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

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 ΩX. 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.

cohomology, infinite loop spaces, spectral sequence
Mathematical Subject Classification 2010
Primary: 18G40, 55P47
Received: 13 August 2015
Revised: 23 February 2016
Accepted: 7 March 2016
Published: 7 November 2016
Rune Haugseng
Department of Mathematical Sciences
University of Copenhagen
Universitetsparken 5
DK-2100 Copenhagen
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