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!

  • 6
    $\begingroup$ If you take the multiplication on framed cobordism to be the product of framed manifolds, and the multiplication on stable homotopy of spheres to be the smash product of maps, then yes its multiplicative. It's a direct argument with little fuss, provided your input maps are transverse to begin with. So it generalizes to other bordism theories. I haven't thought about the Adams Spectral Sequence since grad school so I'll leave that to others. $\endgroup$ Aug 14, 2015 at 22:16
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    $\begingroup$ The first statement may be 2.3.3 in Ravenel's complex cobordism book. $\endgroup$ Aug 15, 2015 at 4:47
  • $\begingroup$ @RyanBudney Thanks. I also thought of this, but it seems it has not been written down anywhere. I wonder what did you think of the case of other bordism theories? $\endgroup$
    – user51223
    Aug 15, 2015 at 6:59
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    $\begingroup$ I don't know if this helps you to find a published reference, but the post mathoverflow.net/questions/173691/… deals with the second question. $\endgroup$
    – user43326
    Aug 17, 2015 at 16:21
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    $\begingroup$ I discuss this as well as `higher products' in MR1831346 Bruner, Robert R. Extended powers of manifolds and the Adams spectral sequence. Homotopy methods in algebraic topology (Boulder, CO, 1999), 41--51, Contemp. Math., 271, Amer. Math. Soc., Providence, RI, 2001. $\endgroup$ Jan 17, 2018 at 0:09


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