Subject: Re: 3 postings
Date: Mon, 28 Jan 2002 15:55:15 -0500 (EST)
From: Walter Neumann <neumann@math.columbia.edu>
> _______________________________
>
> Subject: Can anyone help me?
> Date: Fri, 25 Jan 2002 14:29:00 -0800
> From: "Zhang, Bin" <bin_zhang2@hp.com>
>
> Dear Experts,
>
> I have a surface S  in a n-dimensional Euclidian space R^n.  I also know
>
> that
> the surface is homomorphic to a hyperplane.  Can I always extend the
> topological homomorphism defined on S to the whole space R^n so that the
>
> extended mapping is a homomorphism from R^n to itself?
>
> Thanks!
>
> Bin

The Alexander Horned Sphere (which Google will cheerfully find) is an
embedded S^2 in S^3 whose complement is non-simply-connected. By removing
a point you get a surface homeomorhic to a hyperplane in R^3 with
non-simply-connected complement, so there is no homeomorphism of R^3
taking it to a hyperplane.

--walter
________________________________________

Subject: Re: 3 postings
Date: Mon, 28 Jan 2002 14:20:50 -0500
From: Clarence Wilkerson <wilker@math.purdue.edu>

Hypersurface: if I interpret this correctly ( homomorphic =
homeomorphic ), I believe the answer is no. I think there
is a need for some local flatness (bi-collared) such as in Mort Brown's
paper from the 60's

] 22 #8470b Brown, Morton A proof of the generalized Schoenflies theorem.  Bull. Amer. Math. Soc.  66  1960 74--76. (Reviewer: S. Eilenberg) 54.00 (57.00)

Clarence Wilkerson