Max Karoubi

• Paris Diderot University

The purpose of this short paper is to investigate relations between various real K-theories. In particular, we show how a real projective bundle theorem implies an unexpected relation between Atiyah’s K R-theory and the usual equivariant K-theory of real vector bundles. This relation has been used recently in a new computation of the Witt group of...

The purpose of this short paper is to investigate relations between various real K-theories. In particular, we show how a real projective bundle theorem implies an unexpected relation between Atiyah's KR-theory and the usual equivariant K-theory of real vector bundles. This relation has been used recently in a new computation of the Witt group of r...

There is an appendix by M. Schlichting

We establish some structural results for the Witt and Grothendieck-Witt groups of schemes over $\mathbb{Z}[1/2]$, including homotopy invariance for Witt groups and a formula for the Witt and Grothendieck-Witt groups of punctured affine spaces over a scheme. All these results hold for singular schemes and at the level of spectra.

We introduce a version of the Brauer–Wall group for Real vector bundles of algebras (in the sense of Atiyah) and compare it to the topological analogue of the Witt group. For varieties over the reals, these invariants capture the topological parts of the Brauer–Wall and Witt groups.

Let $V$ be an algebraic variety defined over $\mathbb R$, and $V_{top}$ the space of its complex points. We compare the algebraic Witt group $W(V)$ of symmetric bilinear forms on vector bundles over $V$, with the topological Witt group $WR(V_{top})$ of symmetric forms on Real vector bundles over $V_{top}$ in the sense of Atiyah, especially when $V$...

We introduce a version of the Brauer--Wall group for Real vector bundles of algebras (in the sense of Atiyah), and compare it to the topological analogue of the Witt group. For varieties over the reals, these invariants capture the topological parts of the Brauer--Wall and Witt groups.

We introduce the notion of the "covering type" of a space, which is more subtle that the notion of Lusternik Schnirelman category. It measures the complexity of a space which arises from coverings by contractible subspaces whose non-empty intersections are also contractible.

We introduce the notion of the "covering type" of a space, which is more subtle that the notion of Lusternik Schnirelman category. It measures the complexity of a space which arises from coverings by contractible subspaces whose non-empty intersections are also contractible.

We introduce a general framework to unify several variants of twisted topological $K$-theory. We focus on the role of finite dimensional real simple algebras with a product-preserving involution, showing that Grothendieck-Witt groups provide interesting examples of twisted $K$-theory. These groups are linked with the classification of algebraic vec...

We introduce a general framework to unify several variants of twisted topological $K$-theory. We focus on the role of finite dimensional real simple algebras with a product-preserving involution, showing that Grothendieck-Witt groups provide interesting examples of twisted $K$-theory. These groups are linked with the classification of algebraic vec...

Let $V$ be an algebraic variety over $\mathbb R$. The purpose of this paper is to compare its algebraic Witt group $W(V)$ with a new topological invariant $WR(V_{\mathbb C})$, based on symmetric forms on Real vector bundles (in the sense of Atiyah) on the space of complex points of $V$, This invariant lies between $W(V)$ and the group $KO(V_{\mathb...

The main purpose of the present article is to establish the real case of "Karoubi's conjecture" in algebraic K-theory. The complex case was proved in 1990-91 by the second author and Andrei Suslin. Compared to the case of complex algebras, the real case poses additional difficulties. This is due to the fact that topological K-theory of real Banach...

As an application of our papers in hermitian K-theory, in favourable cases we prove the periodicity of hermitian K-groups with a shorter period than previously obtained. We also compute the homology and cohomology with field coeffcients of infinite orthogonal and symplectic groups of specific rings of integers in a number field.

Bott periodicity for the unitary, orthogonal and symplectic groups is fundamental to topological K-theory. Analogous to unitary topological K-theory, for algebraic K-groups with finite coefficients similar periodicity results are consequences of the Milnor and Bloch-Kato conjectures, affirmed by Voevodsky, Rost and others. More generally, we prove...

