Algebraic & Geometric Topology

Formes différentielles généralisées sur une opérade et modèles algébriques des fibrations

David Chataur

Full-text: Open access


We construct functors of generalized differential forms. In the case of nilpotent spaces of finite type, they determine the weak homotopy type of the spaces. Moreover they are equipped, in an elementary and natural way, with the action of cup-i products. Working with commutative algebras up to homotopy (viewed as algebras over a cofibrant resolution of the operad of commutative algebras), we show using these functors that the model of the fiber of a simplicial map is the cofiber of the algebraic model of this map.

Resum é

On construit des foncteurs de formes différentielles généralisées. Ceux-ci, dans le cas d’espaces nilpotents de type fini, déterminent le type d’homotopie faible des espaces. Ils sont munis, d’une manière élémentaire et naturelle, de l’action de cup-i produits. Pour les algèbres commutatives à homotopit prés (algèbres sur une résolution cofibrante de l’opérade des algèbres commutatives), on démontre en utilisant les formes différentielles généralisées que le modèle de la fibre d’une application simpliciale est la cofibre du modèle de ce morphisme.

Article information

Algebr. Geom. Topol., Volume 2, Number 1 (2002), 51-93.

Received: 17 October 2001
Accepted: 1 February 2002
First available in Project Euclid: 21 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 18D50: Operads [See also 55P48]
Secondary: 55P43: Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.) 55P48: Loop space machines, operads [See also 18D50] 55T99: None of the above, but in this section

modèles algébriques formes différentielles opérades suites spectrales


Chataur, David. Formes différentielles généralisées sur une opérade et modèles algébriques des fibrations. Algebr. Geom. Topol. 2 (2002), no. 1, 51--93. doi:10.2140/agt.2002.2.51.

Export citation


  • H.J. Baues, M. Jibladze, A. Tonks: Cohomology of monoids in monoidal categories, Operads: Proceedings of Renaissance Conferences (Hartford, CT; Luminy, 1995), 137–165, Contemp. Math., 202, Amer. Math. Soc., Providence, RI, 1997.
  • C. Berger, B. Fresse: Combinatorial operad actions on cochains, preprint (2001).
  • J.M. Boardman, R.M. Vogt: Homotopy invariant structures on topological spaces, Lecture Notes in Math., Springer-Verlag.
  • A.K. Bousfield, V.K.A.M. Gugheneim: On P.L. de Rham theory and rational homotopy type, Memoirs A.M.S, t.8, 179 (1976).
  • H. Cartan: Théories cohomologiques, Inventiones Mathematicae 3, 172–178 (1967).
  • A. Dold: Ueber die Steenrodschen Kohomologieoperationen, Ann. of Math (2) 73, 258–294 (1961).
  • N. Dupont, K. Hess: Noncommutative algebraic models for fiber squares, Math. Ann. 314, no. 3, 449–467 (1999).
  • A. Dress: Zur Spectralsequenz von Faserungen, Inventiones Mathematicae 35, 261–271 (1976).
  • W.G. Dwyer, J. Spalinski: Homotopy theory and model categories, Handbook of algebraic topology 73–126 (North-Holland) (1995).
  • S. Eilenberg, S. Maclane: Relations between homology and homotopy groups, Ann. of Math. 46, 480–509 (1945).
  • S. Eilenberg, S. Maclane: Acyclic models, Amer. J. of Math. 79, 189–199 (1953).
  • Y. Felix, S. Halperin, J.C. Thomas: Differential graded algebras in topology, Handbook of algebraic topology 829–865 (North Holland) (1995).
  • Y. Felix, S. Halperin, J.C. Thomas: Rational Homotopy Theory, Graduate Texts in Mathematics 205, Springer Verlag (2000).
  • B. Fresse: Cogroups in algebras over an operad are free algebras, Comment. Math. Helv. 73, 637–676 (1998).
  • E. Getzler, J.D.S Jones: Operads, homotopy algebra and iterated integrals for double loop spaces, preprint (1994).
  • V. Ginzburg, M. Kapranov: Koszul duality for operads, Duke J.Math.(1)76, 203–272 (1994).
  • P.P. Grivel, Formes différentielles et suites spectrales, Ann. de Inst. Fourier (Grenoble) 29, 17–37 (1979).
  • S. Halperin: Lectures on minimal models, Mémoire de la S.M.F. 9/10 (1983).
  • V. Hinich: Homological algebra of homotopy algebras, Comm. Algebra 25 no. 10, 3291–3323 (1997).
  • V. Hinich: Virtual operad algebras and realization of homotopy types, J. Pure Appl. Algebra 159, no. 2–3, 173–185, (2001).
  • V. Hinich, V. Schechtmann: Homotopy limits of homotopy algebras, in “K-theory: algebra, geometry, arithmetic", Lecture Notes in Math. 1289, 240–264.
  • P.S. Hirschorn: Localization of models categories, Preprint (1998).
  • M. Hovey; Model categories, Mathematical Surveys and Monographs, 63. American Mathematical Society, 1999.
  • M. Karoubi: Formes différentielles non commutatives et cohomologie à coefficients arbitraires, Transactions of the A.M.S 347, 4277–4299 (1995).
  • M. Karoubi: Formes différentielles non commutatives et opérations de Steenrod, Topology 34, 699–715 (1995).
  • I. Kriz, J.P. May: Operads, algebras, modules and motives, Astérisque, 233 (1995).
  • D. Lehmann: Théorie homotopique des formes différentielles, Astérisque, 45 (1977).
  • J.L. Loday: La renaissance des opérades, Séminaire Bourbaki 1994–1995, Astérisque, 236 47–74 (1996).
  • J. McCleary: A user's guide to spectral sequences, Mathematics Lecture series 12, Publish or Perish (1985).
  • M. Majewski: Rational homotopical models and uniquess, Memoir of the American Mathematical Society, 143 (2000).
  • M. Mandell: $E_\infty$-algebras and p-adic homotopy theory, Topology (1) 40, 43–94 (2001).
  • M. Mandell: Cochains and homotopy type, preprint (2001).
  • M. Markl: Models for operads, Comm.Algebra.(24)4, 1471–1500 (1996).
  • J.P. May: The geometry of iterated loop spaces, Lecture Notes in Math., Springer-Verlag 271 (1972).
  • J.P. May: A general algebraic approach to Steenrod operations, Lecture Notes in Math. 168 153–231.
  • D. Quillen: Homotopical Algebra, Lecture Notes in Math.
  • N. Spaltenstein: Resolutions of unbounded complexes, Compositio Mathematica 65 (1988) 121–154.
  • E.H. Spanier: Algebraic Topology, McGraw-Hill series in higher mathematics (1966).
  • D. Stanley: Determining closed model category structure, Preprint (1998).
  • J. Smith: Iterating the cobar construction, Mem. Amer. Math. Soc. 109, 524 (1994).
  • V.A. Smirnov: Homotopy theory of coalgebras, Izv. Akad. Nauk SSSR Ser. Mat. 49, $n^0$6, 1302–1321 (1985).
  • V.A. Smirnov: Lie algebras over operads and their application to homotopy theory, Izv. Math. 62, $n^0$3, 549–580 (1998).
  • D. Sullivan: Infinitesimal computations in topology, Publ. I.H.E.S. 47 (1977), 269–331.
  • R. Swan: Thom's theory of differential forms on simplicial sets, Topology 14, 271–273 (1975).