On non-realization results and conjectures of N. Kuhn

Nguyen The Cuong ¹   ; Gérald Gaudens ²   ; Geoffrey Powell ²   ; Lionel Schwartz ³

¹ LIA CNRS Formath Vietnam and Université Paris 13 93430 Villetaneuse, France
² LAREMA, UMR 6093 CNRS and Université d’Angers 49045 Angers, France
³ LAGA, UMR 7539 CNRS and Université Paris 13 93430 Villetaneuse, France

Fundamenta Mathematicae, Tome 234 (2016) no. 2, pp. 139-161

Résumé

We discuss two extensions of results conjectured by Nick Kuhn about the non-realization of unstable algebras as the mod-$p$ singular cohomology of a space, for $p$ a prime. The first extends and refines earlier work of the second and fourth authors, using Lannes’ mapping space theorem. The second (for the prime $2$) is based on an analysis of the $-1$ and $-2$ columns of the Eilenberg–Moore spectral sequence, and of the associated extension. In both cases, the statements and proofs use the relationship between the categories of unstable modules and functors between ${\mathbb F}_p$-vector spaces. The second result in particular exhibits the power of the functorial approach.

Détail

DOI : 10.4064/fm95-10-2015

Keywords: discuss extensions results conjectured nick kuhn about non realization unstable algebras mod p singular cohomology space prime first extends refines earlier work second fourth authors using lannes mapping space theorem second prime based analysis columns eilenberg moore spectral sequence associated extension cases statements proofs relationship between categories unstable modules functors between mathbb p vector spaces second result particular exhibits power functorial approach