homotopy with a contractible domain
Theorem. Assume that is an arbitrary topological space and is a contractible topological space. Then all maps are homotopic if and only if is path connected.
Proof: Assume that all maps are homotopic. In particular constant maps are homotopic, so if , then there exists a continous map such that and for all . Thus the map defined by the formula for a fixed is the wanted path.
On the other hand assume that is path connected. Since is contractible, then for any there exists a continous homotopy connecting the identity map and a constant map . Let be an arbitrary map. Define a map by the formula: . This map is a homotopy from to a constant map . Thus every map is homotopic to some constant map.
The space is path connected, so for all there exists a path from to . Therefore constant maps are homotopic via the homotopy .
Finaly for any continous maps and any point we get:
which completes the proof.
Corollary. If is a contractible space, then for any topological space there exists a bijection between the set of homotopy classes of maps from to and the set of path components of .
Proof: Assume that , where are path components of . It is well known that contractible spaces are path connected, thus the image of any continous map is contained in for some . It follows from the theorem that two maps from to are homotopic if and only if their images are contained in the same . Thus we have a well defined, injective map
where is such that . This map is also surjective, since for any there exists , so the class of the constant map is mapped into .
|Title||homotopy with a contractible domain|
|Date of creation||2013-03-22 18:02:12|
|Last modified on||2013-03-22 18:02:12|
|Last modified by||joking (16130)|