---
tag: GX8K2SCWVQ
kind: concept
name: Image of J
msc: [55Q50, 55Q45]
sources:
  - url: https://en.wikipedia.org/wiki/J-homomorphism
---

## Summary

The image of J is the image of the stable [[J-homomorphism|YYJPRD1HYW]] $J \colon \pi_r(SO) \to \pi_r^s$ inside the [[stable homotopy groups of spheres|HGZAYPQBXW]].
It is the part of the stable stems that comes from rotations, and it is known completely.
In degree $4k - 1$ it is cyclic of order equal to the denominator of $B_{2k}/4k$, where $B_{2k}$ is a [[Bernoulli number|0KC00K7KY4]].
It is the most regular and best understood part of the stable homotopy groups of spheres.

## Explanations

### L1 human

The stable homotopy groups of spheres look chaotic, but a part of them follows a clear rule.
That part comes from twisting spheres by rotations, and it is called the image of J.
Its sizes are given by Bernoulli numbers.
For instance, it has $24$ elements in degree $3$ and $240$ elements in degree $7$.
The rest of the stable groups is much harder to understand.