homology of the sphere

Every loop on the sphere S2 is contractibleMathworldPlanetmath to a point, so its fundamental groupMathworldPlanetmathPlanetmath, π1(S2), is trivial.

Let Hn(S2,) denote the n-th homology groupMathworldPlanetmath of S2. We can compute all of these groups using the basic results from algebraic topology:

  • S2 is a compact orientable smooth manifoldMathworldPlanetmath, so H2(S2,)=;

  • S2 is connected, so H0(S2,)=;

  • H1(S2,) is the abelianizationMathworldPlanetmath of π1(S2), so it is also trivial;

  • S2 is two-dimensional, so for k>2, we have Hk(S2,)=0

In fact, this pattern generalizes nicely to higher-dimensional spheres:


This also provides the proof that the hyperspheresMathworldPlanetmath Sn and Sm are non-homotopic for nm, for this would imply an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath between their homologiesMathworldPlanetmathPlanetmath.

Title homology of the sphere
Canonical name HomologyOfTheSphere
Date of creation 2013-03-22 13:46:49
Last modified on 2013-03-22 13:46:49
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 14
Author mathcam (2727)
Entry type Derivation
Classification msc 51M05
Related topic sphere
Related topic HomologyTopologicalSpace
Related topic Sphere