Image of J

Remarks

By Bott periodicity, for $r \ge 1$ the group $\pi_r(SO)$ is $\Z/2$ if $r \equiv 0, 1 \pmod 8$, $\Z$ if $r \equiv 3, 7 \pmod 8$, and $0$ otherwise. For $r = 4k - 1$, the image is cyclic of order the denominator of $B_{2k}/4k$, where $B_{2k}$ is the Bernoulli number. For $r \equiv 0, 1 \pmod 8$ with $r \ge 1$, $J$ is injective and the image is $\Z/2$.

Example: in degree $3$, $B_2 = 1/6$ gives $B_2 / 4 = 1/24$, and the image of J is all of $\pi_3^s \cong \Z/24$.

Non-example: in degree $2$, $\pi_2(SO) = 0$, so the image of J is $0$; the nonzero element $\eta^2$ of $\pi_2^s \cong \Z/2$ is not in the image of J.

J. F. Adams determined the image, with the upper bound depending on the Adams conjecture, which was later proved by Quillen and by Sullivan.