Contractible space

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Illustration of some contractible and non-contractible spaces. Spaces A, B, and C are contractible; spaces D, E, and F are not. Contractibility figure.png
Illustration of some contractible and non-contractible spaces. Spaces A, B, and C are contractible; spaces D, E, and F are not.

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 within that space.

Contents

Properties

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:

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 X is locally contractible at a pointx if for every neighborhood U of x there is a neighborhood V of x contained in U such that the inclusion of V is nulhomotopic in U. A space is locally contractible if it's locally contractible at every point. This definition is occasionally referred to as the "geometric topologist's locally contractible," though is the most common usage of the term. In Hatcher's standard Algebraic Topology text, this definition is referred to as "weakly locally contractible," though that term has other uses.

If every point has a local base of contractible neighborhoods, then we say that X is strongly locally contractible. 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. The circle is (strongly) locally contractible but not contractible.

Strong local contractibility is a strictly stronger property than local contractibility; the counterexamples are sophisticated, the first being given by Borsuk and Mazurkiewicz in their paper Sur les rétractes absolus indécomposables, C.R.. Acad. Sci. Paris 199 (1934), 110-112).

There is some disagreement about which definition is the "standard" definition of local contractibility; the first definition is more commonly used in geometric topology, especially historically, whereas the second definition fits better with the typical usage of the term "local" with respect to topological properties. Care should always be taken regarding the definitions when interpreting results about these properties.

Examples and counterexamples

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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.