Higher homotopy excision and Blakers-Massey theorems for structured ring spectra (Q291788)

As an important tool for his influential work on the ``Calculus of functors'', \textit{T. G. Goodwillie} [\(K\)-Theory 5, No. 4, 295--332 (1992; Zbl 0776.55008)] has established generalizations of the classical Blakers-Massey theorem. These ``higher Blakers-Massey theorems'' state that maps extracted from certain higher-dimensional cubical diagrams of spaces induce isomorphisms in a range of homotopy groups.NEWLINENEWLINEIt is an obvious question to what extent Goodwillie's results hold for cubical diagrams in other categories than spaces. In the paper under review, the authors provide higher Blakers-Massey theorems for various categories arising in the context of structured ring spectra. More specifically, they provide such theorems for cubical diagrams in the categories of \(\mathcal O\)-algebras and left \(\mathcal O\)-modules for an operad \(\mathcal O\) in the category of \(R\)-modules over a commutative symmetric ring spectrum \(R\). Letting \(\mathcal O\) be the associativity or the commutativity operad, their result specializes to give higher Blakers-Massey theorems for associative and commutative \(R\)-algebra spectra. One technical backbone of the paper is an analysis of certain pushout diagrams in \(\mathcal O\)-algebras and left \(\mathcal O\)-modules which is based on a filtration studied by \textit{A. D. Elmendorf} and \textit{M. A. Mandell} [Adv. Math. 205, No. 1, 163--228 (2006; Zbl 1117.19001)].NEWLINENEWLINEAs the authors point out, their results ``establish an important part of the foundations for the theory of Goodwillie calculus in the context of structured ring spectra''.