every map into sphere which is not onto is nullhomotopic
Since is contractible, then there is such that is homotopic to the constant map in (denoted with the same symbol ). Let be a map such that (note that is not the inverse of because is not onto) and take any homotopy from to . Then we have a homotopy defined by the formula . It is clear that
Thus is a homotopy from to a constant map.
Corollary. If is a deformation retract of , then .
Proof. If then by deformation retraction (associated to ) we understand a map such that for all , for all and for all . Thus a deformation retract is a subset such that there is a deformation retraction associated to .
Assume that is a deformation retract of and . Let be a deformation retraction. Then such that is homotopic to the identity map (by definition of a deformation retract), but on the other hand it is homotopic to a constant map (it follows from the proposition, since is not onto, because is a proper subset of ). Thus the identity map is homotopic to a constant map, so is contractible. Contradiction.
|Title||every map into sphere which is not onto is nullhomotopic|
|Date of creation||2013-03-22 18:31:41|
|Last modified on||2013-03-22 18:31:41|
|Last modified by||joking (16130)|