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 $E_2$-term is computable as certain non-abelian derived functors evaluated at $H^{\ast }(X)$ 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.

36 pages, v2: substantially rewritten, with a stronger version of the main result, v3: accepted version

References in corpus (2)

• The mod 2 homology of infinite loopspaces
• Operations in the homology spectral sequence of a cosimplicial infinite loop space

Cited by in corpus (1)

• Quillen homology of spectral Lie algebras with application to mod $p$ homology of labeled configuration spaces