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 $\pi_6(S^3)$?
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2There is another Hopf fibration $S^{1} \rightarrow S^{3} \rightarrow S^{2}$, so the long exact sequence reduces your problem to computing $\pi_{6}(S^{2})$. – tharris Jul 02 '13 at 10:43
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I don't know how to go about computing unstable homotopy groups of spheres in general, but known computations (including yours, the answer is $\mathbb{Z}/12$) are in Toda's book "Composition methods in homotopy groups of spheres". – tharris Jul 02 '13 at 10:55
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related: third stable homotopy group of spheres via geometry – Grigory M Jul 02 '13 at 11:02
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You can find the 2-component calculation in Mosher-Tangora, in the Appendix to Chapter 12. There is also a way to do this using the EHP sequence I think – Drew Jul 02 '13 at 12:37
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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.
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Do you know how can I prove that for all $ n \ge 3$ $\pi_{n+1}(S^n) \simeq \mathbb{Z}_2$? – ArthurStuart Jul 02 '13 at 13:35
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As such groups are in the stable range, we denote them by $\pi_1^S$ and call it the stable 1-stem. I would suggest reading any book that mentions the calculations of the stable homotopy groups of spheres. You can always ask this as a question again on this site. – Dan Rust Jul 02 '13 at 13:52