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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 &lt;7vnh26$isn$1@panix3.panix.com&gt;,
>   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 &lt;mdi11@hotmail.com&gt; 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

The Mating Dance of the Alexander Horned Spheres

by Tim Poston

mating.gif

alexanders_horned_sphere.txt · Last modified: 2007/07/11 18:08 by nik