Alexander horned sphere

The Alexander horned sphere is one of the most famous pathological examples in mathematics.

The horned sphere was introduced by James Waddell Alexander in 1924 as a counterexample to his previous claim of a three-dimensional Jordan-Schönflies 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 the 3-dimensional Euclidean space 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.

Other different constructions exist for constructing such "wild" spheres. Another famous example, also due to Alexander, is Antoine's horned sphere, which is based on Antoine's necklace, a pathological embedding of the Cantor set into the 3-sphere.
An animated fractal visualization is available here.

See also

*Fox-Artin arc

External links

* J. W. Alexander. An Example of a Simply Connected Surface Bounding a Region which is not Simply Connected. Proceedings of the National Academy of Sciences 1924; 10(1): 8-10.
*Eric W. Weisstein. Alexander's Horned Sphere. From MathWorld - A Wolfram Web Resource. [1] - Gives a figure
*Zbigniew Fiedorowicz. Math 655 - Introduction to Topology. [2] - Lecture notes