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

TL;DR

This paper proves Kuhn's conjecture on the mod p cohomology of spaces as unstable modules over the Steenrod algebra, using a new method applicable to all characteristics, and addresses gaps in previous work for odd primes.

Contribution

It introduces a novel method to prove Kuhn's conjecture that works in all characteristics, completing previous incomplete proofs for odd primes.

Findings

Proves Kuhn's conjecture for all primes.

Introduces a new method applicable in any characteristic.

Completes the proof for odd prime cases.

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 caracteristique. De cette mani\'ere on r\'etablit un 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.