We construct invariants of relative K-theory classes of multiparameter dependent pseudodifferential operators, which recover and generalize Melrose's divisor flow and its higher odd-dimensional versions of Lesch and Pflaum. These higher divisor flows are obtained by means of pairing the relative K-theory modulo the symbols with the cyclic cohomological characters of relative cycles constructed out of the regularized operator trace together with its symbolic boundary. Besides giving a clear and conceptual explanation to the essential features of the divisor flows, namely homotopy invariance, additivity and integrality, this construction allows to uncover the previously unknown even-dimensional counterparts. Furthermore, it confers to the totality of these invariants a purely topological interpretation, that of implementing the classical Bott periodicity isomorphisms in a manner compatible with the suspension isomorphisms in both K-theory and in cyclic cohomology. We also give a precise formulation, in terms of a natural Clifford algebraic suspension, for the relationship between the higher divisor flows and the spectral flow.

Let X be a geometrically irreducible smooth projective curve defined over a field k. Assume that X has a k–rational point; fix a k–rational point x ε X. From these data we construct an affine group scheme X defined over the field k as well as a principal X–bundle over the curve X. The group scheme X is given by a ℚ–graded neutral Tannakian category built out of all strongly semistable vector bundles over X. The principal bundle is tautological. Let G be a linear algebraic group, defined over k, that does not admit any nontrivial character which is trivial on the connected component, containing the identity element, of the reduced center of G. Let EG be a strongly semistable principal G–bundle over X. We associate to EG a group scheme M defined over k, which we call the monodromy group scheme of EG, and a principal M–bundle EM over X, which we call the monodromy bundle of EG. The group scheme M is canonically a quotient of X, and EM is the extension of structure group of . The group scheme M is also canonically embedded in the fiber Ad(EG)x over x of the adjoint bundle.

Calmès and Fasel have shown that the twisted Witt groups of split flag varieties vanish in a large number of cases. For flag varieties over algebraically closed fields, we sharpen their result to an if-and-only-if statement. In particular, we show that the twisted Witt groups vanish in many previously unknown cases. In the non-zero cases, we find that the twisted total Witt group forms a free module of rank one over the untwisted total Witt group, up to a difference in grading.

Our proof relies on an identification of the Witt groups of flag varieties with the Tate cohomology groups of their K-groups, whereby the verification of all assertions is eventually reduced to the computation of the (twisted) Tate cohomology of the representation ring of a parabolic subgroup.

We study the K′-theory of a CM Henselian local ring R of finite Cohen-Macaulay type. We first describe a long exact sequence involving the groups K′i(R) and the K-groups of certain other rings, including the Auslander algebra. By examining the terms and maps in the sequence, we obtain information about K′(R).

We give a brief introduction to two fundamental papers by Daniel Quillen appearing in the Annals, 1971. These papers established the foundations of equivariant cohomology and gave a qualitative description of the cohomology of an arbitrary finite group. We briefly describe some of the influence of these seminal papers in the study of cohomology and representations of finite groups, restricted Lie algebras, and related structures.

For every regular cardinal α, we construct a cofibrantly generated Quillen model structure on a category whose objects are essentially dg categories which are stable under suspensions, cosuspensions, cones and α-small sums.

Using results of Porta, we show that the category of well-generated (algebraic) triangulated categories in the sense of Neeman is naturally enhanced by our Quillen model category.

This article has two purposes. In [15] we showed that the FIC (Fibered Isomorphism Conjecture for pseudoisotopy functor) for a particular class of 3-manifolds (we denoted this class by C) is the key to prove the FIC for 3-manifold groups in general. And we proved the FIC for the fundamental groups of members of a subclass of C. This result was obtained by showing that the double of any member of this subclass is either Seifert fibered or supports a nonpositively curved metric. In this article we prove that for any M ε C there is a closed 3-manifold P such that either P is Seifert fibered or is a nonpositively curved 3-manifold and π1(M) is a subgroup of π1(P). As a consequence it is obtained that the FIC is true for any B-group (see definition 4.2 in [15]). Therefore, the FIC is true for any Haken 3-manifold group and hence for any 3-manifold group (using the reduction theorem of [15]) provided we assume the Geometrization conjecture. The above result also proves the FIC for a class of 4-manifold groups (see [14]).

The second aspect of this article is to relax a condition in the definition of strongly poly-surface group ([13]) and define a new class of groups (we call them weak strongly poly-surface groups). Then using the above result we prove the FIC for any virtually weak strongly poly-surface group. We also give a corrected proof of the main lemma of [13].

We generalize the construction of the category of 1–motives with torsion in [2] as well as the construction of the category of 1–motivic sheaves Shv1 in [1] to perfect fields k (without inverting the exponential characteristic). For k transcendental over the prime field we extend a result in [1] showing that and Shv1 have equivalent bounded derived categories.

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].

In this paper we prove the p-adic analogue of a result of Hamida [11], namely that the p-adic Borel regulator introduced by Huber and Kings for the K-theory of a p-adic number field equals Karoubi's p-adic regulator up to an explicit rational factor.

