the sphere is indecomposable as a topological space

Proposition. If for any topological spacesMathworldPlanetmath X and Y the n-dimensional sphere 𝕊n is homeomorphic to X×Y, then either X has exactly one point or Y has exactly one point.

Proof. Recall that the homotopy groupMathworldPlanetmath functor is additive, i.e. πn(X×Y)πn(X)πn(Y). Assume that 𝕊n is homeomorphic to X×Y. Now πn(𝕊n) and thus we have:


Since is an indecomposable group, then either πn(X)0 or πn(Y)0.

Assume that πn(Y)0. Consider the map p:X×YY such that p(x,y)=y. Since X×Y is homeomorphic to 𝕊n and πn(Y)0, then p is homotopicMathworldPlanetmathPlanetmath to some constant map. Let y0Y and H:I×X×YY be such that


Consider the map F:I×X×YX×Y defined by the formula


Note that F(0,x,y)=(x,y) and F(1,x,y)=(x,y0) and thus X×{y0} is a deformation retractMathworldPlanetmath of X×Y. But X×Y is a sphere and spheres do not have proper deformation retracts (please see this entry ( for more details). Therefore X×{y0}=X×Y, so Y={y0} has exactly one point.

Title the sphere is indecomposable as a topological space
Canonical name TheSphereIsIndecomposableAsATopologicalSpace
Date of creation 2013-03-22 18:31:45
Last modified on 2013-03-22 18:31:45
Owner joking (16130)
Last modified by joking (16130)
Numerical id 8
Author joking (16130)
Entry type Theorem
Classification msc 54F99