Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-28T08:50:13.308Z Has data issue: false hasContentIssue false
  • English
  • Français

Daniel Quillen, the father of abstract homotopy theory

Published online by Cambridge University Press:  01 March 2013

Jean-Claude Thomas  and
Micheline Vigué-Poirrier
Jean-Claude Thomas
Affiliation:
LAREMA, UMR CNRS 6093, Université d'Angers, 2, Boulevard Lavoisier, 49045 Angers, [email protected]
Micheline Vigué-Poirrier
Affiliation:
LAGA, UMR CNRS 7539, Université de Paris 13, Institut Galilée, 93430 Villetaneuse, [email protected]
Get access

Abstract

In this short paper we try to describe the fundamental contribution of Quillen in the development of abstract homotopy theory and we explain how he uses this theory to lay the foundations of rational homotopy theory.

Keywords

Model categoryQuillen functorsHomotopical algebraRational homotopy type55-0155P6255P65
Type
Research Article
Information
Journal of K-Theory , Volume 11 , Issue 3: The Legacy of Daniel Quillen , June 2013 , pp. 479 - 491
Copyright
Copyright © ISOPP 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Adams, J. F. and Hilton, P. J., On the cochain algebra of a loop space, Comment. Math. Helvetici 30 (1956) 305–330.Google Scholar
2.Allday, C. and Halperin, S., Lie group actions on spaces of finite ranks, Quart. J. Math. Oxford 29 (1978) 69–76.Google Scholar
3.Anick, D., A counterexample to a conjecture of Serre, Annals of Mathematics 115 (1982) 1–33 and 116 (1982) 661.Google Scholar
4.Anick, D., Non commutative graded algebras and their Hilbert series, Journal of Algebra 78 (1982) 120–140.Google Scholar
5.Anick, D., Hopf algebras up to homotopy, Journal of the American Mathematical Society 2 (1989) 417–453.Google Scholar
6.Anick, D. J., Inert sets and the Lie algebra associated to a group, Journal of Algebra 111 (1987) 154–165.Google Scholar
7.Anick, D. J., Differential algebras in topology. Research Notes in Mathematics 3. A K Peters, Ltd., Wellesley, MA, 1993.Google Scholar
8.Anick, D. J. and Halperin, S., Commutative rings, algebraic topology, graded Lie algebras and the work of Jan-Erik Roos, J. Pure Appl. Algebra 38 (1985) 103–109.Google Scholar
9.Aubry, M. and Lemaire, J.-M., Homotopie d'algèbres de Lie et de leurs algèbres enveloppantes, Lectures Notes in Mathematics 1318, Springer Verlag (1988) 26–30.Google Scholar
10.Avramov, L., Free Lie subalgebra of the cohomology of a local ring, Trans. Amer. Math. Soc. 270 (1982) 589–608.CrossRefGoogle Scholar
11.Barge, J., Structures différentiables sur les types d’homotopie rationnelle simplement connexes. Ann. Sci. École Norm. Sup. 9 (1976) 469–501.CrossRefGoogle Scholar
12.Baues, H. J., Algebraic homotopy theory, Cambridge University Press 1989.Google Scholar
13.Baues, H. J. and Lemaire, J.-M., Minimal models in homotopy theory, Math. Ann. 225 (1977) 219–242.CrossRefGoogle Scholar
14.Bousfield, A.K. and Gugenheim, V.K.A.M., On PL de Rham theory and rational homotopy type, Memoirs of the Amer. Math. Society 179 1976.Google Scholar
15.Buijs, U. and Murillo, A., The rational homotopy Lie algebra of function spaces, Comment. Math. Helv. 83 (2008) 723–739.CrossRefGoogle Scholar
16.Buijs, U., Félix, Y. and Murillo, A., Lie models for the components of sections of nilpotent fibrations, Trans. Amer. Math. Soc. 361 (2009) 5601–5614.Google Scholar
17.Buijs, U., Félix, Y. and Murillo, A., L∞-model for mapping spaces, Journal of the Math. Soc. of Japan, to appearGoogle Scholar
18.Cartan, H. and Eilenberg, S., Homological Algebra, Princeton University Press, 1956.Google Scholar
