Every loop on the sphere $S^2$ is contractible to a point, so its fundamental group, $\pi_1(S^2)$ , is trivial.

Let $H_n(S^2,\Z)$ 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 $H_2(S^2,\Z)=\Z$ ;
• $S^2$ is connected, so $H_0(S^2,\Z)=\Z$ ;
• $H_1(S^2,\Z)$ is the abelianization of $\pi_1(S^2)$ , so it is also trivial;
• $S^2$ is two-dimensional, so for $k>2$ , we have $H_k(S^2,\Z)=0$

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