### Alexander's Horned Sphere

http://www.math.ohio-state.edu/~fiedorow/math655/Jordan.html

From: lrudolph@panix.com (Lee Rudolph Subject: Re: Alexander's 'wild sphere'? Date: 2 Nov 1999 15:20:22 -0500 Newsgroups: sci.math.research Keywords: horned sphere and other wild embeddings of S^2 into S^3 Marco de Innocentis <mdi11@hotmail.com> writes: >What is Alexander's "wild sphere"? From what little I've >heard, it seems to be a manifold which changes orientation >under homeomorphism. It's hard to know what you mean by "a manifold which changes orientation under homeomorphism". My best guess is that you are referring to what is more commonly called "Alexander's horned sphere". It is a certain closed subset A of the 3-sphere (3-space R^3 compactified by adding a single point at infinity), such that A is homeomorphic to the 2-sphere but is so "wildly" embedded in the 3-sphere that the two components of the complement of A are not homeomorphic (one is homeomorphic to an open 3-ball, and the other isn't even simply-connected). In particular, there is no orientation-reversing homeomorphism of the 3-sphere whose fixed points are A (whereas, of course, there is such a homeomorphism whose fixed points are the standard 2-sphere). There are other "wild" embeddings of the 2-sphere in the 3-sphere which do admit such a symmetry. Does any of what I'm saying relate to what you were thinking about when you wrote "a manifold which changes orientation under homeomorphism"? Lee Rudolph ============================================================================== From: "David L. Johnson" <david.johnson@lehigh.edu> Subject: Re: Alexander's 'wild sphere'? Date: Wed, 03 Nov 1999 17:34:02 -0500 Newsgroups: sci.math.research Marco de Innocentis wrote: > > In article <7vnh26$isn$1@panix3.panix.com>, > lrudolph@panix.com (Lee Rudolph) wrote: > > > Does any of what I'm saying relate to what you were thinking > > about when you wrote "a manifold which changes orientation > > under homeomorphism"? > > Sorry, my stupid mistake. What I meant was a manifold whose > _orientability_ changes under homeomorphism. Thanks a lot for > the detailed explaination. I still don't thnk that was what you meant. No homeomorphsim can change orientability of a manifold, since that is a topological invariant. The interesting thing about the horned sphere is that the exterior is not simply-connected, even though the manifold is topologically a sphere embedded in space. -- David L. Johnson david.johnson@lehigh.edu Department of Mathematics http://www.lehigh.edu/~dlj0/dlj0.html Lehigh University, 14 E. Packer Avenue, Bethlehem, PA 18015-3174 You will say Christ saith this and the apostles say this; but what canst thou say? -- George Fox. ============================================================================== From: toby@ugcs.caltech.edu (Toby Bartels Subject: Re: Alexander's 'wild sphere'? Date: 3 Nov 1999 04:41:41 GMT Newsgroups: sci.math.research Marco de Innocentis <mdi11@hotmail.com> wrote: >What is Alexander's "wild sphere"? From what little I've >heard, it seems to be a manifold which changes orientation >under homeomorphism. You already have an explanation of Alexander's horned sphere. I thought I'd give you a picture, if you don't have one yet -- and if you do you can check that it's the same as the wild sphere. It's a fractal, so there are pictures of it everywhere; the first one I found is http://math.math.sunysb.edu/~tony/archive/top/pix/horn.gif For a discussion repeating much of what you already got, but which has pictures embedded in it, try http://www.math.ohio-state.edu/~fiedorow/math655/Jordan.html. For fun, see also http://www.treasure-troves.com/math/aimg375.gif and http://www.math.ohio-state.edu/~fiedorow/math655/mating.html -- Toby toby@ugcs.caltech.edu