Applications depuis K(Z/p, 2) et une conjecture de N. Kuhn

TL;DR

This paper proves a conjecture by N. Kuhn on the mod p cohomology of spaces as unstable modules over the Steenrod algebra, introducing a new method applicable across all characteristics and fixing gaps in previous work.

Contribution

It provides a new, characteristic-independent proof of Kuhn's conjecture, extending known results from characteristic 2 to all primes and correcting earlier incomplete proofs.

Findings

Confirmed Kuhn's conjecture for all primes.

Developed a new method applicable in any characteristic.

Fixed gaps in previous proofs for odd primes.

Abstract

On d\'emontre une conjecture due \`a N. Kuhn concernant la cohomologie singuli\`ere \`a coefficients mod p des espaces, comme module instable sur l'alg\`ebre de Steenrod. Notre d\'emonstration de ce r\'esultat, d\'ej\`a connu en caract\'eristique 2, fait appel \`a une m\'ethode nouvelle, qui fonctionne en toute caract\'eristique. De cette mani\`ere on r\'etablit le r\'esultat de [S98] dont la preuve est incompl\`ete dans le cas d'un nombre premier impair. We settle a conjecture due to N. Kuhn about the mod p cohomology of spaces considered as unstable modules over the Steenrod algebra. This result is already known to hold in characteristic 2. The method presented here is essentially new and works for all characteristics. In doing so we fix a gap in [S98] concerning the odd prime case.