On a spectral sequence for the cohomology of infinite loop spaces

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 E₂-term is computable as certain nonabelian derived functors evaluated at H* (X) as a module over the Steenrod algebra, and it converges to the cohomology of Ω ∞ X. We provide general methods for computing the E₂-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 E₂ when X is a suspension spectrum.