Around conjectures of N. Kuhn
Résumé
The following note discuss two ways to extend conjectures of N. Kuhn about the non realization of certain unstable algebras as the mod p singular cohomology of spaces. The first case (established for all primes p) extends and refines preceding work the second and the third author [GS13]. The second one (only for the prime 2) discuss another approach, not depending on Lannes' mapping space theorem and relies on an analysis of the columns −1 and −2 of the Eilenberg-Moore spectral sequence, and the triviality of the associated extension in the category of unstable modules. In both case the statements and proofs will use the links between the categories of unstable modules and functors. In both cases it would be possible to work entirely within the category of unstable modules, however it is much more cleaner and efficient to use both categories, thus this note is propaganda for this point of view.
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