#### 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
##### Publication
Received: 13 August 2015
Revised: 23 February 2016
Accepted: 7 March 2016
Published: 7 November 2016
##### Authors
 Rune Haugseng Department of Mathematical Sciences University of Copenhagen Universitetsparken 5 DK-2100 Copenhagen Denmark http://sites.google.com/site/runehaugseng Haynes Miller Department of Mathematics Massachusetts Institute of Technology Building 2, Room 106 %Rm 2-237 this seems to be old office number 77 Massachusetts Avenue Cambridge, MA 02139-4307 United States http://math.mit.edu/~hrm