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Algèbres quasi-commutatives et carrés de Steenrod
Published online by Cambridge University Press: 20 November 2018
Résumé
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Soit 
$k$ un corps de caractéristique 
$p$ quelconque. Nous définissons la catégorie des $k$-algèbres de cochaînes fortement quasi-commutatives et nous donnons une condition nécessaire et suffisante pour que l’algèbre de cohomologie à coefficients dans
${{\text{Z}}_{2}}$ d’un objet de cette catégorie soit un module instable sur l’algèbre de Steenrod à coefficients dans
${{\text{Z}}_{2}}$.
A tout c.w. complexe simplement connexe de type fini 
$X$ on associe une $k$-algèbre de cochaînes fortement quasi-commutative; la structure de module sur l’algèbre de Steenrod définie sur l’algèbre de cohomologie de celle-ci coïncide avec celle de
${{H}^{*}}(X;\,{{\text{Z}}_{2}})$.
Abstract
We define the category of strongly quasi-commutative cochain $k$-algebras, where $k$ is a field of any characteristic $p$. We give a necessary and sufficient condition which enables the cohomology algebra with
${{\text{Z}}_{2}}$-coefficients of an object in this category to be an unstable module on the
${{\text{Z}}_{2}}$-Steenrod algebra.
To each simply connected c.w. complex of finite type $X$ is associated a strongly quasi-commutative model and the module structure over the
${{\text{Z}}_{2}}$-Steenrod algebra defined on the cohomology of this model is the usual structure on
${{H}^{*}}(X;\,{{\text{Z}}_{2}})$.
Keywords
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- Copyright © Canadian Mathematical Society 1999
References
[1]
Adams, J. F. and Hilton, P. J., On the chain algebra of loop space.
Comment. Math. Helv. 30 (1955), 305–330.Google Scholar
[2]
Anick, D. J., Hopf algebras up to homotopy.
J. Amer. Math. Soc. (3)2 (1989), 417–451.Google Scholar
[3]
Benson, D. J., Representations and cohomology II: cohomology of groups and modules.
Cambridge Stud. Adv. Math. 31 .Google Scholar
[4]
Ndombol, Bitjong, Algèbres de cochaînes quasi-commutatives et fibrations algébriques. J. Pure Appl. Algebra 125 (1998), 261–273.Google Scholar
[6]
Bullet, S. R. and Macdonald, I. G., On the Adem relations.
Topology (3) 21 (1982), 329–332.Google Scholar
[5]
Bredon, G. E., Topology and geometry.
Graduate Texts in Math. 139 , Springer Verlag, 1993.Google Scholar
[8]
Halperin, S. and Lemaire, J. M., Notions of category in differential algebra.
Lecture Notes in Math. 1318 (1988), 138–154.Google Scholar
[9]
Karoubi, M., Formes différentielles non commutatives et cohomologie à coefficients arbitraires. Trans. Amer. Math. Soc. (11) 347 (1995), 4277–4299.Google Scholar
[10]
Karoubi, M., Formes différentielles non commutatives et opérations de Steenrod. Topology (3) 34 (1995), 699–715.Google Scholar
[11]
Karoubi, M., Formes topologiques non commutatives. Ann. Sci. E´ cole Norm. Sup. Se´r. 4 t. 28 fasc. 4(1995), 477–492.Google Scholar
[12]
Majewski, M., A cellular Lie algebramodel for spaces and its equivalence with the models of Quillen and Sullivan.
Thèse Freie Universit¨at Berlin, 1996.Google Scholar
[13]
Munkholm, H. J., The Eilenberg-Moore spectral sequence and strongly homotopy multiplicative maps. J. Pure Appl. Algebra 5 (1974), 1–50.Google Scholar
[14]
Steenrod, S. and Epstein, D., Cohomology operations. Ann. of Math. Stud. 50 , Princeton Univ. Press, Princeton, NJ
1962.Google Scholar
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