Let $\theta$ be the generator of $\pi_{21}(Sp(3))\cong \mathbb{Z}_3$, (localized at 3). How to show the composition $$S^{24}\longrightarrow S^{21}\overset{\theta}\longrightarrow Sp(3)$$ is non-trivial (3-loca).
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3$\begingroup$ A little bit of background/motivation for the question can help... $\endgroup$– user43326Commented Jun 16, 2023 at 8:36
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$\begingroup$ We have, 3-Local, $\pi_{24}(S^{21})\cong \mathbb{Z}_3$ and also $\pi_{24}(Sp(3))\cong \mathbb{Z}_3$. Probably by a contradiction or Toda brackets, we can show it. $\endgroup$– Sajjad MohammadiCommented Jun 16, 2023 at 8:48
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$\begingroup$ How is $\theta$ detected? If you know the answer then some cohomology operation on higher order may do the job! $\endgroup$– user51223Commented Jun 20, 2023 at 17:20
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