Contractible space

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 : YX are homotopic.
  • Any map f : YX 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

References


Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • Classifying space — In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG (i.e. a topological space for which all its homotopy groups are trivial) by a free action of G. It… …   Wikipedia

  • Classifying space for U(n) — In mathematics, the classifying space for the unitary group U(n) is a space B(U(n)) together with a universal bundle E(U(n)) such that any hermitian bundle on a paracompact space X is the pull back of E by a map X → B unique up to homotopy. This… …   Wikipedia

  • Contractibility of unit sphere in Hilbert space — In topology, it is a surprising fact that the unit sphere in (infinite dimensional) Hilbert space is a contractible space, sinceno finite dimensional spheres are contractible.This can be demonstrated in several different ways. Topological proof… …   Wikipedia

  • Connected space — For other uses, see Connection (disambiguation). Connected and disconnected subspaces of R² The green space A at top is simply connected whereas the blue space B below is not connected …   Wikipedia

  • Acyclic space — In mathematics, an acyclic space is a topological space X in which cycles are always boundaries, in the sense of homology theory. This implies that the integral homology groups in all dimensions of X are isomorphic to the corresponding homology… …   Wikipedia

  • Quotient space — In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or gluing together certain points of a given space. The points to be identified are specified …   Wikipedia

  • classifying space — noun A topological space that is the quotient of a free action (of the specified group) on a weakly contractible space …   Wiktionary

  • CAT(k) space — In mathematics, a CAT( k ) space is a specific type of metric space. Intuitively, triangles in a CAT( k ) space are slimmer than corresponding model triangles in a standard space of constant curvature k . In a CAT( k ) space, the curvature is… …   Wikipedia

  • Weakly contractible — In mathematics, a topological space is said to be weakly contractible if all of its homotopy groups are trivial.PropertyIt follows from Whitehead s Theorem that if a CW complex is weakly contractible then it is contractible.ExampleDefine S^infty… …   Wikipedia

  • Locally simply connected space — In mathematics, a locally simply connected space is a topological space that admits a basis of simply connected sets. Every locally simply connected space is also locally path connected and locally connected.The circle is an example of a locally… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”