Algebraic K-Theory and the Conjectural Leibniz K-Theory

The analogy between algebraic K-theory and cyclic homology is used to build a program aiming at understanding the algebraic K-theory of fields and the periodicity phenomena in algebraic K-theory. In particular, we conjecture the existence of a Leibniz K-theory which would play the role of Hochschild homology. We propose a motivated presentation for the Leibniz K ₂-group ofa field.     

E -Theoretic Duality for Higher Rank Graph Algebras

We prove that there is a Poincaré type duality in E-theory between higher rank graph algebras associated with a higher rank graph and its opposite correspondent. We obtain an r-duality, that is the fundamental classes are in E^r. The basic tools are a higher rank Fock space and higher rank Toeplitz algebra which has a more interesting ideal structure than in the rank 1 case. The K-homology fundamental class is given by an r-fold exact sequence whereas the K-theory fundamental class is given by a homomorphism. The E-theoretic products are essentially pull-backs so that the computation is done at the level of exact sequences. Mathematics Subject Classification (2000): 46L80.     

KK-groups for generalized operator algebras. II

The extension of Kasparovs bivariant K-theory to inverse limits of C ^* -algebras admits exact Puppe sequences in both variables. Two exact sequences generalizing Milnor's lim-lim¹ sequences are established. For CW complexes the extended K-theory is representable K-theory.     

Semi-Topological K-Homology and Thomason''s Theorem

In this paper, we introduce the 'semi-topological K-homology' of complex varieties, a theory related to semi-topological K-theory much as connective topological K-homology is related to connective topological K-theory. Our main theorem is that the semi-topological K-homology of a smooth, quasi-projective complex variety Y coincides with the connective topological K-homology of the associated analytic space Y ^an. From this result, we deduce a pair of results relating semi-topological K-theory with connective topological K-theory. In particular, we prove that the 'Bott inverted' semi-topological K-theory of a smooth, projective complex variety X coincides with the topological K-theory of X ^an. In combination with a result of Friedlander and the author, this gives a new proof, in the special case of smooth, projective complex varieties, of Thomason's celebrated theorem that 'Bott inverted' algebraic K-theory with /n coefficients coincides with topological K-theory with /n coefficients.     

Dihedral homology and Hermitian K-theory

We establish a rational isomorphism between certain relative versions of Hermitian K-theory and the dihedral homology of simplicial Hermitian rings. This is the dihedral analogue of Goodwillie's result for cyclic homology and algebraic K-theory. In particular, we describe involutions on (negative) cyclic homology and the K-theory of simplicial rings. We show that Goodwillie's map from K-theory to negative cyclic homology can be chosen to preserve involutions. By work of Burghelea and Fiedorowicz the invariants of the involution on K-theory can be identified with symmetric Hermitian K-theory. Finally, we show how the author's chain complex defining dihedral homology can be extended to the left to capture the invariants of the involution on negative cyclic homology.Supported by New Mexico State University, grant No. RC90-051.     

The Algebraic K-Theory Spectrum of a 2-Adic Local Field

We explicitly determine the homotopy type of the 2-completed algebraic K-theory spectrum KF, where F is an arbitrary finite extension of the 2-adic rational numbers. The answer is formulated in terms of topological complex K-theory and the K-theory of suitable finite fields, suspended copies of which are glued together by connecting maps that depend on the Iwasawa theory of F.     

Topological K-Theory of Algebraic K-Theory Spectra

Let KX denote the algebraic K-theory spectrum of a regular Noetherian scheme X. Under mild additional hypotheses on X, we construct a spectral sequence converging to the topological K-theory of KX. The spectral sequence starts from the étale homology of X with coefficients in a certain copresheaf constructed from roots of unity. As examples we consider number rings, number fields, local fields, smooth curves over a finite field, and smooth varieties over the complex numbers.     

Continuously controlledA-theory

To anycontrolled space over the metric spaceB we can associate itsboundedly controlled algebraic K-theory, a functor designed to give information about the space of stable bounded concordances of manifolds homotopy equivalent toX. Generalizing a construction of D. R. Anderson, F. X. Connolly, S. Ferry, and E. K. Pedersen, we define another functor, calledcontinuously controlled A-theory, which depends only on thetopology of the control space, not itsmetric properties. In the special case whereB=R ₊, this functor is (more or less by definition) the same asproper A-theory. We prove that under certain conditions on the controlled space the natural transformation from boundedly controlledA-theory to continuously controlledA-theory is a weak homotopy equivalence, and hence defines a generalized homology theory. Continuously controlledK-theory is used in the approaches of G. Carlsson, E. K. Pedersen, and S. Ferry, S. Weinbergervto theK-theory Novikov conjecture.     

Real C*-Algebras, United KK-Theory, and the Universal Coefficient Theorem

We define united KK-theory for real C*-algebras A and B such that A is separable and B is -unital, extending united K-theory in the sense that KK^CRT( , B) = K^CRT(B). United KK-theory combines real, complex, and self-conjugate KK-theory; but unlike unaugmented KK-theory for real C*-algebras, it admits a Universal Coefficient Theorem. For all separable A and B in which the complexification of A is in the bootstrap category, KK^CRT(A,B) appears as the middle term of a short exact sequence whose outer terms involve the united K-theory of A and B. As a corollary, we prove that united K-theory classifies KK-equivalence for real C*-algebras whose complexification is in the bootstrap category.Mathematics Subject Classification (2000): 19K35, 46L80.     

