A new model of smooth K0-theory ([5] [1]) is constructed, with the help of the total Chern class (contrary to the theories considered in ]1], [5], [12] and [13] which use the Chern character). The correspondence with the earlier model [1] is obtained by adapting, at the level of transgression forms, the usual formulae which express the Chern character in terms of the Chern classes and vice versa.

The advantage of this new model is that it allows constructing Chern classes with values in integral Chern-Simons characters in a natural way: this construction answers a question asked by U. Bunke [4].

We show that all pairings defined in the literature extending the Connes-Moscovici characteristic map in Hopf cyclic cohomology are isomorphic as natural transformations of derived double functors.

Let G be a locally compact group and ρ a non-unitary finite dimensional representation of G. We consider tensor products of ρ by some unitary representations of G in order to define two Banach algebras analogous to the group C*-algebras, C*(G) and C*r(G). We calculate the K-theory of such algebras for a large class of groups satisfying the Baum-Connes conjecture.

We construct explicit generators of the K-theory and K-homology of the coordinate algebras of ‘functions’ on the quantum projective spaces. We also sketch a construction of unbounded Fredholm modules, that is to say Diraclike operators and spectral triples of any positive real dimension.

In this paper we introduce a new notion of weight structure (w) for a triangulated category C; this notion is an important natural counterpart of the notion of t-structure. It allows extending several results of the preceding paper [Bon09] to a large class of triangulated categories and functors.

The heart of w is an additive category Hw ⊂ C. We prove that a weight structure yields Postnikov towers for any X ∈ ObjC (whose 'factors' Xi ∈ ObjHw). For any (co)homological functor H : C → A (A is abelian) such a tower yields a weight spectral sequenceT : H(Xi [j]) ⇒ H(X[i + j]); T is canonical and functorial in X starting from E2. T specializes to the usual (Deligne) weight spectral sequences for 'classical' realizations of Voevodsky's motives DMeffgm (if we consider w = wChow with Hw = Choweff ) and to Atiyah-Hirzebruch spectral sequences in topology.

We prove that there often exists an exact conservative weight complex functor C → K(Hw). This is a generalization of the functor t : DMeffgm → Kb(Choweff) constructed in [Bon09] (which is an extension of the weight complex of Gillet and Soulé). We prove that K0(C) ≅ K0(Hw) under certain restrictions.

We also introduce the concept of adjacent structures: a t-structure is adjacent to w if their negative parts coincide. This is the case for the Postnikov t-structure for the stable homotopy category SH (of topological spectra) and a certain weight structure for it that corresponds to the cellular filtration. We also define a new (Chow) t-structure tChow for DMeff_ ⊃ DMeffgm which is adjacent to the Chow weight structure. We have HtChow ≅ AddFun(Choweffop,Ab); tChow is related to unramified cohomology. Functors left adjoint to those that are t-exact with respect to some t-structures are weight-exact with respect to the corresponding adjacent weight structures, and vice versa. Adjacent structures identify two spectral sequences converging to C(X,Y): the one that comes from weight truncations of X with the one coming from t-truncations of Y (for X,Y ∈ ObjC). Moreover, the philosophy of adjacent structures allows expressing torsion motivic cohomology of certain motives in terms of the étale cohomology of their 'submotives'. This is an extension of the calculation of E2 of coniveau spectral sequences (by Bloch and Ogus).

We complete the calculation of the twisted cyclic homology of the quantised coordinate ring = ℂq [SL(2)] of SL(2) that we began in [14]. In particular, a nontrivial cyclic 3-cocycle is constructed which also has a nontrivial class in Hochschild cohomology and thus should be viewed as a noncommutative geometry analogue of a volume form.

We study the coniveau spectral sequence for quadrics defined by Pfister forms. In particular, we explicitly compute the motivic cohomology of anisotropic quadrics over ℝ, by showing that their coniveau spectral sequences collapse from the -term

For compact hyperbolic 3-manifolds we lift the Bloch invariant defined by Neumann and Yang to an integral class in K3(ℂ). Applying the Borel and the Bloch regulators, one gets back the volume and the Chern-Simons invariant of the manifold. We perform our constructions in stable homotopy theory, pushing a generalized orientation of the manifold directly into K-theory. On the way we give a purely homotopical construction of the Bloch-Wigner exact sequence which allows us to explain the ℚ/ℤ ambiguity that appears in the non-compact case.

The aim of these notes is to prove that any right exact functor between reasonable Waldhausen categories, that induces an equivalence at the level of homotopy categories, gives rise to a homotopy equivalence between the corresponding K-theory spectra. This generalizes a well known result of Thomason and Trobaugh. The ingredients, for this proof, are a generalization of the Waldhausen approximation theorem, and a simple combinatorial caracterization of derived equivalences. We also study simplicial localization of Waldhausen categories. We prove that a (homotopy) right exact functor induces an equivalence of homotopy categories if and only if it induces anequivalence of simplicial localizations. This allows to make the link with the K-theory of simplicial categories introduced by Toën and Vezzosi.

We describe structural properties of globally defined Mackey functors related to the stratification theory of algebras. We show that over a field of characteristic zero they form a highest weight category and we also determine precisely when this category is semisimple. This approach is used to show that the Cartan matrix is often symmetric and non-singular, and we are able to compute finite parts of it in some instances. We also develop a theory of vertices of globally defined Mackey functors in the spirit of group representation theory, as well as giving information about extensions between simple functors.

The topic of this paper is the construction of a finite set of generators for a subgroup of finite index in the unit group u(ℤG) of the integral group ring of a finite group G. The present paper is a continuation of earlier research by Bass and Milnor, Jespers and Leal, and Ritter and Sehgal who constructed such generators provided that the group G does not have a non-abelian fixed-point free epimorphic image and the rational group algebra ℚG does not have simple epimorphic images that are two-by-two matrices over either the rationals, a quadratic imaginary extension of the rationals or a non-commutative division algebra. In this paper we allow simple images of the type M2(ℚ). We will do so by introducing new additional generators using Farey symbols, which are in one to one correspondence with fundamental polygons of congruence subgroups of PSL2(ℤ). Furthermore, for each simple Wedderburn component M2(ℚ) of ℚG, the new generators give a free subgroup that is embedded in M2(ℤ).

We define generalized Atiyah-Patodi-Singer boundary conditions of product type for Dirac operators associated to C*-vector bundles on the product of a compact manifold with boundary and a closed manifold. We prove a product formula for the K-theoretic index classes, which we use to generalize the product formula for the topological signature to higher signatures.

We begin the paper with a simple formula for the second integral homology of a range of Artin groups. The formula is derived from a polytopal classifying space. We then introduce the notion of a twisted Artin group and obtain polytopal classifying spaces for a range of such groups. We demonstrate that these explicitly constructed spaces can be implemented on a computer and used in homological calculations.

This paper presents, by example, an index theory appropriate to algebras without trace. Whilst we work exclusively with Cuntz algebras the exposition is designed to indicate how to develop a general theory. Our main result is an index theorem (formulated in terms of spectral flow) using a twisted cyclic cocycle where the twisting comes from the modular automorphism group for the canonical gauge action on each Cuntz algebra. We introduce a modified K1-group for each Cuntz algebra which has an index pairing with this twisted cocycle. This index pairing for Cuntz algebras has an interpretation in terms of Araki's notion of relative entropy.