Volume 16, issue 5 (2016)

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On a spectral sequence for the cohomology of infinite loop spaces

Rune Haugseng and Haynes Miller

Algebraic & Geometric Topology 16 (2016) 2911–2947
Abstract

We study the mod-$2$ cohomology spectral sequence arising from delooping the Bousfield–Kan cosimplicial space giving the $2$–nilpotent completion of a connective spectrum $X$. Under good conditions its ${E}_{2}$–term is computable as certain nonabelian derived functors evaluated at ${H}^{\ast }\left(X\right)$ as a module over the Steenrod algebra, and it converges to the cohomology of ${\Omega }^{\infty }X$. We provide general methods for computing the ${E}_{2}$–term, including the construction of a multiplicative spectral sequence of Serre type for cofibration sequences of simplicial commutative algebras. Some simple examples are also considered; in particular, we show that the spectral sequence collapses at ${E}_{2}$ when $X$ is a suspension spectrum.

Keywords
cohomology, infinite loop spaces, spectral sequence
Mathematical Subject Classification 2010
Primary: 18G40, 55P47