Bivariant K-theory via correspondences

We use correspondences to define a purely topological equivariant bivariant K-theory for spaces with a proper groupoid action. Our notion of correspondence differs slightly from that of Connes and Skandalis. Our construction uses no special features of equivariant K-theory. To highlight this, we construct bivariant extensions for arbitrary equivariant multiplicative cohomology theories.We formulate necessary and sufficient conditions for certain duality isomorphisms in the topological bivariant K-theory and verify these conditions in some cases, including smooth manifolds with a smooth cocompact action of a Lie group. One of these duality isomorphisms reduces bivariant K-theory to K-theory with support conditions. Since similar duality isomorphisms exist in Kasparov theory, the topological and analytic bivariant K-theories agree if there is such a duality isomorphism.     

Higher algebraic K-theory for actions of diagonalizable groups

We study the K-theory of actions of diagonalizable group schemes on noetherian regular separated algebraic spaces: our main result shows how to reconstruct the K-theory ring of such an action from the K-theory rings of the loci where the stabilizers have constant dimension. We apply this to the calculation of the equivariant K-theory of toric varieties, and give conditions under which the Merkurjev spectral sequence degenerates, so that the equivariant K-theory ring determines the ordinary K-theory ring. We also prove a very refined localization theorem for actions of this type. Mathematics Subject Classification (2000) 19E08, 14L30     

The homotopy limit problem for Hermitian K-theory, equivariant motivic homotopy theory and motivic Real cobordism

The homotopy limit problem for Karoubi?s Hermitian K-theory (Karoubi, 1980) [26] was posed by Thomason (1983) [44]. There is a canonical map from algebraic Hermitian K-theory to the Z/2-homotopy fixed points of algebraic K-theory. The problem asks, roughly, how close this map is to being an isomorphism, specifically after completion at 2. In this paper, we solve this problem completely for fields of characteristic 0 (Theorems 16, 20). We show that the 2-completed map is an isomorphism for fields F of characteristic 0 which satisfy cd₂(F[i])<∞, but not in general.     

Simplicial groups that are models for algebraic K-theory

In this paper, we present and compare some simplicial groups, functorially associated to a ring R, whose homotopy groups are Quillens K-groups of R. The first such simplicial group is the group (NQPR), where is the loop space construction of Clemens Berger, applied to the simplicial set NQPR (the nerve of Quillens category QPR). The second is a subgroup GR of the simplicial group (NQPR). This second group is compared to Kans construction [12] of a loop group for a connected simplicial set, and shown to be isomorphic to it as a simplicial group. Other simplicial groups that are models for algebraic K-theory are also presented; in particular, the subgroup G(s.PR) of (s.PR); here, s.PR is Waldhausens simplicial set [25], [26]. We initially give an exposition of Bergers construction in general; then, we present the construction of GR and a summary of Kans construction. Next, we point out that GR is an infinite loop object in the category of simplicial groups, and draw some corollaries. We then compare directly the homotopy groups thus constructed with the classical K-theory in degrees 0 and 1. The final section compares various models.     

The Leibniz formula in algebraic K-theory

The paper may be viewed as an addendum to a paper of Thomason and Throbaugh, where the K-theory of algebraic varieties is equipped with relative K-groups. It is proved that this enriched K-theory satisfies the Panin—Smirnov axioms for ring cohomology theories of algebraic varieties. In particular, it is proved that the Leibniz formula, describing the interaction between multiplication and differential, holds in this case. The language of symmetric spectra and of monoidal model categories is used. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 319, 2004, pp. 264–292.     

The K-theory of p-compact groups

In this paper, we show that the p-adic K-theory of a connected p-compact is the ring of invariants of the Weyl group action on the K-theory of a maximal torus. We apply this result to show that a connected finite loop space admits a maximal torus if and only if its complex K-theory is -isomorphic to the K-theory of some BG, where G is a compact connected Lie group. Received: November 9, 1996     

K-theory of Prüfer domains

Archiv der Mathematik - In this article, we show that the homotopy invariance of K-theory holds for rings of weak global dimension at most one. Prüfer domains are examples of such rings. We...     

Algebraic K-theory of generalized free products and functors Nil

In this paper, we extend Waldhausen's results on algebraic K-theory of generalized free products in a more general setting and we give some properties of the Nil functors. As a consequence, we get new groups with trivial Whitehead groups.     

On the K-theory of the stable C-algebras from substitution tilings

We describe a method to compute the K-theory of the C^?-algebra arising from the stable equivalence relation in the Smale space associated to a substitution tiling, and give detailed computations for one- and two-dimensional examples. We prove that for one-dimensional tilings the group K₀ is always torsion free and give an example of a two-dimensional tiling such that K₀ has torsion.     

Algebraic K-theory

One gives a survey of the fundamental methods and results of the algebraic K-theory obtained in the past decade. One presents the basic constructions of the K-theory of rings and of the K-theory of exact categories. A special attention is given to the K-theory of schemes.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 20, pp. 71–152, 1982.     

