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.     

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.     

Lambda-structure on Grothendieck groups of Hermitian vector bundles

We define a “compactification” of the representation ring of the linear group scheme over Specℤ, in the spirit of Arakelov geometry. We show that it is a λ-ring which is canonically isomorphic to a localized polynomial ring and that it plays a universal role with respect to natural operations on theK ₀-theory of hermitian bundles defined by Gillet-Soulé. As a byproduct, we prove that the natural pre-λ-ring structure of theK ₀-theory of hermitian bundles is a λ-ring structure. This last result plays a key role in the proof of the main results of [18] and [12].     

A Descent Theorem in Topological K-Theory

We prove the Lichtenbaum–Quillen conjecture in the topological context: in other words, real K-theory can be deduced from complex K-theory via the usual descent spectral sequence. More precise results are proved, however, and new applications are stated. The main ingredients in the proof are Atiyah's KR-theory and the definition of higher K-groups via Clifford algebras.     

K-Theory for Triangulated Categories 3 1/2 (A): A Detailed Proof of the Theorem of Homological Functors Dedicated to Daniel Quillen and to the Memory of Robert Thomason

Let A and B be Abelian categories. Let H: A B be a bounded -functor. We prove that H induces a natural map in higher K-theory. From a more precise analysis of the proof, we deduce that it is possible to define a K-theory of the bounded derived category of A, which contains Quillen's K-theory of A as a retract.     

A K-theoretic note on geometric quantization

We give a purely K-theoretic proof of a case of the “quantization commutes with reduction” result, conjectured by Guillemin and Sternberg and proved by Meinrenken and Vergne. We show that the quantization is simply a pushforward in K-theory, and use Lerman's symplectic cutting and the localization theorem in equivariant K-theory to prove that quantization commutes with reduction. The case where G=S ¹ and the action is free on the zero level set of the moment map is addressed. Received: 9 March 1999     

Symmetric K-Theory Spectra of C*-Categories

We define K-theory groups and symmetric K-theory spectra associated to ₂-graded C ^*-categories and show that the exterior product of K-theory groups can be expressed in terms of the smash product of symmetric spectra.     

A Fourier–Mukai approach to the K-theory of compact Lie groups

Let G be a compact, connected, simply-connected Lie group. We use the Fourier–Mukai transform in twisted K-theory to give a new proof of the ring structure of the K-theory of G.     

On the <Emphasis Type="Italic">KO</Emphasis> Characteristic Cycle of a Spin<Superscript><Emphasis Type="Italic">c</Emphasis></Superscript> Manifold

We compute the KO-characteristic numbers of a characteristic submanifold of a Spin^c manifold in terms of its K-characteristic numbers. The proof is based on the geometry of the Thom class in K-theory and is simpler than the existing proofs of several previously known special cases.     

On products in algebraic K-theory

This paper investigates the product structure in algebraic K-theory of rings. The first objective is to understand the relationships between products and the kernel of the Hurewicz homomorphism relating the algebraic K-theory of any ring to the integral homology of its linear groups. The second part of the paper is devoted to the ring of integers . Using recent results of V. Voevodsky we completely determine the products in tensored with the ring of 2-adic integers. Received: January 3, 1999.     

Bivariant K-Theories for C*-Algebras

A new framework for bivariant K-theory is developed. Various types of homology-cohomology theories are discussed. Our techniques can be used for producing natural elements in E-theory out of continuous fields with non-isomorphic fibers. An alternative definition for the Kasparov product in E-theory is proposed.     

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.     

The Algebraic K-Theory of Operator Algebras

We the study the algebraic K-theory of C *-algebras, forgetting the topology. The main results include a proof that commutative C*-algebras are K-regular in all degrees (that is, all theirN ^T K _iand extensions of the Fischer-Prasolov Theorem comparing algebraic and topological K-theory with finite 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.     

An elementary proof of a distribution relation¶on elliptic curves

We give another elementary proof of a certain identity of elliptic functions arising from the K-theory of elliptic curves and Wildeshaus's generalisation of Zagier's conjectures. This proof consists of a calculation with the q-expansions, and is offered in the hope that its more explicit flavour may be generalised to other situations. Received: 7 December 1999 / Revised version: 3 July 2000     

K-Theory for Triangulated Categories 3 1/2 (B): A Detailed Proof of the Theorem of Homological Functors Dedicated to Daniel Quillen and to the Memory of Robert Thomason

This article is a sequel to K-theory for triangulated categories 3 1/2 (A). In this article, we complete the proof of the main theorem of the previous article.     

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.     

TheK-theory of finite fields, revisited

A new, short proof is given for the Quillen theorem that calculates theK-theory of finite fields. This proof uses the Gabber rigidity theorem and the homotopy theory of simplicial presheaves.Research supported by NSERC.     

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.     

Relative hermitian algebraic k-theory and dihedral homology

The main result of this paper is that there is, under a certain hypothesis, an isomorphism between the rational relative hermitian algebraic K-theory and the rational relative dihedral homology. The general line of the proof of the main theorem is an imitation of Goodwillie's paper, Relative algebraic K-theory and cyclic homology, but in a more complicated setting whose details require developing some new techniques. In the proof of the main theorem we use the hermitian Volodin model and some combinatorial calculations using the classical invariant theory of Loday and Procesi. We believe the computations used to prove the main result may be of independent interest.