# What is the 31th homotopy group of the 2 - sphere ?

What is $\pi_{31}(S^2)$, the 31th homotopy group of the 2 - sphere ?

This question has a physics motivation:

1) There are relations between (2nd and 3rd) Hopf fibrations and (2- and 3-) qbits (quantum bits) entanglement, see this reference

2) Maybe there are relations between classification of qbits entanglements and sphere homotopy groups, and we are interested in the classification of 4-qbits entanglements.

I tried fo find the solution on the net, with help of math fans, but without success.

Wikipedia gives only to the 22th group homotopy of the 2-sphere

This article of John Baez gives interesting references, like Allen Hatcher, Stable homotopy groups of spheres or a link with braids. One speak of a book of Kochman Stanley O. : Stable Homotopy Groups of Spheres A Computer-Assisted Approach

But I am totally unable to find the answer.

A subsidiary question would be : Until what rank do we know these high homotopy group of the 2-sphere ?

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But there is no fourth Hopf fibration! –  Mariano Suárez-Alvarez May 17 '13 at 7:45
@David: $\pi_{31}(S^2)$ is well out of the stable range. There is a homomorphism to the 29th stable group, but it is unlikely to be injective or surjective. –  Neil Strickland May 17 '13 at 7:54
You could deduce some information on $\pi_{32}(S^3)$ from Bob Bruners unstable chart on the bottom of math.wayne.edu/~rrb/cohom/index.html (if that's of any help) –  Christian Nassau May 17 '13 at 8:08
Actually, I remember seeing more unstable charts in the "Oaxtapec proceedings" (CONM146) (Appendix 2 by Paul Shick), but I don't know if this relates to $S^2$. Be warned that it's usually nontrivial to deduce the homotopy from these charts, so this info might not be of much use to you. –  Christian Nassau May 17 '13 at 8:27
«For separable states, the original Hilbert space $S^7$ simplifies to $S^2\times S^2$» Reading physics papers requires a lot of restraint :-) –  Mariano Suárez-Alvarez May 17 '13 at 8:31

One simple observation is that $\pi_{31}(S^2)\cong\pi_{31}(S^3)$, by the long exact sequence of the Hopf fibration.

The homotopy groups $\pi_i(S^3)$ for $i\le 64$ are apparently computed in:

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.

Unfortunately I wasn't able to get a hold of that reference to check for an explicit answer. Maybe someone has it on their shelf and can check.

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Thanks for the reference –  Trimok May 17 '13 at 9:12
The beginning of the article can apparently be read on Google Books –  Trimok May 17 '13 at 9:34
I don't know if it is the correct result, but it seems that the answer should be $Z_2 + Z_2 + Z_2 + Z_2$ (Found page 204, in 1st paper reference below) Nobuyuki Oda, On the 2.Components of the Unstable Homotopy Groups of Spheres I. Source: Proc. Japan Acad. Ser. A Math. Sci. Volume 53, Number 7 (1977), 215-218. 1st paper –  Trimok May 17 '13 at 10:29
@Trimok: Good find! Note that this is only supposed to be the $2$_primary part; there may be other summands of the form $\mathbb{Z}_{p^n}$ with $p$ an odd prime. –  Mark Grant May 17 '13 at 11:01
@Mark : Yes, I understand this is not the final answer... –  Trimok May 17 '13 at 11:04