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 two-dimensional, so for $k>2$ , we have $H_{k}(S^{2},\mathbb{Z})=0$

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

\displaystyle H_{k}(S^{n},\mathbb{Z})=\begin{cases}\mathbb{Z}&k=0,n\\ 0&{\rm else}\end{cases}

This also provides the proof that the hyperspheres $S^{n}$ and $S^{m}$ are non-homotopic for $n\neq m$ , for this would imply an isomorphism between their homologies.

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