Algebraic & Geometric Topology

Higher cohomology operations and $R$–completion

David Blanc and Debasis Sen

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Let R be either F p or a field of characteristic 0 . For each R –good topological space  Y , we define a collection of higher cohomology operations which, together with the cohomology algebra H ( Y ; R ) , suffice to determine Y up to R –completion. We also provide a similar collection of higher cohomology operations which determine when two maps f 0 , f 1 : Z Y between R –good spaces (inducing the same algebraic homomorphism H ( Y ; R ) H ( Z ; R ) ) are R –equivalent.

Article information

Algebr. Geom. Topol., Volume 18, Number 1 (2018), 247-312.

Received: 5 September 2016
Revised: 4 May 2017
Accepted: 22 May 2017
First available in Project Euclid: 1 February 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55P60: Localization and completion
Secondary: 55N99: None of the above, but in this section 55P15: Classification of homotopy type 55P20: Eilenberg-Mac Lane spaces

higher cohomology operation Toda bracket cosimplicial resolution


Blanc, David; Sen, Debasis. Higher cohomology operations and $R$–completion. Algebr. Geom. Topol. 18 (2018), no. 1, 247--312. doi:10.2140/agt.2018.18.247.

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  • J F Adams, On the non-existence of elements of Hopf invariant one, Ann. of Math. 72 (1960) 20–104
  • H-J Baues, D Blanc, Comparing cohomology obstructions, J. Pure Appl. Algebra 215 (2011) 1420–1439
  • H-J Baues, D Blanc, S Gondhali, Higher Toda brackets and Massey products, J. Homotopy Relat. Struct. 11 (2016) 643–677
  • H-J Baues, M Jibladze, Suspension and loop objects in theories and cohomology, Georgian Math. J. 8 (2001) 697–712
  • G Biedermann, G Raptis, M Stelzer, The realization space of an unstable coalgebra, preprint (2014)
  • D Blanc, Higher homotopy operations and the realizability of homotopy groups, Proc. London Math. Soc. 70 (1995) 214–240
  • D Blanc, CW simplicial resolutions of spaces with an application to loop spaces, Topology Appl. 100 (2000) 151–175
  • D Blanc, Realizing coalgebras over the Steenrod algebra, Topology 40 (2001) 993–1016
  • D Blanc, Homotopy operations and rational homotopy type, from “Categorical decomposition techniques in algebraic topology” (G Arone, J Hubbuck, R Levi, M Weiss, editors), Progr. Math. 215, Birkhäuser, Basel (2004) 47–75
  • D Blanc, W G Dwyer, P G Goerss, The realization space of a $\Pi$–algebra: a moduli problem in algebraic topology, Topology 43 (2004) 857–892
  • D Blanc, M W Johnson, J M Turner, On realizing diagrams of $\Pi$–algebras, Algebr. Geom. Topol. 6 (2006) 763–807
  • D Blanc, M W Johnson, J M Turner, Higher homotopy operations and cohomology, J. K–Theory 5 (2010) 167–200
  • D Blanc, M W Johnson, J M Turner, Higher homotopy operations and André–Quillen cohomology, Adv. Math. 230 (2012) 777–817
  • D Blanc, M W Johnson, J M Turner, Higher invariants for spaces and maps, preprint (2015)
  • D Blanc, M Markl, Higher homotopy operations, Math. Z. 245 (2003) 1–29
  • D Blanc, G Peschke, The fiber of functors between categories of algebras, J. Pure Appl. Algebra 207 (2006) 687–715
  • D Blanc, D Sen, Mapping spaces and $R$–completion, preprint (2013)
  • F Borceux, Handbook of categorical algebra, II: Categories and structures, Encyclopedia of Mathematics and its Applications 51, Cambridge Univ. Press (1994)
  • A K Bousfield, Cosimplicial resolutions and homotopy spectral sequences in model categories, Geom. Topol. 7 (2003) 1001–1053
  • A K Bousfield, D M Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics 304, Springer (1972)
  • W Chachólski, J Scherer, Homotopy theory of diagrams, Mem. Amer. Math. Soc. 736, Amer. Math. Soc., Providence, RI (2002)
  • C Ehresmann, Esquisses et types des structures algébriques, Bul. Inst. Politehn. Iaşi 14 (1968) 1–14
  • Y Félix, Modèles bifiltrés: une plaque tournante en homotopie rationnelle, Canad. J. Math. 33 (1981) 1448–1458
  • Y Félix, S Halperin, J-C Thomas, Rational homotopy theory, Graduate Texts in Mathematics 205, Springer (2001)
  • P G Goerss, J F Jardine, Simplicial homotopy theory, Progress in Mathematics 174, Birkhäuser, Basel (1999)
  • J R Harper, Secondary cohomology operations, Graduate Studies in Mathematics 49, Amer. Math. Soc., Providence, RI (2002)
  • K Hess, Rational homotopy theory: a brief introduction, from “Interactions between homotopy theory and algebra” (L L Avramov, J D Christensen, W G Dwyer, M A Mandell, B E Shipley, editors), Contemp. Math. 436, Amer. Math. Soc., Providence, RI (2007) 175–202
  • P S Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs 99, Amer. Math. Soc., Providence, RI (2003)
  • F W Lawvere, Functorial semantics of algebraic theories, Proc. Nat. Acad. Sci. U.S.A. 50 (1963) 869–872
  • C R F Maunder, Cohomology operations of the $N^{\text{{th}}}$ kind, Proc. London Math. Soc. 13 (1963) 125–154
  • J P May, Simplicial objects in algebraic topology, Van Nostrand Mathematical Studies 11, Van Nostrand, Princeton, NJ (1967)
  • D G Quillen, Homotopical algebra, Lecture Notes in Mathematics 43, Springer (1967)
  • D Quillen, Rational homotopy theory, Ann. of Math. 90 (1969) 205–295
  • L Schwartz, Unstable modules over the Steenrod algebra and Sullivan's fixed point set conjecture, Univ. of Chicago Press (1994)
  • E Spanier, Secondary operations on mappings and cohomology, Ann. of Math. 75 (1962) 260–282
  • E Spanier, Higher order operations, Trans. Amer. Math. Soc. 109 (1963) 509–539
  • H Toda, Composition methods in homotopy groups of spheres, Annals of Mathematics Studies 49, Princeton Univ. Press (1962)
  • C A Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge Univ. Press (1994)