11
$\begingroup$

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.

Improve this question
$\endgroup$
2
  • $\begingroup$ 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.
    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
    Jul 25, 2014 at 14:17

1 Answer 1

Reset to default
9
$\begingroup$

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.

Improve this answer
$\endgroup$
1
  • $\begingroup$ very nice article! Harper's push out formula for $\mathsf{O}-\mathsf{Alg}$ seems to be very useful! $\endgroup$
    – Ilias A.
    Jul 25, 2014 at 19:29

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.