Abstract
The main object of this note is to prove the following generalisation of a theorem of Serre. A simply connected space of finite type whose mod. 2 cohomology is nilpotent (and non-trivial) has infinitely many homotopy groups which are not of odd torsion. Incidentally we show that for every fibrationF( ί→ )E ( p→ )B, satisfying certain mild conditions, the following holds. If a classx in the mod. 2 cohomology ofE belongs to the kernel ofi*, then some power ofx belongs to the ideal generated by the image underp* of the mod. 2 reduced cohomology ofB.
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Références
W. G. Dwyer,Strong convergence of the Eilenberg-Moore spectral sequence, Topology13 (1974), 255–265.
C. McGibbon and J. Neisendorfer,On the homotopy groups of a finite dimensional space, Comment. Math. Helv.59 (1984), 253–257.
S. Halperin, Y. Félix, J.-M. Lemaire et J. C. Thomas, Mod-p loop space homology, PUB-IRMA Lille, 1987.
J. Lannes,Sur la cohomologie modulo p des p-groupes abéliens élémentaires, Proc. Durham Symposium on Homotopy Theory 1985, L.M.S., Cambridge University Press, 1987, pp. 97–116.
J. Lannes,Sur les espaces fonctionnels dont la source est le classifiant d’un p-groupe abélien élémentaire, en préparation.
J. Lannes et L. Schwartz,A propos de conjectures de Serre et Sullivan, Invent. Math.83 (1986), 593–603.
J. Lannes et L. Schwartz,Sur la structure des A-modules instables injectifs, à paraître dans Topology.
J. Lannes et S. Zarati,Sur les foncteurs dérivés de la déstabilisation, Math. Z.194 (1987), 25–59.
H. R. Miller,The Sullivan conjecture on maps from classifying spaces, Ann. of Math.120 (1984), 39–87.
D. Quillen,The spectrum of an equivariant cohomology ring, I, II, Ann. of Math.94 (1971), 549–572, 573–602.
D. Rector,Steenrod operations in the Eilenberg-Moore spectral sequence, Comment. Math. Helv.45 (1970), 540–552.
L. Schwartz, La filtration nilpotente de la catégorieA et la cohomologie des espaces de lacets, Proc. Louvain la Neuve 1986, Lecture Notes in Math.1318, Springer-Verlag, Berlin, 1986.
J.-P. Serre,Cohomologie modulo 2des complexes d’Eilenberg-MacLane, Comment. Math. Helv.27 (1953), 198–232.
L. Smith,On Künneth theorem I. The Eilenberg-Moore spectral sequence, Math. Z.116 (1970), 94–140.
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Lannes, J., Schwartz, L. Sur Les Groupes D’Homotopie Des Espaces Dont La Cohomologie Modulo 2 Est Nilpotente. Israel J. Math. 66, 260–273 (1989). https://doi.org/10.1007/BF02765897
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DOI: https://doi.org/10.1007/BF02765897