---
tag: YYJPRD1HYW
kind: concept
name: J-homomorphism
msc: [55Q50]
sources:
  - url: https://en.wikipedia.org/wiki/J-homomorphism
same_as:
  wikidata: Q6103838
  wikipedia: J-homomorphism
---

## Summary

The J-homomorphism $J \colon \pi_r(SO(n)) \to \pi_{r+n}(S^n)$ turns a family of rotations of $\R^n$ parametrised by an $r$-sphere into a map of spheres.
It connects the homotopy of the rotation groups, known through Bott periodicity, with the homotopy groups of spheres.
In the stable range it gives a homomorphism $\pi_r(SO) \to \pi_r^s$ into the [[stable homotopy groups of spheres|HGZAYPQBXW]], whose image is the [[image of J|GX8K2SCWVQ]].

## Explanations

### L1 human

A loop of rotations of the plane, such as one full turn, can be used to twist a sphere around another sphere.
The J-homomorphism makes this precise in every dimension.
Its value on one full turn of the plane is the Hopf map $S^3 \to S^2$, up to sign.
It explains a regular part of the otherwise chaotic homotopy groups of spheres.