topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

, a branch of mathematics, a

topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

''X'' is said to be simply connected at infinity if for any

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

subset ''C'' of ''X'', there is a compact set ''D'' in ''X'' containing ''C'' so that the induced map on

fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...

s :

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

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 In mathematics, the Whitehead manifold is an open 3-manifold that is contractible, but not homeomorphic to \R^3. discovered this puzzling object while he was trying to prove the Poincaré conjecture, correcting an error in an earlier paper where ...

is an example of a 3-

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

that is

contractible In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within t ...

but not simply connected at infinity. Since this property is invariant under

homeomorphism In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...

, this proves that the Whitehead manifold is not homeomorphic to R³. However, it is a theorem of John R. Stallings that for

n \geq 5

, a contractible ''n''-manifold is homeomorphic to R^''n'' precisely when it is simply connected at infinity.

References

Algebraic topology Properties of topological spaces {{topology-stub