deformation retract

Let X and Y be topological spacesMathworldPlanetmath such that YX. A deformation retractMathworldPlanetmath of X onto Y is a collectionMathworldPlanetmath of mappings ft:XX, t[0,1] such that

  1. 1.

    f0=idX, the identity mapping on X,

  2. 2.


  3. 3.

    Y is a retractMathworldPlanetmath of X via f1 (that is, f1 restricted to Y is the identity on Y)

  4. 4.

    the mapping X×IX, (x,t)ft(x) is continuousPlanetmathPlanetmath, where the topology on X×I is the product topology.

Of course, by condition 3, condition 2 can be improved: f1(X)=Y.

A deformation retract is called a strong deformation retract if condition 3 above is replaced by a stronger form: Y is a retract of X via ft for every t[0,1].


  • Let X and Y be as in the above definition. Then a collection of mappings ft:XX, t[0,1] is a deformation retract (of X onto Y) if and only if it is a homotopyMathworldPlanetmath ( rel Y idX and some retraction r of X onto Y.


  • If x0n, then ft(x)=(1-t)x+tx0, xn shows that n deformation retracts onto {x0}. Since {x0}n, it follows that deformation retract is not an equivalence relationMathworldPlanetmath.

  • The same map as in the previous example can be used to deformation retract any star-shaped set in n onto a point.

  • we obtain a deformation retraction of n\{0} onto the Sn-1 by setting


    where xn\{0}, n>0,

  • Theöbius strip deformation retracts onto the circle S1.

  • The 2-torus with one point removed deformation retracts onto two copies of S1 joined at one point. (The circles can be chosen to be longitudinal and latitudinal circles of the torus.)

  • The charactersMathworldPlanetmath E,F,H,K,L,M,N, and T all deformation retract onto the charachter I, while the letter Q deformation retracts onto the letter O.

Title deformation retract
Canonical name DeformationRetract
Date of creation 2013-03-22 13:31:44
Last modified on 2013-03-22 13:31:44
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 14
Author mathcam (2727)
Entry type Definition
Classification msc 55Q05
Related topic Retract
Defines strong deformation retract