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I'm looking for a reference for analogues of the Blakers-Massey triad connectivity theorem (and its higher-order generalization) for ring spectra. That is:

Suppose that $A\to A_1$ is a $k_1$-connected map of (associative) ring spectra and $A\to A_2$ is a $k_2$-connected map of ring spectra, and that the maps are cofibrations so that the pushout, call it $A_{12}$, is a homotopy pushout. Then as long as all of the rings are connective the map of spectra $$ A\to holim (A_1\to A_{12}\leftarrow A_2) $$ is $(k_1+k_2)$-connected.

Has anyone worked out a detailed proof of this? I would be happy to see this in any reasonable theory of structured ring spectra.

EDIT: I know how a proof should go: Filter the spectrum $A_{12}$ by "word length" and examine the sequence of subquotients (the "associated graded object"), which are wedges of "tensor products" of the spectra $A_i/A$ regarded as bimodules over $A$. But I do not want to delve into technicalities if the details are already out there somewhere.

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  • $\begingroup$ I was just wondering if the examples you have in mind are of the form $A_{i}=\Sigma^{\infty}\Omega X^{i}_{+}$ and maps of ring spectra are induced by $X\rightarrow X^{i}$ ? If I'm not wrong in this case we can reduce every thing to the classical lakers-Massey triad connectivity theorem. $\endgroup$
    – Ilias A.
    Commented Jul 25, 2014 at 13:56
  • $\begingroup$ @Fedotov: I need it not only for examples like that, but also for examples like ordinary discrete rings viewed as (Eilenberg-MacLane) ring spectra. $\endgroup$
    – Tom Goodwillie
    Commented Jul 25, 2014 at 14:17

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Theorems 1.4–1.11 in Ching and Harper's paper “Higher homotopy excision and Blakers-Massey theorems for structured ring spectra” (arXiv:1402.4775) give higher homotopy excision and Blakers-Massey (and their dual versions) for structured ring spectra and more generally, for algebras over operads.

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  • $\begingroup$ very nice article! Harper's push out formula for $\mathsf{O}-\mathsf{Alg}$ seems to be very useful! $\endgroup$
    – Ilias A.
    Commented Jul 25, 2014 at 19:29

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