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
In this work, we begin studying the classification, up to isomorphism, of unstable H*V-A-modules E such that the tensor product of F_2 and E over H*V is isomorphic to a given unstable A-module.
Unstable modules over the Steenrod algebra with only the top $k$ operations are introduced in the language of ringoids. We prove the category of such modules has homological dimension at most $k$. A pratical method, which generalizes the $Λ$ complex, to compute the $\mathrm{Ext}$ group from such modules to spheres is proposed. We are also able to establish several functors to relate such modules and unstable modules over the Steenrod algebra, and to describe the connections between the $\mathrm{Ext}$ groups in them.
A new and natural description of the category of unstable modules over the Steenrod algebra as a category of comodules over a bialgebra is given; the theory extends and unifies the work of Carlsson, Kuhn, Lannes, Miller, Schwartz, Zarati and others. Related categories of comodules are studied, which shed light upon the structure of the category of unstable modules at odd primes. In particular, a category of bigraded unstable modules is introduced; this is related to the study of modules over the motivic Steenrod algebra.
Let A be a DGA over a field and X a module over H_*(A). Fix an $A_\infty$-structure on H_*(A) making it quasi-isomorphic to A. We construct an equivalence of categories between A_{n+1}-module structures on X and length n Postnikov systems in the derived category of A-modules based on the bar resolution of X. This implies that quasi-isomorphism classes of A_n-structures on X are in bijective correspondence with weak equivalence classes of rigidifications of the first n terms of the bar resolution of X to a complex of A-modules. The above equivalences of categories are compatible for different v…
In this paper we describe and continue the study begun by the author, Jones, and Segal, of the homotopy theory that underlies Floer theory. In that paper the authors addressed the question of realizing a Floer complex as the celluar chain complex of a CW -spectrum or pro-spectrum, where the attaching maps are determined by the compactified moduli spaces of connecting orbits. The basic obstructions to the existence of this realization are the smoothness of these moduli spaces, and the existence of compatible collections of framings of their stable tangent bundles. In this note we describe a gen…
A relation between Schur algebras and Steenrod algebra is shown in [Hai10] where to each strict polynomial functor the author associates an unstable module. We show that the restriction of Hai's functor to the subcategory of strict polynomial functors of a given degree is fully faithfull.
We introduce a notion of unstable algebra over an operad in general characteristic. We show that the unstable algebra freely generated by an unstable module is itself a free algebra under suitable conditions. We introduce a family of '$q$-level' operads which allows us to identify unstable modules studied by Brown--Gitler, Miller and Carlsson to free unstable $q$-level algebras.
In the early 1990's, Lionel Schwartz gave a lovely characterization of the Krull filtration of U, the category of unstable modules over the mod p Steenrod algebra. Soon after, this filtration was used by the author as an organizational tool in posing and studying some topological nonrealization conjectures. In recent years the Krull filtration of U has been similarly used by Castellana, Crespo, and Scherer in their study of H--spaces with finiteness conditions, and Gaudens and Schwartz have given a proof of some of my conjectures. In light of these topological applications, it seems timely to …
Conjugation spaces are topological spaces equipped with an involution such that their fixed points have the same mod $2$ cohomology (as a graded vector space, a ring, and even an unstable algebra) but with all degrees divided by two, generalizing the classical examples of complex projective spaces under complex conjugation. Spaces which are constructed from unit balls in complex Euclidean spaces are called spherical and are very well understood. Our aim is twofold. We construct "exotic" conjugation spaces and study the realization question: which spaces can be realized as real loci, i.e., fixe…
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 th…
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 $\otimes$ …
In the category $\mathcal{P}_{d}$ of strict polynomial functors, the morphisms between extension groups induced by the Frobenius twist are injective. In \cite{Cuo14a}, the category $\mathcal{P}_{d}$ is proved to be a full sub-category of the category $\mathcal{U}$ of unstable modules \textit{via} Hai's functor. The Frobenius twist is extended to the category $\mathcal{U}$ but remains mysterious there. This article aims to study the Frobenius twist and its effects on the extension groups of unstable modules. We compute explicitly several extension groups and show that in these cases, the morphi…
Let K 0 (Fp GLn(Fp)-proj) denote the Grothendieck group of finitely generated pro-jective Fp GLn(Fp)-modules. We show that the algebra C $\otimes$ n$\ge$0 K 0 (Fp GLn(Fp)-proj) with multiplication given by induction functors, is a polynomial algebra. We explicit generators and their relation with Deligne-Lusztig characters.
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.
