---
tag: 212MDDPN8A
kind: definition
name: Homotopy groups of spheres
msc: [55Q40, 55Q05]
defines: YD6VAXXSBM
uses: [P976WYEFMB]
sources:
  - url: https://en.wikipedia.org/wiki/Homotopy_groups_of_spheres
---

## Definition

Let $n \ge 1$, and let $S^n = \set{x \in \R^{n+1} : \norm{x} = 1}$ be the unit $n$-sphere with base point $s_0 = (1, 0, \ldots, 0)$.
For $k \ge 1$, the group $\pi_k(S^n)$ is the [[homotopy group|P976WYEFMB]] $\pi_k(S^n, s_0)$.
The *homotopy groups of spheres* are the groups $\pi_k(S^n)$ for all $k, n \ge 1$.

## Remarks

Changing the base point gives isomorphic groups, because $S^n$ is path-connected; this is why it is usually omitted.
Example: $\pi_1(S^1) \cong \Z$, and the class of a loop is its winding number.
Example: $\pi_k(S^n) = 0$ for $1 \le k < n$.
Give $S^n$ the CW structure with one $0$-cell and one $n$-cell.
By cellular approximation, a based map $S^k \to S^n$ is homotopic to a map into the $k$-skeleton, which is a point.

Non-example: the groups are not the homology groups of spheres.
The homology $H_3(S^2)$ is zero, but $\pi_3(S^2) \cong \Z$, generated by the Hopf map.

Further values: $\pi_n(S^n) \cong \Z$ for all $n \ge 1$, $\pi_k(S^1) = 0$ for $k \ge 2$, and $\pi_{n+1}(S^n) \cong \Z/2$ for $n \ge 3$.