---
tag: HGZAYPQBXW
kind: concept
name: Stable homotopy groups of spheres
msc: [55Q45, 55Q10]
sources:
  - url: https://en.wikipedia.org/wiki/Homotopy_groups_of_spheres
---

## Summary

The stable homotopy groups of spheres $\pi_k^s$ are the groups $\pi_{n+k}(S^n)$ for $n$ large, where they no longer depend on $n$.
The [[Freudenthal suspension theorem|7DM2KSS2DB]] shows that the suspension maps become isomorphisms once $n \ge k + 2$.
Together they form a graded ring, the coefficient ring of stable homotopy theory, and they are the homotopy groups of the sphere spectrum.
Their computation is a central problem of algebraic topology, studied with the Adams spectral sequence and chromatic methods.

## Explanations

### L1 human

The homotopy groups of spheres change wildly as the dimensions change.
If both dimensions are raised together by the same amount, however, the groups settle down after a while.
These settled values are the stable groups.
The first ones are $\Z$, $\Z/2$, $\Z/2$, $\Z/24$, $0$, $0$, $\Z/2$, $\Z/240$, and their pattern is still not understood in general.