homotopy invariance

Let be a functorMathworldPlanetmath from the category of topological spaces to some categoryMathworldPlanetmath 𝒞. Then is called homotopy invariant if for any two homotopic maps f,g:XY between topological spacesMathworldPlanetmath X and Y the morphismsMathworldPlanetmath f and g in 𝒞 induced by are identical.

Suppose is a homotopy invariant functor, and X and Y are homotopy equivalent topological spaces. Then there are continuous maps f:XY and g:YX such that gfidX and fgidY (i.e. gf and fg are homotopicMathworldPlanetmath to the identity maps on X and Y, respectively). Assume that is a covariant functor. Then the homotopy invariance of implies




From this we see that X and Y are isomorphicPlanetmathPlanetmathPlanetmath in 𝒞. (The same argument clearly holds if is contravariant instead of covariant.)

An important example of a homotopy invariant functor is the fundamental groupMathworldPlanetmathPlanetmath π1; here 𝒞 is the category of groups.

Title homotopy invariance
Canonical name HomotopyInvariance
Date of creation 2013-03-22 14:24:51
Last modified on 2013-03-22 14:24:51
Owner pbruin (1001)
Last modified by pbruin (1001)
Numerical id 4
Author pbruin (1001)
Entry type Definition
Classification msc 55Pxx
Related topic HomotopyEquivalence
Defines homotopy invariant