---
tag: YD6VAXXSBM
kind: concept
name: Homotopy groups of spheres
msc: [55Q40, 55Q05]
sources:
  - url: https://en.wikipedia.org/wiki/Homotopy_groups_of_spheres
same_as:
  wikidata: Q3118090
  wikipedia: Homotopy groups of spheres
---

## Summary

The homotopy groups of spheres are the groups $\pi_k(S^n)$ of homotopy classes of based maps from the $k$-sphere to the $n$-sphere.
Some are simple: $\pi_k(S^n) = 0$ for $k < n$, and $\pi_n(S^n) \cong \Z$, detected by the degree.
For $k > n$ they are mostly finite but highly irregular, and no complete formula is known.
Their computation has driven much of algebraic topology, through fibrations, spectral sequences and stable homotopy theory.

## Explanations

### L1 human

These groups ask in how many essentially different ways one sphere can be wrapped around another.
Wrapping a circle around a circle is counted by the winding number, an integer.
Surprisingly, a $3$-sphere can be wrapped around a $2$-sphere in infinitely many different ways, by the Hopf map and its multiples, even though the $3$-sphere has higher dimension.
Small changes in the dimensions give very different answers, which is why these groups are famous.