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What is $\pi_{31}(S^2)$ – high 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 : http://arxiv.org/pdf/0904.4925v1.pdf

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2) Maybe there are relations between classification of qbits entanglements and sphere homotopy groups ?

3) We are interested in the classification of 4-qbits entanglements.

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In general, finding the isomorphism class of higher homotopy groups of the spheres is an extremely hard problem. Luckily however, there seem to be some techniques available to the low dimensional spheres which aren’t available to the higher dimensional spheres. In particular, there appears to be an isomorphism between $\pi_n(S^2)$ and the co-kernel of a homomorphism from the Brunnian braid group in $n$ strings on the disk to the equivalent Brunnian braid group on the sphere. The homomorphism is induced by the canonical geometric embedding of the disk in the sphere.

That is, let $f\colon D\rightarrow S^2$ be the canonical embedding of the disk in to the sphere and let $C(n,D)$ and $C(n,S^2)$ be the configuration space of $n$ points in the the disk $D$ and sphere $S^2$ respectively. It’s a well-known theorem that $\pi_1(C(n,M))$ is isomorphic to the pure braid group on $n$ strings in the manifold $M$ and so $f$ induces a homomorphism $f_*\colon\pi_1(C(n,D))\rightarrow\pi_1(C(n,S^2))$ on braid groups.

Let $BB_n\leq\pi_1(C(n,D))$ be the subgroup of Brunnian braids in the disk and similarly let $BB_N^{S^2}\leq \pi_1(C(n,S^2))$ be the subgroup of Brunnian braids in the sphere. A Brunnian braid is a pure braid such that the removal of any one string leaves the remaining braid isotopic to the identity on $n-1$ strings. We may restrict $f_*$ to the Brunnian braids and so get a homomorphism $f_*’\colon BB_n\rightarrow BB_n^{S^n}$. For $n$ large enough (greater than 3 i think), there is an isomorphism $$\pi_n(S^2)\cong \mbox{coker}f_*’=BB_n^{S^2}/\mbox{Im}f_*’.$$

This theorem is attributed to Berrick, Cohen, Wong, Wu in their paper Configurations, braids, and homotopy groups.

What makes this theorem so useful is that it reduces the hard problem of finding homotopy groups of the sphere to what is essentially computational group theory. This is because we have pretty good knowledge of both the disk and sphere braid groups thanks to work done in the 60s stemming from the analysis of the Fadell-Neuwirth fibrations between configuration spaces of $n$ points in a manifold. I’ve not done any computational group theory before so I don’t know if identifying the isomorphism class of the above mentioned cokernel is computationally efficient, but it at least gives us a known algorithm for calculating it (it’s possible that identifying the Brunnian subgroup of the respective braid groups might be difficult).

The following is just to keep a comprehensive collection of information we have, and I take no credit for finding these references.

For those that haven’t seen the cross-posted question on MathOverflow Mark Grant gives a reference to a paper which appears to give computations of $\pi_i(S^3)$ for all $i\leq 64$ which are of course isomorphic to the $\pi_i(S^2)$ for suitably large $i$ by the isomorphism induced by the hopf fibration:

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.

In the comments, the OP also gives a reference to a paper in which the $2$ primary part of $\pi_{31}(S^3)$ is given as isomorphic to $\mathbb{Z}_2\oplus\mathbb{Z}_2\oplus\mathbb{Z}_2\oplus\mathbb{Z}_2$:

Nobuyuki Oda, *On the 2.Components of the Unstable Homotopy Groups of Spheres I*, Proc. Japan Acad. Ser. A Math. Sci. Volume 53, Number 7 (1977), 215-218. (link)

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