Hopf fibration and homotopy of spheres

Question

Let $$ S^3 \to S^7 \to S^4 $$ an the Hopf fibration. We con consider the induced sequence in homotopy $$ \pi_i(S^3) \to \pi_i(S^7) \to \pi_i(S^4) \to \pi_{i-1}(S^3) \to \pi_{i-1}(S^7) \to \cdots $$ So we have, using the suspension homomorphism that if $i=7$ and Freudental theorem $$ \pi_7(S^4) \simeq \mathbb{Z} \times \pi_6(S^3) $$ How can I compute $\pi_6(S^3)$?

Answer 1

In general, it is not easy to compute the unstable homotopy groups of spheres. However, some techniques do exist.

I gave an answer to a similar question (about $\pi_{31}(S^2)=\pi_{31}(S^3)$) a few weeks ago which would seem appropriate here.

In particular, an answer given by Mark Grant on this mathoverflow question gives a reference to a paper which appears to give computations of $\pi_i(S^3)$ for all $i\leq 64$:

Curtis, Edward B.,Mahowald, Mark, The unstable Adams spectral sequence for $S^3$, Algebraic topology (Evanston, IL, 1988), 125–162, Contemp. Math., 96, Amer. Math. Soc., Providence, RI, 1989.

It's also worth mentioning that the part about using the Brunnian braid groups should offer an effective method for calculating homotopy groups of the sphere, although I have not seen this done, and so there may be some computational obstruction.

Finally, as Tom Harris mentioned in the comments, $\pi_6(S^3)$ has already been calculated and is isomorphic to the cyclic group $\mathbb{Z}/12\mathbb{Z}$. For reference, see:

Hirosi Toda, Composition methods in homotopy groups of spheres, Annals of Mathematics Studies, No. 49, Princeton University Press, Princeton, N.J., 1962.