Simply connected at infinity

Short description: Mathematical property

In topology, a branch of mathematics, a topological space X is said to be simply connected at infinity if for any compact subset C of X, there is a compact set D in X containing C so that the induced map

[math]\displaystyle{ \pi_1(X-D) \to \pi_1(X-C) }[/math]

is the zero map. 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 that for [math]\displaystyle{ n \geq 5 }[/math], a contractible n-manifold is homeomorphic to Rⁿ precisely when it is simply connected at infinity.