Homotopy of spheres

This article describes the value (and the process used to compute it) of some homotopy invariant(s) for a topological space or family of topological spaces. The invariant is homotopy group and the topological space/family is sphere
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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 $k\geq 2$.

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 $k\neq 2n-1$.

Proof

The case $n=0$

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The case $n=1$

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For higher $n$

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