On a spectral sequence for the cohomology of infinite loop spaces (Q341777)

The paper under review studies the relationship between the mod \(2\) cohomology \(H^*(E)\) of a spectrum \(E\) and the mod \(2\) cohomology \(H^{*}(\Omega^{\infty} E)\) of its zeroth space \(\Omega^{\infty} E\). Here, \(E\) will be a \(0\)-connected spectrum of finite type, i.e., such that \(\pi_k E\) is trivial for \(k \leq 0\) and a finitely generated abelian group for \(k \geq 1\).NEWLINENEWLINELet \(A\) denote the mod \(2\) Steenrod algebra. There is a natural map of unstable \(A\)-algebras \(\theta : U D H^*(E) \to H^*(\Omega^{\infty} E)\). Here, \(D\) is the \textit{destabilization} functor, i.e., \(D H^*(E)\) is the unstable \(A\)-module obtained from \(H^*(E)\) by modding out the instability relations; \(U\) denotes the free unstable \(A\)-algebra functor. By results of Serre, \(\theta\) is an isomorphism if \(E\) is an \(H\mathbb{F}_2\)-module spectrum of finite type. A Bousfield-Kan cosimplicial resolution of \(E\) then gives rise to a convergent spectral sequence NEWLINE\[NEWLINE E_2^{s,t} = \mathbb{L}_{-s}(UD)(H^*E)^t \Rightarrow H^{t+s}(\Omega^{\infty} E) NEWLINE\]NEWLINE (Theorem~2.1). The paper focuses on computing this \(E_2\)-term.NEWLINENEWLINEAs a technical tool, the authors construct a multiplicative ``Serre spectral sequence'' for a cofiber sequence of simplicial augmented graded commutative \(\mathbb{F}_2\)-algebras (Corollary~3.41).NEWLINENEWLINEFor a simplicial unstable \(A\)-module \(M\), they compute the homotopy groups \(\pi_* (U M)\) of the free unstable \(A\)-algebra on \(M\) in terms of \(\pi_* M\) (Theorems~4.8 and 4.20). Specializing to the \(E_2\)-term yields an expression in terms of derived functors of destabilization (Corollaries~4.17 and 4.21).NEWLINENEWLINEAs an example, the authors use the spectral sequence with the Eilenberg-MacLane spectra \(E = \Sigma^{k} H\mathbb{Z}\) and \(\Sigma^k H\mathbb{Z} / 2^n\) for \(k \geq 1\), recovering calculations of Serre. For the suspension spectrum \(E = \Sigma^{\infty} X\) of a connected space of finite type \(X\), they prove that the spectral sequence collapses at \(E_2\) (Corollary~5.5). The proof uses work on the derived functors of destabilization [\textit{N. J. Kuhn} and \textit{J. B. McCarty}, Algebr. Geom. Topol. 13, No. 2, 687--745 (2013; Zbl 1333.55008)], along with the homology of infinite loop spaces in terms of Dyer-Lashof operations [\textit{F. R. Cohen} et al., The homology of iterated loop spaces. Berlin-Heidelberg-New York: Springer-Verlag (1976; Zbl 0334.55009)].