Image of J

Summary

The image of J is the image of the stable J-homomorphism $J \colon \pi_r(SO) \to \pi_r^s$ inside the stable homotopy groups of spheres. 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. It is the most regular and best understood part of the stable homotopy groups of spheres.