Let A be a commutative ring with 1/2 in A. In this paper, we define new characteristic classes for finitely generated projective A-modules V provided with a non degenerate quadratic form. These classes belong to the usual K-theory of A. They generalize in some sense the classical "cannibalistic" Bott classes in topological K-theory, when A is the r...

We offer here a more direct approach to twisted K-theory, based on the notion of twisted vector bundles (of finite or infinite dimension) and of twisted principal bundles. This is closeely related to the classical notion ot torsors and bundles of modules over an algebra bundle. Twisted K-theory is simply defined as the Grothendieck group of twisted...

Let X be a noetherian scheme of finite Krull dimension, having 2 invertible in its ring of regular functions, an ample family of line bundles, and a global bound on the virtual mod-2 cohomological dimensions of its residue fields. We prove that the comparison map from the hermitian K-theory of X to the homotopy fixed points of K-theory under the na...

Dans leurs recherches sur un théorème de l’indice pour les variétés à coins, Pierre-Yves Le Gall et Bertrand Monthubert [9] ont été amenés à étudier le groupe de K-théorie équivariante * n K ( R), où le groupe |n symétrique opère naturellement sur R | n n par permutation des coordonnées. Soit p le nombre de partitions de n du type n = λ +... + λ av...

Let us first consider an abeUan monoid M, i.e. a set provided with a composition law (denoted +) which satisfies all the properties of an abelian group except possibly the existence of inverses. Then we can associate an abelian group S(M) with M and a homomorphism of the underlying monoids s: M → S(M), having the following universal property. For a...

The purpose of this section is to define an isomorphism K ℂq(X) ≈ K ℂq(V), for any complex vector bundle V over a locally compact space X (note that K ℂq(V) ≈ K ℂq(B(V),S(V)), with respect to any metric on V; cf. II.5.12). For V trivial, we again obtain Bott periodicity in complex K-theory (cf. III.1.3 and III.2.1); however, Bott periodicity is...

Let k be the field of real numbers of the complex numbers1) and let X be a topological space.

In this appendix to R. Hazrat and N. Vavilov [J. K-Theory 4, No. 1, 1–65 (2009; Zbl 1183.19001)], we state a periodicity conjecture using form parameters. This conjecture contains as particular case the fundamental theorem in hermitian K-theory proved by M. Karoubi [Ann. Math. (2) 112, 259–282 (1980; Zbl 0483.18008)] but also some results of A. Bak...

The K-theory of a functor may be viewed as a relative version of the K-theory of a ring. After its description, we prove a Mayer–Vietoris exact sequence in this framework.In the case of a Galois extension of a number field F/L with rings of integers A, B respectively, this K-theory of the “norm functor” is an extension of a subgroup of the ideal cl...

We completely determine the 2-primary torsion subgroups of the hermitian K-groups of rings of 2-integers in totally real 2-regular number fields. The result is almost periodic with period 8. Moreover, the 2-regular case is precisely the class of totally real number fields that have homotopy cartesian “Bökstedt square”, relating the K-theory of the...

We introduce a new algebraic concept of a difierential graded al- gebra which is "almost" commutative (ADGQ, in French). We associate to any simplicial set X an ADGQ - called D(X) - and show how we can recover the homotopy type of the topological realization of X from this algebraic structure (assuming some flniteness conditions). The theory is su-...

Bott periodicity plays an important role in topological K-theory. The purpose of this paper is to extend the periodicity theorem in a discrete context, where all classical groups are involved and not just the general linear group. The present paper generalizes previous results of the author [K1] and [K2], where 2 was assumed to be invertible in the...

The purpose of this short paper is to make the link between the fundamental work of Atiyah, Bott and Shapiro [1] and twisted K-theory as defined by P. Donovan, J. Rosenberg and the author [2] [8] [7]. This link was implicit in the literature (for bundles over spheres as an example) but was not been explicitly defined before. The setting is the foll...

We give a necessary and sufficient condition for lagrangians in a symplectic vector bundle to be deformed stably into transversal lagrangians. In the case of three lagrangians, we show that the associated Grothendieck group can be identified with a Hermitian K-theory group.

