the sphere is indecomposable as a topological space
Proof. Recall that the homotopy group functor is additive, i.e. . Assume that is homeomorphic to . Now and thus we have:
Since is an indecomposable group, then either or .
Assume that . Consider the map such that . Since is homeomorphic to and , then is homotopic to some constant map. Let and be such that
Consider the map defined by the formula
Note that and and thus is a deformation retract of . But is a sphere and spheres do not have proper deformation retracts (please see this entry (http://planetmath.org/EveryMapIntoSphereWhichIsNotOntoIsNullhomotopic) for more details). Therefore , so has exactly one point.
|Title||the sphere is indecomposable as a topological space|
|Date of creation||2013-03-22 18:31:45|
|Last modified on||2013-03-22 18:31:45|
|Last modified by||joking (16130)|