Algebraic and Geometric Topology 3 (2003), paper no. 13, pages 399-433.

Espaces profinis et problemes de realisabilite

Francois-Xavier Dehon and Gerald Gaudens

Abstract. 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.

Keywords. Steenrod operations, nilpotent modules, realization, Eilenberg-Moore spectral sequence, profinite spaces

AMS subject classification. Primary: 55S10. Secondary: 55T20, 57T35.

DOI: 10.2140/agt.2003.3.399

E-print: arXiv:math.AT/0306271

Submitted: 29 November 2002. (Revised: 3 May 2003.) Accepted: 14 January 2003. Published: 8 May 2003.

Notes on file formats

Francois-Xavier Dehon and Gerald Gaudens

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


AGT home page

EMIS/ELibM Electronic Journals

Outdated Archival Version

These pages are not updated anymore. They reflect the state of 21 Apr 2006. For the current production of this journal, please refer to