Alexander horned sphere

From Academic Kids

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Alexander-horned-sphere.jpg
A drawing of Alexander's horned sphere
The Alexander horned sphere is one of the most famous pathological examples in mathematics. A closely related object is Antoine's horned sphere.

The horned sphere was introduced by James Waddell Alexander in 1924 as a counterexample to his previous claim of a three-dimensional Schoenflies theorem. At the same time, he proved the piecewise linear/smooth versions of the Schoenflies theorem in dimension three. This is one of the earliest examples where the need for distinction between the TOP (topological), DIFF (differentiable), and PL (piecewise linear) categories was noticed.

Alexander's horned sphere is a particular embedding of the 2-sphere into the 3-sphere, considered as the one point compactification of ℝ³. (Sometimes the embedding is considered to be into ℝ³, but in this article we do not do so.) There are two complementary domains, with one being a 3-ball (and so is simply-connected) and the other being non-simply connected. Notice that if we wished, we could make both domains non-simply-connected by growing more "horns" into the 3-ball domain.

The closure of the non-simply connected domain is called the Alexander horned ball. Although the horned ball is not a manifold, RH Bing showed that its double is in fact the 3-sphere. One can consider other gluings of the horned ball to a copy of it, arising from different homeomorphisms of the boundary sphere to itself. This was shown by others to also be the 3-sphere. Besides the horned ball, there are many other examples of crumpled cubes arising from other embeddings of 2-spheres into the 3-sphere. In particular, one can generalize Alexander's construction to generate other horned spheres.

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