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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 | |||
: <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}} | |||
Revision as of 16:32, 13 January 2014
In topology, a branch of mathematics, a topological space X is said to be simply connected at infinity if for all compact subsets C of X, there is a compact set D in X containing C so that the induced map
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 R3.
However, it is a theorem of John R. Stallings [1] that for , a contractible n-manifold is homeomorphic to Rn precisely when it is simply connected at infinity.
References
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