---
aliases: ["image of J"]

---

Tags:
#homotopy/stable-homotopy 

# J-homomorphism

# References

- [https://www.youtube.com/watch?v=Ix4pg87LKVk](https://www.youtube.com/watch?v=Ix4pg87LKVk)

# What is the $J$ homomorphism? 

The $J$ homomorphism seems to link the [Homotopy Groups of Spheres](Homotopy%20Groups%20of%20Spheres.md). 
For example $J$ takes $\pi_k(SL^n(R))$ to $\pi_{n}^k S^n$.

Look at the map
\[
SO(n) &\to \Omega^n S^n\\
A: (\RR^n \to \RR^n) &\mapsto A^+
\]

AKA
$$
J: \pi_n(SO(k)) \to \pi_{n+k}(S_k)
$$

Where we view a matrix as a linear function on $\RR^n$, and take it to its compactification which is a map $S^n\to S^n$. Taking the limit yields a map from $\SO^\infty \to QS^0 = \Omega^\infty \Sigma^\infty S^0$, and taking $\pi_0$ of both sides induces the $J$ homomorphism. The RHS is equal to $\pi_*^s$, the [stable homotopy groups of spheres](stable%20homotopy%20groups%20of%20spheres.md).
But the homotopy groups of $SO$ were computed by Bott, and have some 8-fold periodicity.

The image of $J$ was found by Adams in '66 or so, it is a finite group with order the denominator of some function involving Bernoulli numbers. However, the pattern is more apparent by looking at the $p$-stems, then the number of connected dots really just depends on the $p$-adic divisibility of the horizontal number plus 1. The image of $J$ is just the bottom row in these stem diagrams.


![](attachments/Pasted%20image%2020220505153111.png)