Geometric branched covers between generalized manifolds
type: Item
instance of: scholarly article
title: Geometric branched covers between generalized manifolds
zbMATH Open document ID: 1017.30023
DOI: 10.1215/S0012-7094-02-11333-7
author: Juha Heinonen
author: Seppo Rickman
published in: Duke Mathematical Journal
publication date: 2002-10-27T00:00:00Z
review text: The authors develop a basic theory of geometrically controlled branched covers between generalized metric manifolds. A generalized \(n\)-manifold is a finite-dimensional locally compact Hausdorff space whose local cohomology groups over the integers in dimensions \(n-1\) and higher are similar to those of an \(n\)-manifold. The authors study the relationship between regular maps and maps of bounded length distortion, or BLD-maps, on the generalized manifolds. It turns out that under quite general circumstances, the sense-preserving regular maps are precisely the BLD-maps of finite maximal multiplicity. Examples that illustrate the relationship between various classes of maps are given. A particular kind of generalized metric manifolds, called spaces of ``type A'', is introduced. It is shown that BLD-maps between such spaces behave in many respects like BLD-maps between Riemannian manifolds. In particular, for maps from spaces of type A into \(\mathbb R^n\), an analytic description akin to the one used by Martio and Väisälä in \(\mathbb R^n\) is available. Some further analytic and metric properties of BLD-maps between spaces of type A, including a result on value distribution are pointed out. A proof of the generalized Berstein-Edmonds theorem together with some examples and applications are given. In particular, the authors carefully study the geometric decomposition spaces of Semmes, and show how they admit BLD-branched coordinates in \(\mathbb R^3\). It is also pointed out how one obtains bounded quasiregular maps in the open 3-ball in \(\mathbb R^3\) which fail to have radial limits in a large set on the boundary. Moreover, Lipschitz maps \(S^3\to S^3\) with nonzero degree having as fibers some strange continua, for example, the Whitehead continuum, but with dilatation function in a local Lebesgue space \(L^p\) for \(p\) arbitrarily close to 2 are constructed. These examples are quite interesting in view of the fact that each Lipschitz map \(S^3\to S^3\) is a br…
Mathematics Subject Classification ID: 30C65
Mathematics Subject Classification ID: 57P99
Mathematics Subject Classification ID: 57M12
zbMATH DE Number: 1820914
zbMATH Keywords: regular maps
zbMATH Keywords: maps of bounded length distortion
zbMATH Keywords: sense-preserving regular maps
zbMATH Keywords: generalized manifolds
zbMATH Keywords: orientation
zbMATH Keywords: local degree
zbMATH Keywords: sense-preserving branched covers
zbMATH Keywords: geometrically controlled branched covers
zbMATH Keywords: change-of-variables formula
zbMATH Keywords: approximate differentiability
zbMATH Keywords: separating complex sheets
zbMATH Keywords: geometric decomposition spaces
zbMATH Keywords: metric orientation
zbMATH Keywords: radial limits
zbMATH Keywords: Lipschitz map with strange fibers
zbMATH Keywords: Poincaré inequalities
MaRDI profile type: Publication
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