Twisted K-theory has its origins in the author's PhD thesis [27] : http://www.numdam.org/item?id=ASENS_1968_4_1_2_161_0 and in the paper with P. Donovan http://www.numdam.org/item?id=PMIHES_1970__38__5_0 The objective of this paper is to revisit the subject in the light of generalizations and new developments inspired by Mathematical Physics. See f...

The K-theory of a functor may be viewed as a relative version of the K-theory of a ring. In the case of a Galois extension of a number field F/L with rings of integers A/B respectively, this K-theory of the "norm functor" is an extension of a subgroup of the ideal class group Cl(A) by the 0-Tate cohomology group with coefficients in A*. The Mayer-V...

This survey paper is an expanded version of lectures given at the Clay Mathematics Academy ; see http://www.claymath.org/programs/outreach/academy/colloquium2005.php These lectures were intended to very young (and motivated) college students with little background. Therefore, they are accessible to a mathematician of any speciality willing to under...

This is a survey paper, starting from the general notion of coordinate bundle taken from Steenrod. Its aim is to provide a motivation for the introduction of cyclic homology (and the closely related noncommutative de Rham cohomology) by Connes, Tsygan and the author. The bridge is made through a generalization of Chern-Weil theory, explained in a v...

We introduce a new morphism between algebraic and hermitian K-theory. The topological analog is the Adams operation in real K-theory. From this morphism, we deduce a lower bound for the higher algebraic K-theory of a ring A in terms of the classical Witt group of the ring A tensored by its opposite ring.

In this Note, using an idea due to Thomason, we define a "homology theory" on the category of rings which satisfies excision, exactness, homotopy (in the algebraic sense) and periodicity of order 4. For regular noetherian rings, we find Balmer's higher Witt groups. For more general rings, this homology is isomorphic to the KT-theory of Hombostel, i...

We prove that if the classical Baum-Connes conjecture in complex K-theory is true (for a given discrete group G), then the conjecture is also true in the real case (for the same group G). The essential ingredients of the proof are the descent theorem in topological K-theory (math.KT/0509396) and a Paschke duality for C*-algebras proved by John Roe...

Let G be a finite group acting on a finite dimensional real vector space V. We denote by P(V) the projective space associated to V. In this paper we compute in a very explicit way the rank of the equivariant complex K-theory of V and P(V), using previous results by Atiyah and the author. The interest of this computation comes from explicit formulas...

We use recent results proved by Berrick and the author (math.KT/0509404) to improve the periodicity theorem in hermitian K-theory. We define also a new filtration of the classical Witt ring W(A), built from non degenerate quadratic forms over any commutative ring A where 2 is invertible. This filtration is linked to the Milnor and Quillen K-groups....

Let A be a Banach algebra and A' its complexification. In this paper we show that the homotopy fixed point set of K(A'), the topological K-theory space of A', under complex conjugation is just K(A), the topological K-theory space of A. This result generalizes the well known fact that BO is BU^hZ/2. The proof uses in an essential way Atiyah's KR the...

Rognes and Weibel used Voevodsky's work on the Milnor conjecture to deduce the strong Dwyer-Friedlander form of the Lichtenbaum-Quillen conjecture at the prime 2. In consequence (the 2-completion of) the classifying space for algebraic K-theory of the integers ℤ[1/2] can be expressed as a fiber product of well-understood spaces BO and BGL(double-st...

We introduce a new algebraic concept of an algebra which is "almost" commutative (more precisely "quasi-commutative differential graded algebra" or ADGQ, in French). We associate to any simplicial set X an ADGQ - called D(X) - and show how we can recover the homotopy type of the topological realization of X from this algebraic structure (assuming s...

Let V be a smooth variety defined over the real numbers. Every algebraic vector bundle on V induces a complex vector bundle on the underlying topological space V(C), and the involution coming from complex conjugation makes it a Real vector bundle in the sense of Atiyah. This association leads to a natural map from the algebraic K-theory of V to Ati...

In this Note, we introduce a new morphism between algebraic and hermitian K-theory. The topological analog is the Adams operation ψ2 in real K-theory. From this morphism, we deduce a lower bound for the higher algebraic K-theory of a ring A in terms of the classical Witt group of the ring A⊗Aop. To cite this article: M. Karoubi, C. R. Acad. Sci. Pa...

