Lionel Schwartz

Sur la structure des A-modules instables injectifs

Notre resultat est que ce dernier exemple donne essentiellement tous les @-injectifs: nous montrons que tout @-injectif est isomorphe a une somme directe de A-modules instables de la forme L @ J(n), L designant un facteur direct indecomposable de H * (iZ lF,), m et n parcourant N. Pour un n lF,) est %!-injective; il montre aussi par les memes mtthodes que H * (B(Z/p)m; F,) est @-injectif (au moins pour m = 1, ce qui suffisait a son propos). Dans [ 163, S. Zarati et le premier auteur etudient les produits tensoriels de @-injectifs et montrent notamment que H *(B(Z/p)“; F,) @I J(n) est %-injectif. L’intCrCt des @-injectifs tient a ce que la a!-injectivite de la cohomologie modulo p dun espace X donne beaucoup d’informations sur l’ensemble [X, Y] des classes d’homotopie d’applications de X dans Y pour une large classe d’espaces Y (classe dont la taille varie en fonction d’autres proprietes de X). Rappelons que la %-injectivite de H * (B(E/p); F,,) joue un role essentiel dans la solution par Miller de la conjecture de Sullivan [18]. Les @-injectifs interviennent aussi, dans la conjecture de Segal pour les p-groupes abtliens llementaires [6], [16], [17], [26], [13], dans la solution de la conjecture de Sullivan “gtneraliste” donnee dans [13], et dans la thtorie des spectres de Brown-Gitler [18], [14], [9]. Voici le sommaire de l’article:

On the Kummer congruences and the stable homotopy of U

We study the torsion-free part of the stable homotopy groups of the space BU, by considering upper and lower bounds. The upper bound is furnished by the ring PK. (BU) of coaction primitives into which nrs(BU) is mapped by the complex K-theoretic Hurewicz homomorphism Xs(BU) -* PK*(BU). We characterize PK* (B U) in terms of symmetric numerical polynomials and describe systematic families of elements by utilizing the classical Kummer congruences among the Bernoulli numbers. For a lower bound we choose the ring of those framed bordism classes which may be represented by singular hypersurfaces in BU. From among these we define families of classes constructed from regular neighborhoods of embeddings of iterated Thom complexes in Euclidean space. Employing techniques of duality theory, we deduce that these two families correspond, except possibly in the lowest dimensions, under the Hurewicz homomorphism, which thus provides a link between the algebra and the geometry. In the course of this work we greatly extend certain e-invariant calculations of J. F. Adams.

A propos de conjectures de Serre et Sullivan

SummaryWe first give a new proof of a conjecture of J.-P. Serre on the homotopy of finite complexes, which was recently proved by C. McGibbon and J. Neisendorfer. The emphasis is on a property of the mod. 2 homology of certain spaces: their “quasi-boundedness” as right modules over the Steenrod algebra. This property is preserved when one goes from a simply connected space to its loop space and also when one takes a covering of anH-space. Then we show that this notion of quasi-boundedness simplifies H. Millers proof of D. Sullivans conjecture on the contractibility of the space of pointed maps from the classifying space of the groupe ℤ/2 into a finite complex.

Unstable Modules Over the Steenrod Algebra, Functors, and the Cohomology of Spaces

This report presents the connections discovered in the 80’s and the 90’s between homotopy theory, specifically unstable modules over the Steenrod algebra and homotopy classes of maps from classifying spaces, and certain categories of functors. These categories of functors are deeply linked to modular representation theory of symmetric or general linear groups.

Sur Les Groupes D’Homotopie Des Espaces Dont La Cohomologie Modulo 2 Est Nilpotente

The main object of this note is to prove the following generalisation of a theorem of Serre. A simply connected space of finite type whose mod. 2 cohomology is nilpotent (and non-trivial) has infinitely many homotopy groups which are not of odd torsion. Incidentally we show that for every fibrationF(→ί)E (→p)B, satisfying certain mild conditions, the following holds. If a classx in the mod. 2 cohomology ofE belongs to the kernel ofi*, then some power ofx belongs to the ideal generated by the image underp* of the mod. 2 reduced cohomology ofB.

Une propriété de la filtration du degré des foncteurs polynomiaux

Resume On montre que la filtration du degre sur les foncteurs polynomiaux, de la categorie des espaces vectoriels sur le corps F 2 dans elle meme, dont le socle est fini est compatible, en un sens approprie, avec la fitration des socles, dite de Loewy. Pour demontrer ce resultat on se ramene a en montrer un equivalent pour la fitration par le poids (cf. [1]) sur la cohomologie modulo 2 des 2-groupes abeliens elementaires et celle des socles obtenue en la considerant comme objet dans la categorie U / N il . Pour citer cet article : L. Piriou, L. Schwartz, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 587–590.

IndecomposableA-module summands inH*V which are unstable algebras

Letp be a prime and denote byA the modp Steenrod algebra. We determine the indecomposableA-module summands ofH*((ℤ/p))d;Fp which admit the structure of an unstableA-algebra. In fact, it turns out that this is equivalent to the problem of determining those indecomposableA-module summands which arise as the modp cohomology of a space (or even a classifying space of a finite group). We reduce this problem to one in modular representation theory, namely for whichd andp is the projective cover of the trivial one dimensional GL(d,Fp) representationFp a permutation module. Our solution of this latter problem makes use of the classification of subgroups of GL(d,Fp) acting transitively on (Fp)d\{0} and hence depends on the classification of finite simple groups (on Feit-Thompsons odd order theorem ifp=2).