On a spectral sequence for the cohomology of infinite loop spaces

Haugseng, Rune; Miller, Haynes

doi:10.2140/agt.2016.16.2911

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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

DOI: 10.2140/agt.2016.16.2911

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 $H^{*} (X)$ as a module over the Steenrod algebra, and it converges to the cohomology of $Ω^{\infty} X$ . 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

References

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

Volume 16, issue 5 (2016)