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Let $\eta=\eta_n\colon S^{n+1}\to S^n$ be the iterated suspension of the Hopf map $\eta\colon S^3\to S^2$. It is known that $\eta$ is detected by the Steenrod square for $n\geq 3$. Is there any (cohomology) operations capable of detecting the composition $\eta^2=\eta_n\eta_{n+1}$? Thanks in advance!

LipCaty
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    Yes. The composition is detected by the secondary cohomology relation based on the relation $Sq^2(Sq^2\rho_2)=0$, where $\rho_2$ is the reduction mod $2$. – Tyrone Jul 12 '22 at 07:26
  • @Tyrone Thank you very much. I learned from page 96 of Harper's book "secondary cohomology operations" that $\eta^2$ can also be detected by the secondary cohomology operation $\Theta$ based on the relation $Sq^3Sq^1+Sq^2Sq^2=0$. I have another question: With $n\geq 2$ varied, is it ${\Theta_n}$ stable, which means $\Omega \Theta_{n+1}=\Theta_n$? I am not sure by Prop. 4.2.8 of this book. – LipCaty Sep 01 '22 at 15:39
  • It is stable. Maybe it is the appearance of the switching map $\tau$ which is off-putting, but this is only in the case of odd primes that the sign has any significance. btw, if you skip ahead to $\S5.1$ Harper will do some of the work for you. – Tyrone Sep 01 '22 at 16:52
  • @Tyrone I am all gratitude. – LipCaty Sep 02 '22 at 03:20

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