On a spectral sequence for the cohomology of infinite loop spaces

TL;DR

This paper investigates a spectral sequence for mod-2 cohomology of infinite loop spaces, providing methods for its computation and conditions for its collapse, advancing understanding of cohomological structures in algebraic topology.

Contribution

It introduces a computable spectral sequence for the cohomology of infinite loop spaces and develops new methods for calculating its E2-term, including a Serre-type spectral sequence.

Findings

Spectral sequence collapses at E2 for suspension spectra.

Provides a method to compute E2-term via non-abelian derived functors.

Constructs a multiplicative spectral sequence for cofibration sequences.

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 $H^*(X)$ 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.