19.Chas, M. and Sullivan, D., String topology, preprint (1999).Google Scholar
20.Cisinski, D.-C.Les préfaisceaux commes modèles des types d'homotopie, Astérisque 308, Société Mathématique de France 2006.Google Scholar
21.Cohen, F. R., Moore, J. C. and Neisendorfer, J. A., Torsion in homotopy groups Ann. of Math. 109 (1979) 121–168.Google Scholar
22.Cohen, F. R., Moore, J. C. and Neisendorfer, J. A., The double suspension and exponent of the homotopy groups of spheres, Ann. of Math. 110 (1979) 549–565.Google Scholar
23.Silveira, Da, Rational homototy types of fibrations, Pacific J. Math. 113 (1984) 1–34.Google Scholar
24.Dupont, N., A counterexample to the Lemaire-Sigrist conjecture, Topology 13 (1974) 255–265.Google Scholar
25.Dwyer, W. G. and Kan, D.M., Function complex in homotopical algebra, Topology 19 (1980) 427–440.Google Scholar
26.Dwyer, W. G. and Spaliński, J., Homotopy theories and model categories, Handbook of algebraic topology, 73–126North-Holland, Amsterdam 1995.Google Scholar
27.Félix, Y. and Halperin, S., Rational LS category and its applications, Trans. Amer. Math. Soc. 273 (1982) 1–38.Google Scholar
28.Félix, Y., Halperin, S. and Thomas, J.-C., Rational Homotopy Theory, Graduate Texts in Mathematics 205, Springer-Verlag, 2000.Google Scholar
29.Félix, Y., Halperin, S. and Thomas, J.-C., Exponential growth and an asymptotic formula for the ranks of homotopy groups of a finite 1-connected complex, Annals of Mathematics 170 (2009) 443–464.Google Scholar
30.Félix, Y. and Thomas, J.-C., Homotopie rationnelle : Dualité et complémentarité des modèles, Bull. Soc. Math. Belgique 33 (1981) 7–19.Google Scholar
31.Félix, Y., Thomas, J.-C. and Vigué-Poirrier, M., Rational String Topology, J. Eur. Math. Soc. 9 (2006) 123–156.CrossRefGoogle Scholar
32.Gabriel, P. and Zisman, M., Calculus of fractions and homotopy theory, Ergenebenisse der Mathematik und ihrer Grenzgebeitz 35, Springer-Verlag New York 1967.Google Scholar
33.Gatsinzi, J.-B., LS-category of classifying spaces, Bull. Belg. Math. Soc. 2 (1995) 121–126.Google Scholar
34.Goerss, P. and Jardine, J. F., Simplicial homotopy theory, Progress in mathematics, Birhhäuser Verlag, Bassel, 1999.Google Scholar
35.Gromoll, D. and Meyer, W., Periodic geodesic on compact Riemannian manifold, J. of Differential Geom. 3 (1969) 493–510.Google Scholar
36.Griffiths, P.A. and Morgan, J.W., Rational homotopy theory and differential forms, Progress in Mathematics 16 Birkhäuser 1981.Google Scholar
37.Hess, K., A proof of Ganea's conjecture for rational spaces, Topology 30 (1991) 205–214.CrossRefGoogle Scholar
38.Hess, K., A history of rational homotopy theory, History of Topology, 757–796North-Holland, Amsterdam, 1999.CrossRefGoogle Scholar
39.Hirschhorn, P., Model categories and their localizations, Mathematical surveys and Monographs 99, American Mathematical society, Providence, RI, 2002.Google Scholar
40.Hovey, M., Models categories, Mathematical surveys and Monographs 63, American Mathematical society, Providence, RI, 1999.Google Scholar
41.Kahl, T., Lambrechts, P. and Vandembroucq, L., Bords homotopiques et modèles de Quillen, Homotopy, Homology and Applications 8 (2006) 1–28.Google Scholar
42.Kontsevich, M., Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003) 157–216.Google Scholar
43.Koszul, J.-L., Homologie et cohomologie des algèbres de Lie, Bull. Soc. Math. France 78 (1950) 65–127.Google Scholar
44.Lada, T. and Markl, M., Strongly homotopy Lie algebras, Comm. in Algebra 23 (1995) 2147–2161.Google Scholar
45.Lazarev, A., The Stasheff model of a simply-connected manifold and the string bracket, Proceedings of the American Mathematical Society 136 (2008) 735–745.Google Scholar
