deformation retract is transitive


Let ZYX be nested topological spacesMathworldPlanetmath. If there exist a deformation retraction ( of X onto Y and a deformation retraction of Y onto Z, then there also exists a deformation retraction of X onto Z. In other words, “being a deformation retractMathworldPlanetmath of” is a transitive relation.


Since Y is a deformation retract of X, there is a homotopyMathworldPlanetmath F:I×XX between idX and a retractMathworldPlanetmath r:XY of X onto Y. Similarly, there is a homotopy G:I×YY between idY and a retract s:YZ of Y onto Z.

First notice that since both r and s fix Z, the map sr:XZ is a retraction.

Now define a map G~:I×XX by G~=iG(idI×r), where i:YX is inclusion. Observe that

  • G~(0,x)=r(x) for any xX;

  • G~(1,x)=sr(x) for any xX; and

  • G~(t,a)=a for any aZ.

Hence G~ is a homotopy between the retractions r and sr.

Finally we must glue together the homotopies ( F and G~ to get a homotopy between idX and sr. To do this, define a function H:I×XX by


Since F(1,x)=G~(0,x)=r(x), the gluing yieds a continuous mapMathworldPlanetmath. By construction,

  • H(0,x)=x for all xX;

  • H(1,x)=sr(x) for all xX; and

  • H(t,a)=a for any aZ.

Hence H is a homotopy between the identity map on X and a retraction of X onto Z. We conclude that H is a deformation retraction of X onto Z. ∎

Title deformation retract is transitive
Canonical name DeformationRetractIsTransitive
Date of creation 2013-03-22 15:43:59
Last modified on 2013-03-22 15:43:59
Owner mps (409)
Last modified by mps (409)
Numerical id 4
Author mps (409)
Entry type Result
Classification msc 55Q05