homotopy with a contractible domain

Theorem. Assume that Y is an arbitrary topological spaceMathworldPlanetmath and X is a contractibleMathworldPlanetmath topological space. Then all maps f:XY are homotopicMathworldPlanetmath if and only if Y is path connected.

Proof: Assume that all maps are homotopic. In particular constant maps are homotopic, so if y1,y2Y, then there exists a continous map H:I×YY such that H(0,y)=y1 and H(1,y)=y2 for all yY. Thus the map α:IY defined by the formulaMathworldPlanetmathPlanetmath α(t)=H(t,y0) for a fixed y0Y is the wanted path.

On the other hand assume that Y is path connected. Since X is contractible, then for any cX there exists a continous homotopyMathworldPlanetmath H:I×XX connecting the identity map and a constant map c. Let f:XY be an arbitrary map. Define a map F:I×XY by the formula: F(t,x)=f(H(t,x)). This map is a homotopy from f to a constant map f(c). Thus every map is homotopic to some constant map.

The space Y is path connected, so for all y1,y2Y there exists a path α:IY from y1 to y2. Therefore constant maps are homotopic via the homotopy H(t,x)=α(t).

Finaly for any continous maps f,g:XY and any point cX we get:


which completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof.

Corollary. If X is a contractible space, then for any topological space Y there exists a bijection between the set [X,Y] of homotopy classes of maps from X to Y and the set π0(Y) of path components of Y.

Proof: Assume that Y=Yi, where Yi are path components of Y. It is well known that contractible spaces are path connected, thus the image of any continous map f:XY is contained in Yi for some i. It follows from the theorem that two maps from X to Y are homotopic if and only if their images are contained in the same Yi. Thus we have a well defined, injective map


where i is such that f(X)Yi. This map is also surjectivePlanetmathPlanetmath, since for any i there exists yYi, so the class of the constant map f(x)=y is mapped into Yi.

Title homotopy with a contractible domain
Canonical name HomotopyWithAContractibleDomain
Date of creation 2013-03-22 18:02:12
Last modified on 2013-03-22 18:02:12
Owner joking (16130)
Last modified by joking (16130)
Numerical id 26
Author joking (16130)
Entry type Theorem
Classification msc 55P99
Related topic homotopy
Related topic contractible