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:

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