Effet d'un attachement cellulaire dans l'homologie de l'espace des lacets

Félix, Yves  ; Thomas, Jean-Claude

Annales de l'Institut Fourier, Tome 39 (1989) no. 1, pp. 207-224.

Résumé (VO)

Un des problèmes historiques de la théorie homotopique des espaces est de mesurer l’effet de l’attachement d’une cellule au niveau des groupes d’homotopie. Le problème de l’attachement inerte reste en particulier un problème ouvert. Nous donnons ici une réponse partielle à ce problème.

Abstract

One of the historical problem in homotopy theory consists to measure the effect of adding a cell in the homotopy groups. In particular, the problem of inert attachment remains an open problem. We give here a partial answer to this question.

Détail

DOI : 10.5802/aif.1164

Bibliographie

[1] I. Adams et P. Hilton, On the chain algebra of a loop space, Comment. Math. Helv., 30 (1955), 305-330. | Zbl

[2] D. Anick, A rational homotopy analog of Whitehead's problem, Proceedings of the 1983 Nordic Conference, Lecture Notes in Math., 1183, Springer Verlag (1983). | Zbl

[3] D. Anick, Non commutative graded algebras and their Hilbert series, J. Algebra, 78 (1982), 120-140. | Zbl | MR

[4] D. Anick, Homotopy exponent for spaces of category two, preprint (1987).

[5] D. Anick, Hopf algebra up to homotopy, preprint (1988). | Zbl

[6] H. Baues, Commutator calculus and groups of homotopy classes, London Math. Soc. Lecture Notes series 50, Cambridge University Press (1981). | Zbl | MR

[7] H. Baues, On the homotopy classification problem, 5, Homotopy theory of topological mapping cones, Bonn (1984).

[8] H. Baues et J.M. Lemaire, Minimal models in homotopy theory, Math. Ann. 225-3 (1977), 219-242. | Zbl | MR

[9] A.K. Bousfield et D.M. Kan, Homotopy limits, completions and localisations, Lecture Notes in Math. 304, Springer (1972). | Zbl

[10] Y. Felix, S. Halperin, C. Jacobson, C. Lofwall et J.C. Thomas, The radical of the homotopy Lie algebra, Amer. J. Math., 110 (1988), 301-322. | Zbl | MR

[11] Y. Felix et J.C. Thomas, Sur l'opération d'holonomie rationnelle. Lecture Notes in Math., 1183, Springer-Verlag (1983), 136-169. | Zbl | MR

[12] Y. Felix et J.C. Thomas, The fibre-cofibre construction and its applications, J.P.A.A., 53 (1988), 59-69. | Zbl | MR

[13] S. Halperin et J.M. Lemaire, Suites inertes dans les algèbres de Lie graduées, Math. Scand., 61 (1987), 39-67. | Zbl | MR

[14] S. Halperin et J.M. Lemaire, Notions of category in differential algebra, Lecture Notes in Math., 1318, Springer-Verlag (1988), 99-123. | Zbl | MR

[15] J.M. Lemaire, “Autopsie d'un meurtre” dans une algèbre de chaînes, Ann. Sc. E.N.S., 4-11 (1978), 93-100. | Zbl | mathdoc-id

[16] C.A. Mcgibbon et C.W. Wilkerson, Loop spaces of finite complexes at large primes, Proc. A.M.S., 96-4 (1986), 698-702. | Zbl | MR

[17] J.W. Milnor et J.C. Moore, On the structure of Hopf algebras, Ann. Math., 821 (1965), 211-264. | Zbl | MR

[18] J.C. Moore, Differential homological algebra, Proc. Int. Cong. Math., I (1970), 1-5. | Zbl | MR

[19] J.C. Moore et L. Smith, Hopf algebras and multiplicative fibrations, II, Amer. J. Math. | Zbl

[20] H. Scheerer, On rationalized H and Co-H spaces with an appendix on decomposable H and Co-H spaces, Manus. Math, 51 (1984), 63-87. | Zbl | MR

[21] J.H.C. Whitehead, On adding relations to homotopy groups, Ann. of Math., 2-42 (1941), 409-428. | Zbl | JFM

[22] G.W. Whitehead, Elements of homotopy theory, G.T.M., 61, Springer-Verlag (1978). | Zbl | MR