Algebraic & Geometric Topology

On a spectral sequence for the cohomology of infinite loop spaces

Rune Haugseng and Haynes Miller

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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.

Article information

Algebr. Geom. Topol., Volume 16, Number 5 (2016), 2911-2947.

Received: 13 August 2015
Revised: 23 February 2016
Accepted: 7 March 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 18G40: Spectral sequences, hypercohomology [See also 55Txx] 55P47: Infinite loop spaces

cohomology infinite loop spaces spectral sequence


Haugseng, Rune; Miller, Haynes. On a spectral sequence for the cohomology of infinite loop spaces. Algebr. Geom. Topol. 16 (2016), no. 5, 2911--2947. doi:10.2140/agt.2016.16.2911.

Export citation


  • A,K Bousfield, Operations on derived functors of nonadditive functors, unpublished manuscript, Brandeis University (1967)
  • A,K Bousfield, The localization of spectra with respect to homology, Topology 18 (1979) 257–281
  • A,K Bousfield, On the homology spectral sequence of a cosimplicial space, Amer. J. Math. 109 (1987) 361–394
  • A,K Bousfield, D,M Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics 304, Springer (1972)
  • H Cartan, Puissances divisées, from “Séminaire Henri Cartan, 1954/1955 (Exposé 7)”, volume 7, Secrétariat Mathématique, Paris (1955)
  • F,R Cohen, T,J Lada, J,P May, The homology of iterated loop spaces, Lecture Notes in Mathematics 533, Springer (1976)
  • A Dold, Homology of symmetric products and other functors of complexes, Ann. of Math. 68 (1958) 54–80
  • A Dold, D Puppe, Homologie nicht-additiver Funktoren: Anwendungen, Ann. Inst. Fourier Grenoble 11 (1961) 201–312
  • W,G Dwyer, Higher divided squares in second-quadrant spectral sequences, Trans. Amer. Math. Soc. 260 (1980) 437–447
  • W,G Dwyer, Homotopy operations for simplicial commutative algebras, Trans. Amer. Math. Soc. 260 (1980) 421–435
  • P,G Goerss, Unstable projectives and stable ${\rm Ext}$: with applications, Proc. London Math. Soc. 53 (1986) 539–561
  • P,G Goerss, On the André–Quillen cohomology of commutative ${\bf F}\sb 2$–algebras, Astérisque 186, Société Mathématique de France, Paris (1990)
  • P,G Goerss, T,J Lada, Relations among homotopy operations for simplicial commutative algebras, Proc. Amer. Math. Soc. 123 (1995) 2637–2641
  • A Grothendieck, Sur quelques points d'algèbre homologique, Tôhoku Math. J. 9 (1957) 119–221
  • P Hackney, Operations in the homology spectral sequence of a cosimplicial infinite loop space, J. Pure Appl. Algebra 217 (2013) 1350–1377
  • S,B Isaacson, Symmetric cubical sets, preprint (2009) (the published version at J. Pure Appl. Algebra 215 (2011) 1146–1173 does not include the relevant result)
  • D,M Kan, Functors involving c.s.s. complexes, Trans. Amer. Math. Soc. 87 (1958) 330–346
  • N Kuhn, J McCarty, The mod 2 homology of infinite loopspaces, Algebr. Geom. Topol. 13 (2013) 687–745
  • J Lannes, S Zarati, Sur les foncteurs dérivés de la déstabilisation, Math. Z. 194 (1987) 25–59
  • S Mac Lane, Categories for the working mathematician, 2nd edition, Graduate Texts in Mathematics 5, Springer (1998)
  • H Miller, A spectral sequence for the homology of an infinite delooping, Pacific J. Math. 79 (1978) 139–155
  • H Miller, The Sullivan conjecture on maps from classifying spaces, Ann. of Math. 120 (1984) 39–87
  • H Miller, Correction to [MillerSullivan?], Ann. of Math. 121 (1985) 605–609
  • G,M,L Powell, On the derived functors of destabilization at odd primes, Acta Math. Vietnam. 39 (2014) 205–236
  • D,G Quillen, Homotopical algebra, Lecture Notes in Mathematics 43, Springer (1967)
  • C Rezk, Every homotopy theory of simplicial algebras admits a proper model, Topology Appl. 119 (2002) 65–94
  • S Schwede, B,E Shipley, Algebras and modules in monoidal model categories, Proc. London Math. Soc. 80 (2000) 491–511
  • S Schwede, B Shipley, Equivalences of monoidal model categories, Algebr. Geom. Topol. 3 (2003) 287–334
  • J-P Serre, Cohomologie modulo $2$ des complexes d'Eilenberg–MacLane, Comment. Math. Helv. 27 (1953) 198–232
  • W,M Singer, Iterated loop functors and the homology of the Steenrod algebra, II: A chain complex for $\Omega \sp{k}\sb{s}M$, J. Pure Appl. Algebra 16 (1980) 85–97
  • W,M Singer, A new chain complex for the homology of the Steenrod algebra, Math. Proc. Cambridge Philos. Soc. 90 (1981) 279–292
  • J,M Turner, On simplicial commutative algebras with vanishing André–Quillen homology, Invent. Math. 142 (2000) 547–558