Geometric branched covers between generalized manifolds
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 branched cover if the integrability exponent for the dilatation exceeds 2. Finally, preserving the Poincaré inequalities by BLD-maps is discussed.
geometrically controlled branched covers
scientific article; zbMATH DE number 1820914
scientific article