46.Lehmann, D., Théorie homotopique des formes différentielles, Astérisque 45, Soc. Math. France 1987.Google Scholar
47.Lemaire, J.-M., Algèbres connexes et homologie des espaces de lacets, Lectures Notes in Math. 422, Springer-Verlag 1974.Google Scholar
48.Lemaire, J.-M., “Autopsie d'un meutre” dans l'homologie d'une algèbre de chaînes, Ann. Scient. Ec. Norm. Sup. 11 (1978) 93–100.Google Scholar
49.Lemaire, J.-M., Sur le type d'homotopie des espaces de Ganéa, Astérisque 113–114 Soc. Math. de France (1984) 238–247.Google Scholar
50.Lemaire, J.-M. and Halperin, S., Suites inertes dans les algèbres de Lie graduées “Autopsie d'un meurte II”, Math. Scand 61 (1987) 39–67.Google Scholar
51.Lemaire, J.-M. and Thomas, J.-C. (eds) Homotopie algébrique et Algèbre locale, Astérisque 113–114, Soc. Math. de France 1984.Google Scholar
52.Löfwall, C. and Roos, J.-E., Cohomologie des algèbres de Lie graduée et série de Poincaré-Betti non rationnelles, C. R. Acad. Sc. Paris (1980) 733–736.Google Scholar
53.Lupton, G. and Smith, S. B., Rationalized evaluation subgroups of a map II: Quillen models and adjoint maps, J. Pure Appl. Algebra 209 (2007) 173–188.Google Scholar
54.Lupton, G. and Smith, S. B., Criteria for components of a function space to be homotopy equivalent, Math. Proc. Camb. Philos. Soc. 145 (2008) 95–106.Google Scholar
55.Majewski, M., A proof of the Baues-Lemaire conjecture in rational homotopy theory, Rend. Circ Mat. Palermo Suppl. 30 (1993) 113–123.Google Scholar
56.Maltsiniotis, G., La théorie de l'homotopie de Grothendieck, Astérisque 301, Société Mathématique de France 2005.Google Scholar
57.Miller, T., Minimal Lie algebras in rational homotopy theory, Thesis University Notre Dame 1976.Google Scholar
58.Munkolm, H.J., DGA-algebras as Quillen model category, J. Pure and Applied Algebra 13 (1978) 221–232.CrossRefGoogle Scholar
59.Oukili, A., Sur l'homologie d'une algèbre de Lie differentielle, Thesis University of Nice 1980.Google Scholar
60.Quillen, D. G., Homotopical algebra, Lecture Notes in Mathematics 43, Springer-Verlag, Berlin-New York 1967.Google Scholar
61.Quillen, D. G., Rational homotopy theory, Ann. of Math. (2) 90 (1969) 205–295.Google Scholar
62.Quillen, D. G., On the (co)homology of commutative rings, Proc. Symp. Pure Math. (1970) 65–87.Google Scholar
63.Retakh, V.S., Massey operations in Lie super algebras and differentials in the Quillen spectral sequence Funct. Analysis and its Applications 12 (1978) 319–321.CrossRefGoogle Scholar
64.Rothenberg, M. and Triantafillou, G., On the classification of G-manifolds up to finite ambiguity Comm. Pure Appl. Math. 44 (1991) 761–788.Google Scholar
65.Salvatore, P., Rational homotopy nilpotency of self-equivalences, Topology Appl. 77 (1997) 37–50.Google Scholar
66.Sbaï, M., Cocatérorie rationnelle d’un espace topologique, Astérisque 113–114, Soc. Math. de France (1984) 288–291.Google Scholar
67.Schlessinger, M. and Stasheff, J., Deformation theory and rational homotopy types, Preprint 1980.Google Scholar
68.Roos, J. E. (ed.), Algebra, Algebraic Topology and their interactions, Lecture Notes in mathematics 1183 1985.Google Scholar
69.Tanré, D., Homotopie rationnelle: modèles de Chen, Quillen, Sullivan, Lecture Notes in Mathematics 1025, Springer-Verlag, Berlin, 1983.Google Scholar
70.Sullivan, D., Infinitesimal computations in topology, Publ. IHES 47 (1977) 269–331.Google Scholar
71.Vigué-Poirrier, M. and Sullivan, D., The homology theory of the closed geodesic problem, Journal of Differential Geometry 11 (1976) 633–644.Google Scholar