Contractible space

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.^[1][2] Intuitively, a contractible space is one that can be continuously shrunk to a point.

A contractible space is precisely one with the homotopy type of a point. It follows that all the homotopy groups of a contractible space are trivial. Therefore any space with a nontrivial homotopy group cannot be contractible. Similarly, since singular homology is a homotopy invariant, the reduced homology groups of a contractible space are all trivial.

For a topological space X the following are all equivalent (here Y is an arbitrary topological space):

• X is contractible (i.e. the identity map is null-homotopic).
• X is homotopy equivalent to a one-point space.
• X deformation retracts onto a point. (However, there exist contractible spaces which do not strongly deformation retract to a point.)
• Any two maps f,g : Y → X are homotopic.
• Any map f : Y → X is null-homotopic.

The cone on a space X is always contractible. Therefore any space can be embedded in a contractible one (which also illustrates that subspaces of contractible spaces need not be contractible).

Furthermore, X is contractible if and only if there exists a retraction from the cone of X to X.

Every contractible space is path connected and simply connected. Moreover, since all the higher homotopy groups vanish, every contractible space is n-connected for all n ≥ 0.

Locally contractible spaces

A topological space is locally contractible if every point has a local base of contractible neighborhoods. Contractible spaces are not necessarily locally contractible nor vice-versa. For example, the comb space is contractible but not locally contractible (if it were, it would be locally connected which it is not). Locally contractible spaces are locally n-connected for all n ≥ 0. In particular, they are locally simply connected, locally path connected, and locally connected.

Examples and counterexamples

• Any star domain of a Euclidean space is contractible.
• The Whitehead manifold is contractible.
• Spheres of any finite dimension are not contractible.
• The unit sphere in an infinite-dimensional Hilbert space is contractible.
• The house with two rooms is standard example of a space which is contractible, but not intuitively so.
• Dunce hat
• The cone on a Hawaiian earring is contractible (since it is a cone), but not locally contractible or even locally simply connected.

• All manifolds and CW complexes are locally contractible, but in general not contractible.

References

1. ^ Munkres, James R. (2000). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2. 
2. ^ Hatcher, Allen (2002). Algebraic Topology. Cambridge University Press. ISBN 0-521-79540-0. http://www.math.cornell.edu/~hatcher/AT/ATpage.html.