Homotopy of spheres

This article gives the key facts about the computation of::homotopy groups of specific information about::spheres.

For $$n = 0$$
$$\pi_0(S^0)$$ is a two-point set. If we think of $$S^0$$ as a group, $$\pi_0(S^0)$$ gets the same group structure, namely the structure of the cyclic group of order two. For all $$k > 0$$, $$\pi_k(S^0)$$ is the trivial group.

For $$n = 1$$
$$\pi_0(S^1)$$ is a one-point set (or trivial group, if we choose to use $$S^1$$'s group structure to induce a group structure on it). $$\pi_1(S^1) \cong \mathbb{Z}$$, and $$\pi_k(S^1)$$ is trivial for all $$k \ge 2$$.

For higher $$n$$
We have that:


 * $$\pi_0(S^n)$$ is a one-point set (which we can interpret as a trivial group in some cases).
 * $$\pi_k(S^n)$$ is the trivial group for $$0 < k < n$$.
 * $$\pi_n(S^n) \cong \mathbb{Z}$$.
 * $$\pi_3(S^2) \cong \mathbb{Z}$$.
 * $$\pi_k(S^n)$$ is a finite group for $$k > n$$ and $$k \ne 2n - 1$$.