In this article, we give some conditions on the structure of an unstable module, which are satisfied whenever this module is the reduced cohomology of a space or a spectrum. First, we study the structure of the sub-modules of Sigma^sH^*(B(Z/2)^{oplus d};Z/2), i.e., the unstable modules whose nilpotent filtration has length 1. Next, we generalise this result to unstable modules whose nilpotent filtration has a finite length, and which verify an additional condition. The result says that under certain hypotheses, the reduced cohomology of a space or a spectrum does not have arbitrary large gaps …
We extend Schwede's work on the unstable global homotopy theory of orthogonal spaces and $\mathcal{L}$-spaces to the category of $*$-modules (i.e., unstable $S$-modules). We prove a theorem which transports model structures and their properties from $\mathcal{L}$-spaces to $*$-modules and show that the resulting global model structure for $*$-modules is monoidally Quillen equivalent to that of orthogonal spaces. As a consequence, there are induced Quillen equivalences between the associated model categories of monoids, which identify equivalent models for the global homotopy theory of $A_\inft…
The spectral sequence associated to the Arone-Goodwillie tower for the n-fold loop space functor is used to show that the first two non-trivial layers of the nilpotent filtration of the reduced mod 2 cohomology of a (sufficiently connected) space with nilpotent cohomology are comparable. This relies upon the theory of unstable modules over the mod 2 Steenrod algebra, together with properties of a generalized class of almost unstable modules which is introduced here. An essential ingredient of the proof is a non-vanishing result for certain extension groups in the category of unstable modules l…
Lannes' T-functor is used to give a construction of the Singer functor R_1 on the category U of unstable modules over the Steenrod algebra A. This leads to a direct proof that the composite functor Fix R_1 is naturally equivalent to the identity. Further properties of the functors R_1 are deduced, especially when applied to reduced and nilclosed unstable modules.
Let $V$ be an elementary abelian $2$-group and $X$ be a finite $V$-CW-complex. In this memoir we study two cochain complexes of modules over the mod2 Steenrod algebra $\mathrm{A}$, equipped with an action of $\mathrm{H}^{*}V$, the mod2 cohomology of $V$, both associated with $X$. The first, which we call the "topological complex", is defined using the orbit filtration of $X$. The second, which we call the "algebraic complex", is defined just in terms of the unstable $\mathrm{A}$-module $\mathrm{H}^*_V X$, the mod2 equivariant cohomology of $X$. Our study makes intensive use of the theory of un…
The algebra S of symmetric invariants over the field with two elements is an unstable algebra over the Steenrod algebra A, and is isomorphic to the mod two cohomology of BO, the classifying space for vector bundles. We provide a minimal presentation for S in the category of unstable A-algebras, i.e., minimal generators and minimal relations. From this we produce minimal presentations for various unstable A-algebras associated with the cohomology of related spaces, such as the BO(2^m-1) that classify finite dimensional vector bundles, and the connected covers of BO. The presentations then show …
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.
We provide a new algebraic description of the structure on the set of all unstable cohomology operations for a suitable generalised cohomology theory, E^*. Our description is as a graded and completed version of a Tall-Wraith monoid. The E^*-cohomology of a space X is a module for this Tall-Wraith monoid. We also show that the corresponding Hopf ring of unstable co-operations is a module for the Tall-Wraith monoid of unstable operations. Further examples are provided by considering operations from one theory to another.
We construct and examine the universal Toda bracket of a highly structured ring spectrum R. This invariant of R is a cohomology class in the Mac Lane cohomology of the graded ring of homotopy groups of R which carries information about R and the category of R-module spectra. It determines for example all triple Toda brackets of R and the first obstruction to realizing a module over the homotopy groups of R by an R-module spectrum. For periodic ring spectra, we study the corresponding theory of higher universal Toda brackets. The real and complex K-theory spectra serve as our main examples.
For a connected Noetherian unstable algebra $R$ over the mod $p$ Steenrod algebra, we prove versions of theorems of Duflot and Carlson on the depth of $R$, originally proved when $R$ is the mod $p$ cohomology ring of a finite group. This recovers the aforementioned results, and also proves versions of them when $R$ is the mod $p$ cohomology ring of a compact Lie group, a profinite group with Noetherian cohomology, a Kac--Moody group, a discrete group of finite virtual cohomological dimension, as well as for certain other discrete groups. More generally, our results apply to certain finitely ge…
We study H^*(P), the mod p cohomology of a finite p--group P, viewed as an Out(P)--module. In particular, we study the conjecture, first considered by Martino and Priddy, that, if e_S in Z/p[Out(P)] is a primitive idempotent associated to an irreducible Z/p[Out(P)]--module S, then the Krull dimension of e_SH^*(P) equals the rank of P. The rank is an upper bound by Quillen's work, and the conjecture can be viewed as the statement that every irreducible Z/p[Out(P)]--module occurs as a composition factor in H^*(P) with similar frequency. In summary, our results are as follows. A strong form of th…
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 c…
This is a sequel to the paper [Cas]. Here, we extend the methods of Farb-Wolfson using the theory of FI_G-modules to obtain stability of equivariant Galois representations of the etale cohomology of orbit configuration spaces. We establish subexponential bounds on the growth of unstable cohomology, and then use the Grothendieck-Lefschetz trace formula to obtain results on arithmetic statistics for orbit configuration spaces over finite fields. In particular, we show that the average value, across polynomials over F_q, of certain Gauss sums over their roots, stabilizes as the degree goes to inf…
The mod p cohomology of a space comes with an action of the Steenrod Algebra. L. Schwartz [A propos de la conjecture de non realisation due a N. Kuhn, Invent. Math. 134, No 1, (1998) 211--227] proved a conjecture due to N. Kuhn [On topologicaly realizing modules over the Steenrod algebra, Annals of Mathematics, 141 (1995) 321--347] stating that if the mod $p$ cohomology of a space is in a finite stage of the Krull filtration of the category of unstable modules over the Steenrod algebra then it is locally finite. Nevertheless his proof involves some finiteness hypotheses. We show how one can re…
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 ho…
