_{\mathcal B},Q)\). NEWLINENEWLINENEWLINE(3) If \({\mathcal A}\) and \({\mathcal B}\) are as in (2), then every two CAT homeomorphisms \(f,g:(M_{\mathcal A},Q) \to(M_{\mathcal B},Q)\) sufficiently close to the identity can be joined by an arbitrarily small CAT isotopy of \((M_{\mathcal A},Q)\) onto \((M_{\mathcal B},Q)\). NEWLINENEWLINENEWLINEFor the manifold \(M\) itself, with \(Q=\emptyset\) (or equivalently with \(q=n)\), this theorem is due to \textit{D. Sullivan} [Geometric topology, Proc. Conf., Athens/Ga 1977, 543-555 (1979; Zbl 0478.57007)]. The main results allow manifolds with boundary, and they include the generalization, due to \textit{P. Tukia} and \textit{J. Väisälä} [Ann. Acad. Sci. Fenn., Ser A I Math. 6, 303-342 (1981; Zbl 0448.30021)], of Sullivan's results (1) and (2) to this case. In the case \(q=n-1\) the author gave a proof for (1) and (2) in [ibid. 8, 107-138 (1983; Zbl 0488.57005)].
description: scientific article; zbMATH DE number 1566038