From: Linus Kramer Subject: Re: compactification of open contractible manifolds Date: Tue, 20 Feb 2001 13:10:42 +0100 Newsgroups: sci.math.research Summary: Poincare conjecture and one-point compactifications of open, contractible manifolds Estelle Souche wrote: > > I'm working on a property which holds for open contractible manifolds > whose one-point compactifications still are manifolds. > > There is the obvious case when the manifold is the Euclidian space $R^n$ > and its one-point compactification is the sphere $S^n$, and there are > results by Stallings and Freedman showing that an open contractible > manifold of dimension $n$ ($n$ equal or greater than four) is homeomorphic > to $R^n$ if and only if it is simply connected at infinity. But I was > wondering if there were examples different from $R^n$. Any reference > (books, articles) would be welcome. > > In general, are there interesting results about the one-point > compactification of an open (not necessarily) contractible > manifold? > > Estelle Souche > Universite de Provence, Marseille, France > esouche@protis.univ-mrs.fr Let M denote the contractible manifold, and X its 1-point compactification. If you assume that X is an n-manifold, then the local homology of the pair (X,M) together with the exact homology sequence of the pair (X,M) show that X has the same integral homology as a sphere S^n. Using Van Kampen's Theorem, one has that X is 1-connected (let's assume that n is at least 3, the low dimensional cases are easy). So X is a homotopy n-sphere, and, by the proof of the generalized Poincare conjecture, a sphere, except possibly if n=3. L. Kramer -- Linus Kramer Mathematisches Institut Universitaet Wuerzburg Am Hubland 97074 Wuerzburg Germany E-mail: kramer@mathematik.uni-wuerzburg.de http://www.mathematik.uni-wuerzburg.de/~kramer