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Good metric spaces without good parameterizations. (English) Zbl 0854.57018

This excellently written paper deals with the question of when a metric space \(M\) (in fact, a subspace of a Euclidean space), that is topologically \(\mathbb{R}^n\), admits a parametrization \(f : \mathbb{R}^n \to M\) with good metric properties, more specifically, a parametrization \(f\) which is quasisymmetric. (A mapping \(f : (X,d) \to (Y,\rho)\) between metric spaces is called quasisymmetric, provided there exists a homeomorphism \(\eta: [0,\infty) \to [0,\infty)\) such that \(d(x,x') \leq td(x,x'')\) implies \(\rho(f(x), f(x')) \leq \eta(t) \rho(f(x), f(x''))\) for all \(t > 0\), \(x, x', x'' \in X).\)
The author provides counterexamples to certain optimistic conjectures, by providing examples of topological copies of \(\mathbb{R}^3\) in \(\mathbb{R}^4\) admitting no quasisymmetric parametrization, but possessing many other nice Euclidean-like properties. (These sets enjoy good bounds on their geodesic functions and good mass bounds (Ahlfors regularity). They are smooth except for reasonably tame degenerations near small sets, they are uniformly rectifiable, and they have good properties in terms of analysis). The constructions of these examples repeat (with some subtle extra analysis) the well-known examples of the Whitehead continuum, Bing’s dogbone space, and Bing doubling.
Reviewer: T.Banakh (Lviv)

MSC:

57N12 Topology of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010)
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
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