Adams Spectral sequence and Pontrjagin-Thom construction [Reference request]

I will be grateful for any reference for the following statements/claims.

1) Let’s consider the case of $p=2$ and the classic Adams spectral sequence with the $E_2$-term given by $\mathrm{Ext}_{A}(\mathbb{F}_2,\mathbb{F}_2)$. If $\alpha$ and $\beta$ are two permanent cycles in the Adams spectral sequence, converging to elements $f\in{_2\pi_i^s}$ and $g\in{_2\pi_j^s}$, then is it true that $\alpha\beta$ does converge to $fg$? If so, then does it follow from the construction of the spectral sequence or it is not that evident?

I suppose, I am really asking if the $\textbf{convergence}$ in the ASS is multiplicative?

Of course, on the other direction we know the answer to the above question is not always positive as there are elements in the stable homotopy ring which are represented by decomposable elements in the $E_2$-term of the ASS, yet they are not decomposable elements in ${_2\pi_*^s}$.

2) Is the isomorphism $\Omega_*^{framed}\to\pi_*^s$ coming from the Pontrjagin-Thom construction multiplicative?

I think I can prove this second statement, but I wonder if it has been worked out, or it is a folklore that comes from the construction. If so, does it generalise to other bordism theories?

Thank you!

Answer

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Source : Link , Question Author : user51223 , Answer Author : Community

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