# Homotopy of spheres

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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 $k \ge 2$.

### 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 $k \ne 2n - 1$.

## Proof

### The case $n = 0$

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### The case $n = 1$

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### For higher $n$

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