Steiner inellipse: Difference between revisions

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In [[topology]], a branch of mathematics, a [[topological space]] ''X'' is said to be '''simply connected at infinity''' if for all [[compact space|compact]] subsets ''C'' of ''X'', there is a compact set ''D'' in ''X'' containing ''C'' so that the induced map
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: <math> \pi_1(X-D) \to \pi_1(X-C)\,</math>
 
is trivial.  Intuitively, this is the property that loops far away from a small subspace of ''X'' can be collapsed, no matter how bad the small subspace is.
 
The [[Whitehead manifold]] is an example of a 3-[[manifold]] that is [[contractible]] but not simply connected at infinity. Since this property is invariant under [[homeomorphism]], this proves that the Whitehead manifold is not homeomorphic to '''R'''<sup>3</sup>.  
 
However, it is a theorem of [[John R. Stallings]] <ref>http://www.math.rutgers.edu/~sferry/ps/geotop.pdf Ch. 10</ref> that for <math>n \geq 5</math>, a contractible ''n''-manifold is homeomorphic to '''R'''<sup>''n''</sup> precisely when it is simply connected at infinity.
 
==References==
{{Reflist}}
 
 
[[Category:Algebraic topology]]
[[Category:Properties of topological spaces]]
 
{{topology-stub}}

Latest revision as of 16:49, 11 November 2014

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