arXiv subjects
Publications and source records attributed to Lionel Schwartz.
This note shows that the $n$-th homology suspension is surjective for certain quotients of finite $((Z/2)^n$-CW.complexes. This is true as soon as the equivariant $((Z/2)^n$-cohomology are quotients is a free $H^*$((Z/2)^n$-module. An application is given to certain Brown-Gitler spectrum.
The present note presents some results about the mod-2 cohomology, modulo nilpotent elements elements of the fiber E of a decomposable map $ψ$ : K(Z, 2) $\rightarrow$ K(Z/2, p). This is more an announcement and a brief description of the tools that are used: Lannes' T functor and the Eilenberg-Moore spectral sequence.
In the 1980's, the magic properties of the cohomology of elementary abelian groups as modules over the Steenrod algebra initiated a long lasting interaction between topology and modular representation theory in natural characteristic. The Adams-Gunawardena-Miller theorem in particular, showed that their decomposition is governed by the modular representations of the semi-groups of square matrices. Applying Lannes' T functor on the summands L P := Hom Mn(Fp) (P, H * (F p) n) defines an intriguing construction in representation theory. We show that T(L P) $\sim$ = L P $\oplus$ H * V 1 $\otimes$ L $δ$(P) , defining a functor $δ$ from F p [M n (F p)]-projectives to F p [M n--1 (F p)]-projectives. We relate this new functor $δ$ to classical constructions in the representation theory of the general linear groups.
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 $\Fp$-vector spaces. The second result in particular exhibits the power of the functorial approach.
This note gives a new construction of Takayasu's cofibrations.
This note is devoted to some questions about the representation theory over the finite field $\mathbb{F}_2$ of the general linear groups $\mathbb{GL_n(F_2)}$ and Poincaré series of unstable modules. The first draft was describing two conjectures. They were presented during talks made at VIASM in summer 2013. Since then one conjecture has been disproved, the other one has been proved. These results naturally lead to new questions which are going to be discussed. In winter 2013, Nguyen Dang Ho Hai proved the second conjecture, he disproved the first one in spring 2014. Up to now, the proof of the second one depends on a major topological result: the Segal conjecture. This discussion could be extended to an odd prime, but we will not do it here, just a small number of remarks will be made.
This note investigate some finiteness properties of the category U of unstable modules. One shows finiteness properties for the injective resolution of finitely generated unstable modules. One also shows a stabilization result under Frobenius twist for Ext-groups.
On démontre une conjecture due á N. Kuhn concernant la cohomologie singuliére á coefficients mod p des espaces, comme module instable sur l'algébre de Steenrod. Notre démonstration de ce résultat, déjá connu en caractéristique 2, fait appel á une m'ethode nouvelle, qui fonctionne en toute caracteristique. De cette maniére on rétablit un r'esultat de [S98] dont la preuve est incompléte 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.
On démontre une conjecture due à N. Kuhn concernant la cohomologie singulière à coefficients mod p des espaces, comme module instable sur l'algèbre de Steenrod. Notre démonstration de ce résultat, déjà connu en caractéristique 2, fait appel à une méthode nouvelle, qui fonctionne en toute caractéristique. De cette manière on rétablit le résultat de [S98] dont la preuve est incomplète 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.
In a preceding article the authors and Tran Ngoc Nam constructed a minimal injective resolution of the mod 2 cohomology of a Thom spectrum. A Segal conjecture type theorem for this spectrum was proved. In this paper one shows that the above mentioned resolutions can be realized topologically. In fact there exists a family of cofibrations inducing short exact sequences in mod 2 cohomology. The resolutions above are obtained by splicing together these short exact sequences. Thus the injective resolutions are realizable in the best possible sense. In fact our construction appears to be in some sense an injective closure of one of Takayasu. It strongly suggests that one can construct geometrically (not only homotopically) certain dual Brown-Gitler spectra. Contents
One constructs minimal injective resolutions for certain unstable modules that appears to be the mod 2 cohomology of Thom spectra. The terms of the resolution are tensor products of Brown-Gitler modules and Steinberg modules introduced by S. Mitchell and S. Priddy. A combinatorial result of Andrews shows that the alternating sum of the Poincare series of the considered modules is zero. One gives homotopical applications of this result.
Paper written in French -- English abstract: This paper proves a particular case of a conjecture of N. Kuhn. This conjecture is as follows. Consider the Gabriel-Krull filtration of the category U of unstable modules. Let U_n, n>=0, be the n-th step of this filtration. The category U is the smallest thick sub-category that contains all sub-categories U_n and is stable under colimit [L. Schwartz, Unstable modules over the Steenrod algebra and Sullivan's fixed point set conjecture, Chicago Lectures in Mathematics Series (1994)]. The category U_0 is the one of locally finite modules, i.e. the modules that are direct limit of finite modules. The conjecture is as follows, let X be a space then : * either H^*X is locally finite, * or H^*X does not belong to U_n, for all n. As an example the cohomology of a finite space, or of the loop space of a finite space are always locally finite. On the other side the cohomology of the classifying space of a finite group whose order is divisible by 2 does belong to any sub-category U_n. One proves this conjecture, modulo the additional hypothesis that all quotients of the nilpotent filtration are finitely generated. This condition is used when applying N. Kuhn's reduction of the problem. It is necessary to do it to be allowed to apply Lannes' theorem on the cohomology of mapping spaces.[N. Kuhn, On topologically realizing modules over the Steenrod algebra, Ann. of Math. 141 (1995) 321-347].