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Let $f:S^n\rightarrow S^n$ be a degree $k$ map. Then $f$ induces maps $\pi_l(S^n)\rightarrow \pi_l(S^n)$.

I believe that in the stable range ($l\leq 2n-2$) this map is multiplication by $k$. Unstably this need not be the case. Pontryagin proved the induced map $\pi_3(S^2)\rightarrow \pi_3(S^2)$ is multiplication by $k^2$.

I am not known as a stable mathematician and therefore I wonder: is it known what happens unstably in general?

I understand that we are very far from computing the unstable homotopy groups of spheres, but one might still know what the induced maps are without knowing the groups themselves.

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    $\begingroup$ See my answer here. The method works in higher dimensions up to knowing certain Whitehead products and Hopf invariants. This doesn't quite answer your question, though. $\endgroup$
    – Tyrone
    Commented Oct 29 at 12:55
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    $\begingroup$ What I can add is that an odd-dimensional sphere localised at an odd prime is an H-space. Thus for an odd-dimensional sphere, the degree $k$ map acts as multiplication by $k$ on odd primary components. The 2-primary case is not completely understood. $\endgroup$
    – Tyrone
    Commented Oct 29 at 13:07
  • $\begingroup$ Dear Tyrone, thank you for your excellent answer, I didn't see that question before. This tells me how to think about these maps. Of course this involves many computations for all unstable groups. This is more complicated than that I hoped for, but I guess that was to be expected. Again, thanks for the pointers! $\endgroup$
    – Thomas Rot
    Commented Oct 30 at 13:07

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