Simply connected at infinity

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

$\pi_1(X-D) \to \pi_1(X-C)\,$

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 $n \geq 5$, a contractible n-manifold is homeomorphic to Rⁿ precisely when it is simply connected at infinity.

References

1. "Theory : Chapter 10" (PDF). Math.rutgers.edu. Retrieved 2015-03-08.