Espaces profinis et problèmes de réalisabilité

Dehon, Francois-Xavier¹ ; Gaudens, Gerald²

¹Laboratoire J.A. Dieudonné, Université de Nice Sophia-Antipolis, Parc Valrose, BP 2053, 06101 Nice, France
²Laboratoire Jean Leray (UMR 6629 du C.N.R.S.), Université de Nantes, BP 92208, 44322 Nantes Cedex 3, France

Algebraic and Geometric Topology, Tome 3 (2003) no. 1, pp. 399-433

Résumé

The mod p cohomology of a space comes with an action of the Steenrod Algebra. L. Schwartz [A propos de la conjecture de non realisation due a N. Kuhn, Invent. Math. 134, No 1, (1998) 211–227] proved a conjecture due to N. Kuhn [On topologicaly realizing modules over the Steenrod algebra, Annals of Mathematics, 141 (1995) 321–347] stating that if the mod p cohomology of a space is in a finite stage of the Krull filtration of the category of unstable modules over the Steenrod algebra then it is locally finite. Nevertheless his proof involves some finiteness hypotheses. We show how one can remove those finiteness hypotheses by using the homotopy theory of profinite spaces introduced by F. Morel [Ensembles profinis simpliciaux et interpretation geometrique du foncteur T, Bull. Soc. Math. France, 124 (1996) 347–373], thus obtaining a complete proof of the conjecture. For that purpose we build the Eilenberg–Moore spectral sequence and show its convergence in the profinite setting.

Détail

DOI : 10.2140/agt.2003.3.399

Keywords: Steenrod operations, nilpotent modules, realization, Eilenberg–Moore spectral sequence, profinite spaces

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