Volume 3, issue 1 (2003)

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Espaces profinis et problèmes de réalisabilité

Francois-Xavier Dehon and Gerald Gaudens

Algebraic & Geometric Topology 3 (2003) 399–433
 arXiv: math.AT/0306271
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
Mathematical Subject Classification 2000
Primary: 55S10
Secondary: 55T20, 57T35