# 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:

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

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

• $H_{1}(S^{2},\mathbb{Z})$ is the abelianization of $\pi_{1}(S^{2})$, so it is also trivial;

• $S^{2}$ 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 HomologyOfTheSphere 2013-03-22 13:46:49 2013-03-22 13:46:49 mathcam (2727) mathcam (2727) 14 mathcam (2727) Derivation msc 51M05 sphere HomologyTopologicalSpace Sphere