Contractible manifold

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

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 \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.