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Let $T$ denote the thom spectrum over $\Omega S^{2}$ defined by the map

$1+3: \Omega S^{2} \to BG_{3}$

where $1 +3$ is a unit in $3$-adics. Here $G_{3}$ is the unit component of $\Omega^{\infty}S_{3}$, where $S_{3}$ is the $3$-adic sphere spectrum.

If we look at the $H^{*}(T; \mathbb{Z}/3)$ as a mudule over Steenrod algebra, does the 4th cell connect to the bottom by $\mathscr{P}^{1}$?

I also want to extend my question for other primes except $2$. Prime $2$ I managed to do, from the knowledge of characteristic classes at $2$, but other primes seems impossible!

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  • $\begingroup$ What is the group $G_3$? $\endgroup$
    – Craig Westerland
    Commented Feb 14, 2013 at 21:26
  • $\begingroup$ Sorry I should have mentioned this in the questioin, $G_{3}$ is the unit component of $ \Omega^{\infty}S_{3}$ where, $S^{3}$ is the $3$-adic sphere spectrum. $\endgroup$
    – Prasit
    Commented Feb 15, 2013 at 0:26
  • $\begingroup$ I suppose you have resolved the above question by now. But, do we know how the completion functor on spectra and the infinite loop space functor interact? $\endgroup$
    – user51223
    Commented Oct 26, 2015 at 7:30

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