From: wgilbert@uwaterloo.ca (Will Gilbert)
Newsgroups: sci.math.research
Subject: Contractible non-compact manifolds
Date: Fri, 6 Jan 1995 15:32:03 -0500
Is the following true and does anybody have a reference to a proof?
"Is every contractible open subset of Euclidean n-space homeomorphic
to Euclidean n-space?"
In dimension 2, this follows from the Riemann Mapping Theorem.
I am prepared to add extra conditions to guarantee the result,
since it is a by-product of a question from engineering.
--
********************************************************
Will Gilbert, Pure Math Dept, Univ of Waterloo, Canada
wgilbert@uwaterloo.ca
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From: geoff@math.ucla.edu (Geoffrey Mess)
Newsgroups: sci.math.research
Subject: Re: Contractible non-compact manifolds
Date: 7 Jan 1995 01:12:02 GMT
In article wgilbert@uwaterloo.ca (Will
Gilbert) writes:
>
> Is the following true and does anybody have a reference to a proof?
>
> "Is every contractible open subset of Euclidean n-space homeomorphic
> to Euclidean n-space?"
Hi!
No. Whitehead discovered this while attempting to prove the Poincare conjecture. A contractible open subset of R^n need not be "simply connected at
infinity". ( " X is simply connected at infinity" means that for each
compact K there is a larger compact L such that the induced map on
pi_1 from X - L to X - K is trivial.)
A contractible open subset of R^n which _is_ simply connected
at infinity is homeomorphic to R^n a) if n > 4: by
J. Stallings, The piecewise linear structure of Euclidean space,
Proc Camb Phil Soc 58(1962) (481-88)
b) n = 4: by M. Freedman - see Topology of 4-Manifolds by Freedman and Quinn.
c) For n = 3 this is a standard exercise - I don't know who gets the credit,
but you oould refer to AMS memoir 411 by Brin and Thickstun. The ingredients
are i) the Loop theorem and ii) Alexander's theorem that a PL sphere in R^3 bounds a 3-ball - you could even get around that by using the generalized Schoenfliess theorem of Morton Brown.
--
Geoffrey Mess
Department of Mathematics, UCLA. geoff@math.ucla.edu
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From: gk00@midway.uchicago.edu (Greg Kuperberg)
Newsgroups: sci.math.research
Subject: Re: Contractible non-compact manifolds
Date: Sun, 8 Jan 1995 03:49:00 GMT
In article wgilbert@uwaterloo.ca (Will Gilbert) writes:
>Is the following true and does anybody have a reference to a proof?
>
>"Is every contractible open subset of Euclidean n-space homeomorphic
>to Euclidean n-space?"
It's false in 3 dimensions. The standard counterexample is the Whitehead
manifold. Briefly, let V be an unknotted solid torus in S^3, let
h:V -> V be an embedding of V in V such that h(V) is knotted in V
but unknotted in S^3, and let X = intersection of all h^i(V).
Then S^3 - X is contractible, and moreover (S^3 - X) x R is homeomorphic
to R^4, but S^3 - X is not homeomorphic to R^3.
See page 82 of Rolfsen, "Knots and Links".
==============================================================================
From: gk00@midway.uchicago.edu (Greg Kuperberg)
Newsgroups: sci.math.research
Subject: Re: Contractible non-compact manifolds
Date: Mon, 9 Jan 1995 00:28:35 GMT
In article <1995Jan8.034900.14944@midway.uchicago.edu> gk00@midway.uchicago.edu writes:
>It's false in 3 dimensions. The standard counterexample is the Whitehead
>manifold. Briefly, let V be an unknotted solid torus in S^3, let
>h:V -> V be an embedding of V in V such that h(V) is knotted in V
>but unknotted in S^3, and let X = intersection of all h^i(V).
>Then S^3 - X is contractible, and moreover (S^3 - X) x R is homeomorphic
>to R^4, but S^3 - X is not homeomorphic to R^3.
No, actually, this isn't quite right. You need to know that the
meridian of V is null-homotopic in S^3 - h(V), e.g. h(V) and
S^3 - V together make the Whitehead link. Sorry for the mistake
there.
==============================================================================
From: ruberman@maths.ox.ac.uk (Prof Daniel Ruberman)
Newsgroups: sci.math.research
Subject: Re: Contractible non-compact manifolds
Date: Mon, 9 Jan 95 09:33:58 GMT
In article ,
Will Gilbert wrote:
>
>Is the following true and does anybody have a reference to a proof?
>
>"Is every contractible open subset of Euclidean n-space homeomorphic
>to Euclidean n-space?"
>
This is false in all dimensions n greater than or equal to 4: Take a "Mazur
Manifold" which is a contractible compact manifold with boundary. (See
Rolfsen's book on knot theory--the chapter called `A high-dimensional
sampler' for a construction.) If you glue together two copies along
the boundary, you get a sphere; if you remove a point, you get the
interior of the Mazur manifold embedded in R^n. In dimension 3, the
complement of the Whitehead continuum in S^3 is contractible. Both
these sorts of examples the contractible manifolds are distinguished
from R^n because they are not simply-connected at infinity.
>In dimension 2, this follows from the Riemann Mapping Theorem.
>I am prepared to add extra conditions to guarantee the result,
>since it is a by-product of a question from engineering.
What extra conditions? If you assume that your open subset is the
interior of a compact manifold, with boundary a sphere, and the sphere
is nicely embedded, then the Schoenfliess theorem (in dimensions other
than 4) implies that the compact manifold is a disc, and its interior
is R^n.