Let G be a finite group and [G] its integral group ring. We prove that the twisted nil groups N([G]) vanish for all i ≤ 1 for G a finite group of square-free order.

Let G be a locally compact group acting smoothly and properly by isometries on a complete Riemannian manifold M, with compact quotient G\M. There is an assembly map which associates to any G-equivariant K-homology class on M, an element of the topological K-theory of a suitable Banach completion of the convolution algebra of continuous compactly supported functions on G. The aim of this paper is to calculate the composition of the assembly map with the Chern character in entire cyclic homology . We prove an index theorem reducing this computation to a cup-product in bivariant entire cyclic cohomology. As a consequence we obtain an explicit localization formula which includes, as particular cases, the equivariant Atiyah-Segal-Singer index theorem when G is compact, and the Connes-Moscovici index theorem for G-coverings when G is discrete. The proof is based on the bivariant Chern character introduced in previous papers.

This paper displays model structures for the category of pro-objects in simplicial presheaves on an arbitrary small Grothendieck site. The first of these is an analogue of the Edwards-Hastings model structure for pro-simplicial sets, in which the cofibrations are monomorphisms and the weak equivalences are specified by comparisons of function complexes. Other model structures are built from the Edwards-Hastings structure by using Bousfield-Friedlander localization techniques. There is, in particular, an n-type structure for pro-simplicial presheaves, and also a model structure in which the map from a pro-object to its Postnikov tower is formally inverted.

Given a number field F and a prime number p; let Fn denote the cyclotomic extension with [Fn : F] = pn; and let denote its ring of integers. We establish an analogue of the classical Iwasawa theorem for the orders of K2i (){p}.

For a 2-category 2C we associate a notion of a principal 2C-bundle. For the 2-category of 2-vector spaces, in the sense of M.M. Kapranov and V.A. Voevodsky, this gives the 2-vector bundles of N.A. Baas, B.I. Dundas and J. Rognes. Our main result says that the geometric nerve of a good 2-category is a classifying space for the associated principal 2-bundles. In the process of proving this we develop powerful machinery which may be useful in further studies of 2-categorical topology. As a corollary we get a new proof of the classification of principal bundles. Another 2-category of 2-vector spaces has been proposed by J.C. Baez and A.S. Crans. A calculation using our main theorem shows that in this case the theory of principal 2-bundles splits, up to concordance, as two copies of ordinary vector bundle theory. When 2C is a cobordism type 2-category we get a new notion of cobordism-bundles which turns out to be classified by the Madsen–Weiss spaces.

Given two equivalent locally compact Hausdorff groupoids, We prove that the Bost conjecture with Banach algebra coefficients is true for one if and only if it is true for the other. This also holds for the Bost conjecture with C*- coefficients. To show these results, the functoriality of Lafforgue's KK-theory for Banach algebras and groupoids with respect to generalised morphisms of groupoids is established. It is also shown that equivalent groupoids have Morita equivalent L1-algebras (with Banach algebra coefficients).

Let X be an algebraic variety with an action of either the additive or multiplicative group. We calculate the additive invariants of X in terms of the additive invariants of the fixed point set, using a formula of Białynicki-Birula. The method is also generalized to calculate certain additive invariants for Chow varieties. As applications, we obtain results on the Hodge polynomial of Chow varieties in characteristic zero and the number of points for Chow varieties over finite fields. As applications, we obtain the l-adic Euler-Poincaré characteristic for the Chow varieties of certain projective varieties over a field of arbitrary characteristic. Moreover, we show that the virtual Hodge (p,0) and (0,q)-numbers of the Chow varieties and affine algebraic group varieties are zero for all p,q positive.

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.

We study cocoverings of triangulated categories, in the sense of Rouquier, and prove that for any regular cardinal α the condition of α-compactness, in the sense of Neeman, is local with respect to such cocoverings. This was established for ordinary compactness by Rouquier. Our result yields a new technique for proving that a given triangulated category is well-generated. As an application we describe the α-compact objects in the unbounded derived category of a quasi-compact and semi-separated scheme.

Let F be a field, let G = Gal(/F) be its absolute Galois group, and let R(G,k) be the representation ring of G over a suitable field k. In this preprint we construct a ring homomorphism from the mod 2 Milnor K-theory k*(F) to the graded ring grR(G,k) associated to Grothendieck's γ-filtration. We study this map in particular cases, as well as a related map involving the W-group of F, rather than G. The latter is an isomorphism in all cases considered.

Naturally this echoes the Milnor conjecture (now a theorem), which states that k*(F) is isomorphic to the mod 2 cohomology of the absolute Galois group G, and to the graded Witt ring grW(F).

The machinery developed to obtain the above results seems to have independent interest in algebraic topology. We are led to construct an analog of the classical Chern character, which does not involve complex vector bundles and Chern classes but rather real vector bundles and Stiefel-Whitney classes. Thus we show the existence of a ring homomorphism whose source is the graded ring associated to the corresponding K-theory ring KO(X) of the topological space X, again with respect to the γ-filtration, and whose target is a certain subquotient of H*(X, F2).

In order to define this subquotient, we introduce a collection of distinguished Steenrod operations. They are related to Stiefel-Whitney classes by combinatorial identities.