homology of the sphere
Every loop on the sphere ${S}^{2}$ is contractible^{} to a point, so its fundamental group^{}, ${\pi}_{1}({S}^{2})$, is trivial.
Let ${H}_{n}({S}^{2},\mathbb{Z})$ denote the $n$th homology group^{} of ${S}^{2}$. We can compute all of these groups using the basic results from algebraic topology:

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${S}^{2}$ is a compact orientable smooth manifold^{}, so ${H}_{2}({S}^{2},\mathbb{Z})=\mathbb{Z}$;

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${S}^{2}$ is connected, so ${H}_{0}({S}^{2},\mathbb{Z})=\mathbb{Z}$;

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${H}_{1}({S}^{2},\mathbb{Z})$ is the abelianization^{} of ${\pi}_{1}({S}^{2})$, so it is also trivial;

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${S}^{2}$ is twodimensional, so for $k>2$, we have ${H}_{k}({S}^{2},\mathbb{Z})=0$
In fact, this pattern generalizes nicely to higherdimensional spheres:
${H}_{k}({S}^{n},\mathbb{Z})=\{\begin{array}{cc}\mathbb{Z}\hfill & k=0,n\hfill \\ 0\hfill & \mathrm{else}\hfill \end{array}$ 
This also provides the proof that the hyperspheres^{} ${S}^{n}$ and ${S}^{m}$ are nonhomotopic for $n\ne m$, for this would imply an isomorphism^{} between their homologies^{}.
Title  homology of the sphere 

Canonical name  HomologyOfTheSphere 
Date of creation  20130322 13:46:49 
Last modified on  20130322 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 