G-theory of group rings for groups of square-free order

The formula for the G-theory of the group ring of a finite group G conjectured by Hambleton, Taylor, and Williams is shown to be valid for ¦G¦ square-free.     

Equivariant–Bivariant Chern Character for Profinite Groups

For the action of a locally compact and totally disconnected group G on a pair of locally compact spaces X and Y we construct, by sheaf theoretic means, a new equivariant and bivariant cohomology theory. If we take for the first space Y an universal proper G-action then we obtain for the second space its delocalized equivariant homology. This is in exact formal analogy to the definition of equivariant K-homology by Baum, Connes, Higson starting from the bivariant equivariant Kasparov KK-theory. Under certain basic finiteness conditions on the first space Y we conjecture the existence of a Chern character from the equivariant Kasparov KK-theory of Y and X into our cohomology theory made two-periodic which becomes an isomorphism upon tensoring the KK-theory with the complex numbers. This conjecture is proved for profinite groups G. An essential role in our construction is played by a bivariant version of Segal localization which we establish for KK-theory.     

The Milnor–Chow homomorphism revisited

The aim of this note is to give a simplified proof of the surjectivity of the natural Milnor–Chow homomorphism between Milnor K-theory and higher Chow groups for essentially smooth (semi-)local k-algebras A with infinite residue fields. It implies the exactness of the Gersten resolution for Milnor K-theory at the generic point. Our method uses the Bloch–Levine moving technique and some properties of the Milnor K-theory norm for fields. Furthermore we give new applications. Supported by Studienstiftung des deutschen Volkes and Deutsche Forschungsgemeinschaft.     

Computations of K- and L-Theory of Cocompact Planar Groups

The verification of the isomorphism conjectures of Baum and Connes and Farrell and Jones for certain classes of groups is used to compute the algebraic K- and L-theory and the topological K-theory of cocompact planar groups (=cocompact N.E.C-groups) and of groups G appearing in an extension where is a finite group and the conjugation -action on ⁿ is free outside . These computations apply, for instance, to two-dimensional crystallographic groups and cocompact Fuchsian groups.     

Structure Theorem of Kummer étale K-group

In this paper we develop a K-theory of log schemes by using vector bundles on the Ket site. Then, for a wide class of log varieties, we describe the structure of their K-groups in terms of the usual algebraic K-groups.     

Morava K-Theories of p-Groups with Cyclic Maximal Subgroups and Other Related Groups

Let P be a non-Abelian finite p-group, p odd, with cyclic maximal subgroups, and let K(n)^*(–) denote the nth Morava K-theory at p. In this paper we determine the algebras K(n)^*(BP) and K(n)^*(BG) for all groups G with Sylow p-subgroups isomorphic to P, giving further evidence for the fact that Morava K-theory as an invariant of finite groups, is finer than ordinary modp cohomology. Mathematics Subject Classifications (2000): 55N20, 55N22.     

Constructions and Dévissage in Hermitian K-Theory

We first give the good definition of the Hermitian K-theory of an exact category with duality. Then we prove a Dévissage theorem which allows us to compare the Hermitian K-theory of a number field with the Hermitian K-theory of its integers.     

The Bass Conjecture and Group von Neumann Algebras

The precise relationship between the Bass conjecture for the Hattori–Stallings trace for projective ZG-modules and the map from reduced K-theory of ZG to reduced K-theory of the von Neumann algebra, NG, of G, of G is determined. As a consequence it is shown this map is zero for all groups G. It is also shown that the map induced on K-theory from the inclusion of NG to the ring of closed, densely defined operators affiliated to NG is an isomorphism. Together with the above result, this gives some positive evidence for the validity of the Division Ring Conjecture for torsion free groups.     

Real C*-Algebras, United K-Theory, and the Künneth Formula

We define united K-theory for real C^*-algebras, generalizing Bousfield's topological united K-theory. United K-theory incorporates three functors – real K-theory, complex K-theory, and self-conjugate K-theory – and the natural transformations among them. The advantage of united K-theory over ordinary K-theory lies in its homological algebraic properties, which allow us to construct a Künneth-type, nonsplitting, short exact sequence whose middle term is the united K-theory of the tensor product of two real C^*-algebras A and B which holds as long as the complexification of A is in the bootstrap category . Since united K-theory contains ordinary K-theory, our sequence provides a way to compute the K-theory of the tensor product of two real C^*-algebras. As an application, we compute the united K-theory of the tensor product of two real Cuntz algebras. Unlike in the complex case, it turns out that the isomorphism class of the tensor product is not determined solely by the greatest common divisor of K and l. Hence, we have examples of nonisomorphic, simple, purely infinite, real C^*-algebras whose complexifications are isomorphic.     

A simple counterexample to the Hambleton-Taylor-Williams conjecture

It is shown that the conjectured formula of Hambleton, Taylor, and Williams for theG-theory of the integral group ring of a finite group does not hold for the symmetric groupsS ₅.