---
tag: 5SY318V6S3
kind: definition
name: Stable homotopy groups of spheres
msc: [55Q45, 55Q10]
defines: HGZAYPQBXW
uses: [YD6VAXXSBM, P976WYEFMB, Q7RFNJH76Q]
sources:
  - url: https://en.wikipedia.org/wiki/Homotopy_groups_of_spheres
---

## Definition

Let $k \ge 0$ be an integer.
For $n \ge 1$, let $E \colon \pi_{n+k}(S^n) \to \pi_{n+k+1}(S^{n+1})$ be the suspension homomorphism, which sends the class of $f$ to the class of its [[suspension|Q7RFNJH76Q]] $\Sigma f$.
The *$k$-th stable homotopy group of spheres* is the colimit
$$\pi_k^s = \operatorname{colim}_{n} \pi_{n+k}(S^n)$$
of the sequence of [[homotopy groups of spheres|YD6VAXXSBM]] $\pi_{k+1}(S^1) \to \pi_{k+2}(S^2) \to \pi_{k+3}(S^3) \to \cdots$ along the maps $E$.

## Remarks

By the [[Freudenthal suspension theorem|7DM2KSS2DB]], $E \colon \pi_{n+k}(S^n) \to \pi_{n+k+1}(S^{n+1})$ is an isomorphism for $n \ge k + 2$.
So $\pi_k^s \cong \pi_{n+k}(S^n)$ for every $n \ge k + 2$.

Example: $\pi_0^s \cong \Z$, because $\pi_n(S^n) \cong \Z$ for all $n$ and $E$ preserves the degree.

Example: $\pi_1^s \cong \Z/2$, generated by the suspensions of the Hopf map $S^3 \to S^2$; the group $\pi_3(S^2) \cong \Z$ itself is not yet in the stable range, since $n = 2 < k + 2 = 3$.

The next values are $\pi_2^s \cong \Z/2$, $\pi_3^s \cong \Z/24$, $\pi_4^s = \pi_5^s = 0$, $\pi_6^s \cong \Z/2$ and $\pi_7^s \cong \Z/240$.
Composition of maps makes $\pi_*^s = \bigoplus_k \pi_k^s$ a graded commutative ring.
For $k \ge 1$, the group $\pi_k^s$ is finite.