Contractible manifold

Definition
A contractible manifold is a topological space satisfying the following equivalent conditions:


 * 1) It is both a contractible space and a manifold.
 * 2) It is both a weakly contractible space and a manifold.

Examples
Any Euclidean space $$\R^n$$ for $$n$$ a nonnegative integer is a contractible manifold. For $$n = 1$$ or $$n = 2$$, these are the only contractible manifolds up to homeomorphism. For $$n = 3$$, however, there exist contractible manifolds (in fact, contractible open subset of $$\R^3$$), such as the Whitehead manifold, that are contractible but not homeomorphic to Euclidean space.

Opposite properties
Compact manifold is opposite in spirit: the only compact contractible manifold is the one-point space. To see this, note that:


 * Any compact connected orientable manifold has top homology equal to $$\mathbb{Z}$$, making it non-contractible if its dimension is positive.
 * If the manifold is not connected, it anyway cannot be contractible.
 * If the manifold is not orientable, it has nontrivial fundamental group (in particular, the fundamental group admits $$\mathbb{Z}/2\mathbb{Z}$$ as a quotient group) and is therefore not contractible.