Homotopy of spheres

This article gives the key facts about the homotopy groups of spheres.

Statement

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 $0<k<n$.
• ${\pi }_n(S^n)\cong \mathbb{Z}$.
• ${\pi }_{2n-1}(S^n)\cong \mathbb{Z}$.
• ${\pi }_k(S^n)$ is a finite group for $k>n$ and $k\ne 2n-1$.

Proof

The case $n=0$

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The case $n=1$

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For higher $n$

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