# How to find stable homotopy group given the quotient group?

If $\Theta_n$ is the group of exotic spheres in dimension $n$ and $\mathrm{bP}_{n+1}$ is the group of spheres that bounds parallelizable $(n+1)$-manifolds, $\pi_n^S$ is the $n$th stable homotopy group of spheres and $J$ is the so-called $J$-homomorphism then assume

$$\Theta_n/\mathrm{bP}_{n+1}\to \pi^s_n/\mathrm{im}(J)$$

is an isomorphism. How can one recover $\pi_n^s$ from knowledge of $\pi^s_n/\mathrm{im}(J)$?

• There appears to be quite a bit known about the image of the $J$-homomorphism. Have you read the wikipedia page on it? en.wikipedia.org/wiki/J-homomorphism Jun 26, 2013 at 10:10
• @DanielRust Yes but knowing the image of the J-homomorphism will not help determine $\pi_n^s$. Jun 27, 2013 at 8:18

$\pi_n^s$ is a direct sum of $\mathrm{im}(J)$ and $\pi_n^s/\mathrm{Im}(J)$ (for $n\geq1$). This is a consequence of the "Adams Conjecture" (which tells you exactly what $\mathrm{Im}(J)$ is), and calculations in stable homotopy localized with respect to $K$-theory.

Explicitly, the composite of the maps $$\mathrm{Im}(J)_n \to \pi_n S \to \pi_n S_K$$ is an inclusion of a direct summand (for $n\geq 1$). (In fact, at any prime other than $2$, it is an isomorphism.) Here $S_K$ is the $K$-localization of the sphere spectrum.

I believe this was originally proved by Mahowald (though not stated precisely in this form).