Dimension globale et classe fondamentale d'un espace
Annales de l'Institut Fourier, Volume 49 (1999) no. 1, p. 333-350

The Pontryagin algebra of a K-elliptic space satisfy the Auslander-Buchsbaum-Serre theorem. We give some characterizations of the K-elliptic spaces with H * (ΩS;K) of finite global dimension and with (S,K) in the Anick range. We also introduce an “ xt -odd” spectral sequence and complete the results obtained by A. Murillo in the rational case.

L’algèbre de Pontryagin d’un espace K-elliptique vérifie le théorème d’Auslander-Buchsbaum-Serre. Nous donnons ici plusieurs caractérisations des espaces K-elliptiques tels que gldim(H * (ΩS;K))< et lorsque (S,K) est dans le domaine d’Anick. Nous introduisons aussi une suite spectrale “impaire des xt ” et complétons les résultats obtenus par A. Murillo dans le cas rationnel.

@article{AIF_1999__49_1_333_0,
     author = {Rami, Youssef},
     title = {Dimension globale et classe fondamentale d'un espace},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {49},
     number = {1},
     year = {1999},
     pages = {333-350},
     doi = {10.5802/aif.1676},
     zbl = {0920.55009},
     mrnumber = {2000c:55012},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_1999__49_1_333_0}
}
Rami, Youssef. Dimension globale et classe fondamentale d'un espace. Annales de l'Institut Fourier, Volume 49 (1999) no. 1, pp. 333-350. doi : 10.5802/aif.1676. http://www.numdam.org/item/AIF_1999__49_1_333_0/

[1] D. Anick, Hopf algebras up to homotopy, J. Amer. Math. Soc., 2 (1989), 417-453. | MR 90c:16007 | Zbl 0681.55006

[2] L. Bisiaux, Depth and Toomer's invariant, à paraître dans, Topology and its Applications. | Zbl 0938.55013

[3] Y. Félix, La dichotomie elliptique-hyperbolique en homotopie rationnelle, Astérisque, 176 (1989). | MR 91c:55016 | Zbl 0691.55001

[4] Y. Félix and S. Halperin, Rational L-S category and its applications, Trans. Amer. Math. Soc., 273 (1982), 1-73. | MR 84h:55011 | Zbl 0508.55004

[5] Y. Félix, S. Halperin and J.-M. Lemaire and J.-C. Thomas, Mod p loop space homology, Invent. Math., 95 (1989), 247-262. | MR 89k:55010 | Zbl 0667.55007

[6] Y. Félix, S. Halperin and J.-M. Lemaire, The Ganea conjecture and the L-S category of Poincaré duality complexes, Preprint Univ. Nice (1997).

[7] Y. Félix, S. Halperin and J.-C. Thomas, The Homotopy Lie algebra for finite complexes, Publ. I.H.E.S., 56 (1983), 89-96. | Numdam

[8] Y. Félix, S. Halperin and J.-C. Thomas, Gorenstein spaces, Adv. in Maths, 71 (1988), 92-112. | MR 89k:55019 | Zbl 0659.57011

[9] Y. Félix, S. Halperin and J.-C. Thomas, Elliptic Hopf algebras, J. London. Math. Soc., (2) 43 (1991), 545-555. | MR 92i:57033 | Zbl 0755.57018

[10] Y. Félix, S. Halperin and J.-C. Thomas, Hopf algebres of polynomial growth, J. Algebra, 125 (1989), 408-417. | MR 90j:16021 | Zbl 0676.16008

[11] Y. Félix, S. Halperin and J.-C. Thomas, Rational Homotopy Theory, Preprint Université d'Angers (1997). | Zbl 0961.55002

[12] Y. Félix, S. Halperin and J.-C. Thomas, Hopf algebras and a counterexample to a conjecture of Anick, J. of Algebra, 169 (1994), 176-193. | MR 95j:16050 | Zbl 0814.16036

[13] Y. Félix, S. Halperin, C. Jacobson, C. Löfwall and J.-C. Thomas, The radical of the homotopy Lie algebra, Amer. J. Math., 110 (1988), 301-322. | MR 89d:55029 | Zbl 0654.55011

[14] W.H. Greub, S. Halperin and J.R. Vanstone, Connexions, Curvatures and Cohomology, Vol. III, Academic Press, New York, 1975. | Zbl 0372.57001

[15] S. Halperin, Finiteness in the minimal models of Sullivan, Trans. Amer. Math. Soc., 230 (1977), 173-199. | MR 57 #1493 | Zbl 0364.55014

[16] S. Halperin, Universal enveloping algebras and loop space homology, J. Pure Appl. Algebra, 83 (1992), 237-282. | MR 93k:55014 | Zbl 0769.57025

[17] S. Halperin and J.-M. Lemaire, Notion of category in differential algebra, in Algebraic Topology — Rational Homotopy, Lecture Notes in Mathematics, 1318 (1988), 138-154. | MR 89h:55023 | Zbl 0656.55003

[18] I. James, On category, in the sense of Lusternik-Schnirelmann, Topology, 17 (1978), 331-348. | Zbl 0408.55008

[19] A. Murillo, The evaluation map of some Gorenstien algebras, J. Pure. Appl. Algebra, 91 (1994), 209-218. | MR 94m:55012 | Zbl 0789.55011

[20] A. Murillo, The Top cohomology class of certain spaces, J. Pure. App. Algebra, 84 (1993), 209-214. | MR 93k:55015 | Zbl 0766.55007

[21] J.-P. Serre, Algèbre locale, Multiplicités, Lecture Notes in Mathematics, 11 (1975). | Zbl 0296.13018

[22] D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math., 47 (1978), 269-331. | Numdam | MR 58 #31119 | Zbl 0374.57002

[23] G.H. Toomer, Lusternik-Schnirelmann category and the Moore spectral sequence, Math. Z., 138 (1974), 123-143. | MR 50 #8509 | Zbl 0287.55007