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I asked this question in MathStackExchange a couple of months ago, with little feedback, hence I try here.

The Hopf invariant $h(f)$ of a mapping $f:\mathbb S^{2m-1}\to \mathbb S^m$ is a homotopy invariant that gives information on $\pi_{2m-1}(\mathbb S^m)$ for $m$ even. Namely, there is always $f$ with $h(f)\ne0$ and such an $f$ generates a cyclic infinite group. However, $h(f)$ maybe zero and $f$ not null homotopic. This happen most often, but not for $m=2$ or $m=6$, and then the homotopy group above is cyclic infinite. My question is whether there are other $m$’s for which this happens. This essentially asks when the Freudenthal homomorphism $\pi_{2m-2}(\mathbb S^{m-1})\to\pi_{2m-1}(\mathbb S^m)$ is trivial.

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    $\begingroup$ Do you know about the EHP sequence? A good reference for this stuff is chapters 4 and 5 of J. Neisendorfer's book Algebraic Methods in Unstable Homotopy Theory. $\endgroup$
    – Tyrone
    Commented Oct 16 at 9:28
  • $\begingroup$ Tyrone: no, I do not know that sequence. I’m not an specialist, I’ve come to the question somehow sideways. In fact, I was expecting some quick additional example or just to know if it is open. But I’ll try to look at that. Thanks. $\endgroup$
    – Jesus RS
    Commented Oct 16 at 14:06

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My question was very modest, and checking some others on unstable homotopy groups of spheres here in OverFlow I've found that for $m=30$ the homotopy group is cyclic infinite. See:

N. Oda, On the 2-components of the unstable homotopy groups of spheres Parts I, II, Proc. Japan Acad. 53, Ser. A(1977), no. 6/7, 202-205/215-218,

where if I understand well it says $\pi_{59}(\mathbb S^{30})=\mathbb Z$. And I guess my question was quite naive and clearly that of an outsider to the area.

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