This paper is devoted to classical Bott periodicity, its history and more recent extensions in algebraic and Hermitian K-theory. However, it does not aim at completeness. For instance, the variants of Bott periodicity related to bivariant K-theory are described by Cuntz in this handbook. As another example, we don’t emphasize here the relation betw...

Suppose that V is a quasiprojective variety, defined over the real numbers R. Complex conjugation defines an involution on the underlying topological space V (C) of complex points. Every algebraic vector bundle on V induces a complex vector bundle E on V (C), and conjugation gives E the structure of a Real vector bundle in the sense of Atiyah [A]....

In this paper we construct a twisted analog of the differential graded algebra of Kahler differential forms on a commutative algebra (provided by an endomorphism α). This construction generalizes the work done in (Contemp. Math. 279 (2001) 177–193) for topological purposes. The main feature of this twisted analog is a braiding which is the substitu...

The book (shool and conference proceedings) was edited by Max Karoubi,Adereni kuku, and Claudio Pedrini. and the three of us were the sole editors. It does not make sens to write Claudio Pedrini (The editors) since the three of us constitut the editors

In a previous Note [1], we suggested a quantum model of the unit interval [0,1], using convergent power series, parametrized by a variable q (a remarkable example is the quantum exponential, defined by Euler). In the present Note, we suggest a simpler model based on functions f=f(x):Z→k (with an arbitrary commutative ring k) which are constant when...

As it is well known in K-theory, stabilization of matrices enables them to commute “up to homotopy”. The purpose of this short paper is to describe an analogous philosophy for cochains on a space. It is in fact a direct application of a paper of Henri Cartan [1], together with a new idea of stabilization for cochains, similar to matrices. The appli...

Using quantum methods, we introduce here the notion of "neo-algebra" which generalizes the notion of a commutative differential graded algebra. Under some mild finiteness conditions, we can associate functorially to a space a neo-algebra over the finite field Fp: its quasi-isomorphism's class determines the p-adic-homotopy type of X. As a matter of...

Let k be an arbitrary commutative ring. We associate fonctorially to any simplicial set X a differential graded algebra Ŵ∗(X) with a globally defined braiding, which is an improvement of a previous work [3,4]. If k=Z and with some mild finiteness conditions on X, we show that the quasi-isomorphisms class of Ŵ∗(X) as a braided differential graded al...

Let A be an arbitrary ring. We introduce a Dennis trace map mod n, from K_1(A;Z/n) to the Hochschild homology group with coefficients HH_1(A;Z/n). If A is the ring of integers in a number field, explicit elements of K_1(A,Z/n) are constructed and the values of their Dennis trace mod n are computed. If F is a quadratic field, we obtain this way non...

Let A be a commutative unitary ring. We introduce a Dennis trace map mod n, from K-1 (A; Z/n) to Omega(dR)(1) (A)/(n), where Omega(dR)(1) (A) is the Kahler-de Rham module of differentials in A. If A is the ring of integers in a number field, explicit elements of K-1 (A; Z/n) are constructed and the values of their Dennis trace mod n are computed. I...

this paper is to compute the periodic locally finite cyclic cohomology HP

Using quantum methods, we introduce here the notion of “neo-algebra” which generalizes the notion of a commutative differential graded algebra. Under some mild finiteness conditions, we can associate functorially to a space a neo-algebra over the finite field Fp: its quasi-isomorphism's class determines the p-adic-homotopy type of X. As a matter of...

Le but de cet article est de compléter le travail commencé en [6] et [13] § VII, en relation avec les résultats postérieurs de Weibel [23] et Hood-Jones [10]. Il peut donc être considéré comme une suite naturelle de [6], [13], [23] et [10] dont nous utiliserons largement les méthodes. De manière plus précise, soit A une algèbre de Fréchet1 unitaire...

The purpose of the paper is to promote a new definition of cohomology, using the theory of non commutative differential forms, introduced already by Alain Connes and the author in order to study the relation between A"-theory and cyclic homology. The advantages of this theory in classical Algebraic Topology are the following: A much simpler multipl...

