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 ============================================================================== 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 ============================================================================== 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.