A contractible manifold is a topological space satisfying the following equivalent conditions:
- It is both a contractible space and a manifold.
- It is both a weakly contractible space and a manifold.
Any Euclidean space for a nonnegative integer is a contractible manifold. For or , these are the only contractible manifolds up to homeomorphism. For , however, there exist contractible manifolds (in fact, contractible open subset of ), such as the Whitehead manifold, that are contractible but not homeomorphic to Euclidean space.
Relation with other properties
|Property||Meaning||Proof of implication||Proof of strictness (reverse implication failure)||Intermediate notions|
|contractible space|||FULL LIST, MORE INFO|
|acyclic manifold||manifold that is an acyclic space|||FULL LIST, MORE INFO|
|simply connected manifold||manifold that is a simply connected space|||FULL LIST, MORE INFO|
|connected manifold||manifold that is a connected space|||FULL LIST, MORE INFO|
- Any compact connected orientable manifold has top homology equal to , 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 as a quotient group) and is therefore not contractible.