# In the cohomology of Thom spectrum over LoopS^{2} and p-adic characteristic classes

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!

• What is the group $G_3$? – Craig Westerland Feb 14 '13 at 21:26
• 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. – Prasit Feb 15 '13 at 0:26
• 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? – user51223 Oct 26 '15 at 7:30