Sur la realisation des modules instables

Details Sur la realisation des modules instables Sur la realisation des modules instables

2002-12-04

arxiv.org/abs/math/0212054v4

Algebraic and Geometric Topology 4 (2004) 151-175

Mathematics

Algebraic Topology

Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-10.abs.html: Version 4: pa

Scientific paper

In this article, we give some conditions on the structure of an unstable module, which are satisfied whenever this module is the reduced cohomology of a space or a spectrum. First, we study the structure of the sub-modules of Sigma^sH^*(B(Z/2)^{oplus d};Z/2), i.e., the unstable modules whose nilpotent filtration has length 1. Next, we generalise this result to unstable modules whose nilpotent filtration has a finite length, and which verify an additional condition. The result says that under certain hypotheses, the reduced cohomology of a space or a spectrum does not have arbitrary large gaps in its structure. This result is obtained by applying Adams' theorem on the Hopf invariant and the classification of the injective unstable modules. This work was carried out under the direction of L. Schwartz. Resume Dans cet article, on donne des restrictions sur la structure d'un module instable, qui doivent etre verifiees pour que celui-ci soit la cohomologie reduite d'un espace ou d'un spectre. On commence par une etude sur la structure des sous-modules de Sigma^sH^*(B(Z/2)^{oplus d};Z/2), i.e., les modules instables dont la filtration nilpotente est de longueur 1. Ensuite, on generalise le resultat aux modules instables dont la filtration nilpotente est de longueur finie, et qui verifient une condition supplementaire. Le resultat dit que sous certaines hypotheses, la cohomologie reduite d'un espace ou d'un spectre ne contient pas de lacunes de longueur arbitrairement grande. Ce resultat est obtenu par application du celebre theoreme d'Adams sur l'invariant de Hopf et de la classification des modules instables injectifs. Ce travail est effectue sous la direction de L. Schwartz.

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