Semi-topological K-theory for certain projective varieties

In this paper we compute Lawson homology groups and semi-topological K-theory for certain threefolds and fourfolds. We consider smooth complex projective varieties whose zero cycles are supported on a proper subvariety. Rationally connected varieties are examples of such varieties. The computation makes use of different techniques of decomposition of the diagonal cycle, of the Bloch–Kato conjecture and of the spectral sequence relating morphic cohomology and semi-topological K-theory.     

Relatively spectral morphisms and applications to K-theory

Spectral morphisms between Banach algebras are useful for comparing their K-theory and their “noncommutative dimensions” as expressed by various notions of stable ranks. In practice, one often encounters situations where the spectral information is only known over a dense subalgebra. We investigate such relatively spectral morphisms. We prove a relative version of the Density Theorem regarding isomorphism in K-theory. We also solve Swan's problem for the connected stable rank, in fact for an entire hierarchy of higher connected stable ranks that we introduce.     

Equivariant eta forms and equivariant differential K-theory

In this paper, for a compact Lie group action, we prove the anomaly formula and the functoriality of the equivariant Bismut-Cheeger eta forms with perturbation operators when the equivariant family index vanishes. In order to prove them, we extend the Melrose-Piazza spectral section and its main properties to the equivariant case and introduce the equivariant version of the Dai-Zhang higher spectral flow for arbitrarydimensional fibers. Using these results, we construct a new analytic model of the equivariant differential K-theory for compact manifolds when the group action has finite stabilizers only, which modifies the Bunke-Schick model of the differential K-theory. This model could also be regarded as an analytic model of the differential Ktheory for compact orbifolds. Especially, we answer a question proposed by Bunke and Schick(2009) about the well-definedness of the push-forward map.     

K-theory and exact sequences of partial translation algebras

In an earlier paper, the authors introduced partial translation algebras as a generalisation of group C^?${\mathrm{C}}^{?}$-algebras. Here we establish an extension of partial translation algebras, which may be viewed as an excision theorem in this context. We apply this general framework to compute the K-theory of partial translation algebras and group C^?${\mathrm{C}}^{?}$-algebras in the context of almost invariant subspaces of discrete groups. This generalises the work of Cuntz, Lance, Pimsner and Voiculescu. In particular we provide a new perspective on Pimsner's calculation of the K-theory for a graph product of groups.     

Jacobi and Kummer’s ideal numbers

In this article we give a modern interpretation of Kummer’s ideal numbers and show how they developed from Jacobi’s work on cyclotomy, in particular the methods for studying “Jacobi sums” which he presented in his lectures on number theory and cyclotomy in the winter semester 1836/37.     

on stability criterion of complete intersections

For protective varieties, it is known that Chow stable implies N-th Hilbert-Mumford stable for N sufficiently large, which follows from the works of J. Fogarty [2, 6]. In this article, we firstly shall provide a simple criterion for Chow stability of complete intersections. The criterion for Chow stability was previously provided by Mumford [5], but our calculation is different from Mumford’s in that ours is based on the results of Zhang’s article [10]. Applying it, we secondly shall give an elementary proof of the above implications in a complete intersections case.     

S-convexity revisited (fuzzy): long version

In this revisional article, we criticize (strongly) the use made by Medar et al., and those whose work they base themselves on, of the name ‘convexity’ in definitions which intend to relate to convex functions, or cones, or sets, but actually seem to be incompatible with the most basic consequences of having the name ‘convexity’ associated to them. We then believe to have fixed the ‘denominations’ associated with Medar’s (et al.) work, up to a point of having it all matching the existing literature in the field [which precedes their work (by long)]. We also expand his work scope by introducing s ₁-convexity concepts to his group of definitions, which encompasses only convex and its proper extension, s ₂-convex, so far. This article is a long version of our previous review of Medar’s work, published by FJMS (Pinheiro, M.R.: S-convexity revisited. FJMS, 26/3, 2007).     

K-Theory of Noncommutative Lattices

Noncommutative lattices have been recently used as finite topological approximations in quantum physical models. As a first step in the construction of bundles and characteristic classes over such noncommutative spaces, we shall study their K-theory. We shall do it algebraically, by studying the algebraic K-theory of the associated algebras of continuous functions which turn out to be noncommutative approximately finite dimensional (AF)C^*. We also work out several examples.     

On the Distributions of Two Classes of Multiple Dependent Aggregate Claims

In this paper we examine two classes of correlated aggregate claims distributions, with univariate claim counts and multivariate claim sizes. Firstly, we extend the results of Hesselager [ASTIN Bulletin, 24： 19-32（1994）] and Wang ＆ Sobrero＇s [ASTIN Bulletin, 24：161-166 （1994）] concerning recursions for compound distributions to a multivariate situation where each claim event generates a random vector. Then we give a multivariate continuous version of recursive algorithm for calculating a family of compound distribution. Especially, to some extent, we obtain a continuous version of the corresponding results in Sundt [ASTIN Bulletin, 29：29-45 （1999）] and Ambagaspitiya [Insurance： Mathematics and Economics, 24：301-308 （1999）]. Finally, we give an example and show how to use the algorithm for aggregate claim distribution of first class to compute recursively the compound distribution.