# Contractible manifold

From Topospaces

*This article describes a property of topological spaces obtained as a conjunction of the following two properties:* contractible space and manifold

## Contents

## Definition

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.

## Examples

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

### Weaker 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 |

### 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 , 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.