# Espaces profinis et problèmes de réalisabilité de structures algébriques comme cohomologie d'un espace topologique

Abstract : \medskip\noindent {\it Abstract.~} We show that a translation into the framework of F. Morel's homotopy theory of profinite spaces of a method due to L. Schwartz leads to a proof at the prime $2$ of N. Kuhn's conjecture that asserts that the Steenrod algebra's action on the finite field cohomology of a space has to be either locally finite or non polynomial. To this end, we introduce an Eilenberg-Moore spectral sequence for profinite spaces (this part is joint work with F.-X. Dehon). \smallskip\noindent By construction, this spectral sequence is always convergent in the naive sense. We show that an analog of W. Dwyer's strong convergence theorem holds in the profinite case. To simplify some technical points, we prove that the homotopical algebra of profinite spaces is {\it proper}. \smallskip\noindent We also show the following algebraic version of N. Kuhn's conjecture~: a connected unstable algebra with non-nilpotent augmentation ideal has an infinite Loewy series 'up to nilpotent elements'. \smallskip\noindent We also prove that some unstable modules are not topologically realizable by elementary considerations.
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Theses
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https://tel.archives-ouvertes.fr/tel-00002406
Contributor : Gérald Gaudens <>
Submitted on : Monday, February 17, 2003 - 11:48:10 AM
Last modification on : Monday, March 25, 2019 - 4:52:05 PM
Long-term archiving on: : Tuesday, September 11, 2012 - 7:55:09 PM

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• HAL Id : tel-00002406, version 1

### Citation

Gérald Gaudens. Espaces profinis et problèmes de réalisabilité de structures algébriques comme cohomologie d'un espace topologique. Mathématiques [math]. Université de Nantes, 2002. Français. ⟨tel-00002406⟩

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