arXiv subjects

Max Karoubi

Publications and source records attributed to Max Karoubi.

27 records · Page 2Linked to original sources

arXiv2005

Stabilization of the Witt group

Using an idea due to R.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 P. Balmer's higher Witt groups. For more general rings, this homology isomorphic to the KT-theory of J. Hornbostel, inspired by the work of B. Williams. For real or complex C*-algebras, we recover - up to 2 torsion - topological K-theory.

math.KT↗

arXiv2005

From algebraic K-theory to hermitian K-theory

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.

math.KT↗

arXiv2005

On the Baum-Connes conjecture in the real case

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 (Q.J. Math. 55 (2004) 325-331). According to S. Stolz, this result is linked with the existence of a Riemannian metric of positive scalar curvature on a compact connected spin manifold with G as fundamental group.

math.OA↗

arXiv2005

Periodicity of hermitian K-theory and Milnor's K-groups

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. Using the solution of Milnor's conjecture by Voevodsky, we show that the non triviality of Milnor's K-groups mod. 2 implies also the non triviality of higher Witt groups (when A is a field).

math.KT↗

arXiv2005

Algebraic and Real K-theory of Algebraic varieties

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 Atiyah's ``Real K-theory'' of V(C). Passing to finite coefficients Z/m, we show that the maps from K_n(V ; Z/m) to KR ^{-n}(V(C);Z/m) are isomorphisms when n is at least the dimension of V, at least when m is a power of two. Our key descent result is a comparison of the K-theory space of V with the homotopy fixed points (for complex conjugation) of the K-theory space of the complex variety V(C). When V is the affine variety of the d-sphere S, it turns out that KR*(V(C))=KO*(S). In this case we show that for all nonnegative n we have K_n(V ; Z/m) = KO^{-n}(S ; Z/m).

math.KT↗

arXiv2005

A descent theorem in topological K-theory

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 theory and the Clifford algebra definition of higher K-groups.

math.KT↗

arXiv2005

Quasi-commutative cochains in algebraic topology

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 some finiteness conditions). The theory is sufficiently general to include also ringed spaces X. The construction is by itself interesting since it uses the difference calculus (instead of the differential calculus of Sullivan's theory) and a new type of tensor product, called "reduced tensor product". Although we dont have a minimal model yet, it is relatively easy to define the cup i-products and the Steenrod operations on the category of ADGQ's. Homotopy groups can be deduced from the "iterated Hochschild homology" of D(X). The determination of the homotopy type from this algebraic structure uses in an essential way recent results of M. Mandell.

math.AT↗

arXiv2001

Twisted Kähler differential forms

We construct a braided structure on the algebra of Kähler differential forms of a commutative algebra twisted by an endomorphism. This generalises the construction done in M. Karoubi, Quantum Methods in Algebraic Topology, see http://www.math.jussieu.fr/~karoubi, for topological purposes.

math.AT↗

arXiv2000

Quelques classes caracteristiques en theorie des nombres

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 trivial elements of the ideal class group of A. If F is a cyclotomic field, this trace is closely related to Kummer logarithmic derivatives; this trace leads to an unexpected relationship between the first case of Fermat last theorem, K-theory and the number of roots of Mirimanoff polynomials.

math.NT↗