This article is about resettled Afghan Hazaras in Australia, many of whom are currently undergoing a complex process of transition (from transience into a more stable position) for the first time in their lives. Despite their permanent residency status, we show how resettlement can be a challenging transitional experience. For these new migrants, w...

L''objet de cet article est de montrer comment laK-thorie multiplicative peut tre utilise pour dfinir de nouvelles classes caractristiques secondaires de fibrs vectoriels munis de structures supplmentaires, dans des contextes gomtriques varis. On y dveloppe des exemples relis des travaux antrieurs divers.The purpose of this paper is to show how mul...

Dans cet article nous dfinissons un nouveau foncteur MK(X), K-thorie multiplicative de X, dans lequel prennent leurs valeurs les classes caractristiques primaires et secondaires connues de fibrs vectoriels munis de structures supplementaires. Il s'agit notamment des classes caractristiques de fibrs plats, feuillets ou holomorphes ainsi que des rgul...

L'objectif essential de cet article est de dfinir un accouplement Ellp (A) Kp + 1 (A) ® \mathbbC* Ell^p (A) \times K_{p + 1} (A) \to \mathbb{C}^* o K(A) dsigne la K-thorie algbrique de A et o Ell p(A) est le groupe engendr par les modules de Fredholm de dimension p. Nous relions cet accouplement au dterminant de Fredholm et aux extensions centra...

In this volume the authors seek to illustrate how methods of differential geometry find application in the study of the topology of differential manifolds. Prerequisites are few since the authors take pains to set out the theory of differential forms and the algebra required. The reader is introduced to De Rham cohomology, and explicit and detailed...

This paper is a continuation of [4] where we computed the homology groups with coefficients of the infinite orthogonal and symplectic groups of an algebraically closed field F of characteristic ≠2 and 0.Since we have also proved in [4] that these homology groups depend only on the characteristic of F (if it is different from 2), in order to deal wi...

Conjecture : Les @roupes Kn(~ A), n e ~ , sont p@riodiques de pSriode 2 par rapport d net sont isomorphes aux groupes Kntopologique8 de l'algCbre de Banach A. Puisque~A n'a pas d'~l@ment unit~, les groupes Kn(~(~A) doivent 8tre interpr~t6s comme ceux figurant dans la suite exacte 0 = Kn+l( ~ ~ A) ~Kn+I(~A ) ~ Kn(~(~ A) > Kn( ~ ~ A) = 0 Donc Kn(~ A)...

CET ARTICLEcomplète sur certains points les théorèms de [5] et [7] et permet d'étendre à la K-théorie équivariante les résultats essentiels d'Atiyah, Bott et Shapiro [2]. De manière précise, soit G un groupe de Lie compact opérant sur un fibré vectoriel réel V de base compacte X et soit W un sous-fibré de V qui est invariant par l'action de G. Soie...

Without Abstract

In this section we will only consider complex K-theory which will be denoted simply by K(X), K(X,Y) etc. instead of K ℂ(X), K ℂ(X,Y), ....

Le but de cette théorie est d'assigner à un ensemble simplicial X un nouveau type de formes différentielles dites "tressées" et définies sur un anneau de base arbitraire (pas seulement R ou C). Leur premier intérêt est de fournir de nombreux exemples de représentations du groupe des tresses. Elles permettent aussi de définir une algèbre différentie...

The purpose of these notes is to give a feeling of "K-theory", a new interdisciplinary subject within Mathematics. This theory was invented by Alexander Grothendieck1 (BS) in the 50's in order to solve some difficult problems in Algebraic Geometry (the letter "K" comes from the German word "Klassen", the mother tongue of Grothendieck). This idea of...

The K-theory of a functor may be viewed as a relative version of the K-theory of a ring. After its description, we prove a Mayer-Vietoris exact sequence in this framework. In the case of a Galois extension of a number field F/L with rings of integers A, B respec- tively, this K-theory of the "norm functor" is an extension of a subgroup of the ideal...