every map into sphere which is not onto is nullhomotopic

PropositionPlanetmathPlanetmath. Let X be a topological spaceMathworldPlanetmath and f:X𝕊n a continous map from X to n-dimensional sphere which is not onto. Then f is nullhomotopic.

Proof. Assume that there is y0𝕊n such that y0im(f). It is well known that there is a homeomorphism ϕ:𝕊n{y0}n. Then we have an induced map


Since n is contractible, then there is cn such that ϕf is homotopic to the constant map in c (denoted with the same symbol c). Let ψ:n𝕊n be a map such that ψ(x)=ϕ-1(x) (note that ψ is not the inversePlanetmathPlanetmathPlanetmath of ϕ because ψ is not onto) and take any homotopyMathworldPlanetmath H:I×Xn from ϕf to c. Then we have a homotopy F:I×X𝕊n defined by the formulaMathworldPlanetmathPlanetmath F=ψH. It is clear that


Thus F is a homotopy from f to a constant map.

Corollary. If A𝕊n is a deformation retractMathworldPlanetmath of 𝕊n, then A=𝕊n.

Proof. If AX then by deformation retraction (associated to A) we understand a map R:I×XX such that R(0,x)=x for all xX, R(1,a)=a for all aA and R(1,x)A for all xX. Thus a deformation retract is a subset AX such that there is a deformation retraction R:I×XX associated to A.

Assume that A is a deformation retract of 𝕊n and A𝕊n. Let R:I×𝕊n𝕊n be a deformation retraction. Then r:𝕊n𝕊n such that r(x)=R(1,x) 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 r is not onto, because A is a proper subsetMathworldPlanetmathPlanetmath of 𝕊n). Thus the identity map is homotopic to a constant map, so 𝕊n is contractible. ContradictionMathworldPlanetmathPlanetmath.

Title every map into sphere which is not onto is nullhomotopic
Canonical name EveryMapIntoSphereWhichIsNotOntoIsNullhomotopic
Date of creation 2013-03-22 18:31:41
Last modified on 2013-03-22 18:31:41
Owner joking (16130)
Last modified by joking (16130)
Numerical id 8
Author joking (16130)
Entry type Theorem
Classification msc 55P99