A classical problem in geometric topology is to recognize when a topological space is a topological manifold. This paper addresses the question of when a rnetric space admits a quasisymmetric parameterization by providing counterexamples to the obvious optimistic conjectures, or, in other words, by providing examples of spaces with many Euclidean-like properties which are nonetheless substantially different from Euclidean geometry. These examples are geometrically self-similar versions of classical topologically self-similar examples from geometric topology, and they can be realized as codimension 1 subsets of Euclidean spaces. Unlike earlier examples going back to Rickman, these sets enjoy good bounds on their geodesic distance functions and good mass bounds (AhIfors regularity). They are also smooth except for reasonably tame degenerations near small sets, they are uniform1y rectifiable, and they have good properties in terms of analysis (like Sobolev and Poincaré inequalities). The construction also produces uniform domains which have many nice properties but which are not quasiconformally equivalent to balls.
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Stephen Semmes, Good metric spaces without good parameterizations. Rev. Mat. Iberoam. 12 (1996), no. 1, pp. 187–275DOI 10.4171/RMI/198