K-Theoretic Indices of Dirac Type Operators on Complete Manifolds and the Roe Algebra

In this paper we study the K-theoretic indices of Dirac Type operators on complete manifolds and their geometric applications.     

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.     

Anti-Wick Symbols for Infinite Products in K-Homology

We consider infinite products in K-homology. We study these products in relation with operators on filtered Hilbert spaces, and infinite iterations of universal constructions on C*-algebras. In particular, infinite tensor power of extensions of pseudodifferential operators on R are considered. We extend anti-Wick pseudodifferential operators to infinite tensor products of spaces of the type L ²(R), and compare our infinite tensor power construction with an extension of pseudodifferential operators on R . We show that the K-theory connecting maps coincide. We propose a natural definition of ellipticity for anti-Wick operators on R, compute the corresponding index, and draw some consequences concerning these operators.     

Index in K-Theory for Families of Fibred Cusp Operators

A families index theorem in K-theory is given for the setting of Atiyah, Patodi, and Singer of a family of Dirac operators with spectral boundary condition. This result is deduced from such a K-theory index theorem for the calculus of cusp, or more generally fibred-cusp, pseudodifferential operators on the fibres (with boundary) of a fibration; a version of Poincaré duality is also shown in this setting, identifying the stable Fredholm families with elements of a bivariant K-group. (Received: February 2006)     

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.     

Relative K-homology and K-homology of an ideal

We provide a straightforward proof of one of the main results of the Baum-Douglas K-homology theory: If A is a separable nuclear C ^*-algebra and I an ideal of A, then the natural restriction map K ⁰(A, I) K ⁰(I) is an isomorphism of Abelian groups.     

The Product Formula in Unitary Deformation K-theory

For finitely generated groups G and H, we prove that there is a weak equivalence G H (G × H) of ku-algebra spectra, where denotes the “unitary deformation K-theory” functor. Additionally, we give spectral sequences for computing the homotopy groups of G and HG in terms of connective K-theory and homology of spaces of G-representations.     

Homotopy Decompositions and K-Theory of Bott Towers

We describe Bott towers as sequences of toric manifolds M^k, and identify the omniorientations which correspond to their original construction as complex varieties. We show that the suspension of M^k is homotopy equivalent to a wedge of Thom complexes, and display its complex K-theory as an algebra over the coefficient ring. We extend the results to KO-theory for several families of examples, and compute the effects of the realification homomorphism; these calculations breathe geometric life into Bahri and Benderskys analysis of the Adams Spectral Sequence [Bahri, A. and Bendersky, M.: The KO-theory of toric manifolds. Trans. Am. Math. Soc. 352 (2000), 1191–1202.] By way of application we consider the enumeration of stably complex structures on M^k, obtaining estimates for those which arise from omniorientations and those which are almost complex. We conclude with observations on the rôle of Bott towers in complex cobordism theory.Mathematics Subject Classification (2000): 55R25, 55R50, 57R77.(Received: August 2004)     

Flabby Strict Deformation Quantizations and K-Groups

We construct examples of flabby strict deformation quantizations not preserving K-groups. This answers a question of Rieffel negatively.     

Boundary Problems for Dirac-Type Operators on Manifolds with Multi-Cylindrical End Boundaries

The goal of this paper is to establish a geometric program to study elliptic pseudodifferential boundary problems which arise naturally under cutting and pasting of geometric and spectral invariants of Dirac-type operators on manifolds with corners endowed with multi-cylindrical, or b-type, metrics and ‘b-admissible’ partitioning hypersurfaces. We show that the Cauchy data space of a Dirac operator on such a manifold is Lagrangian for the self-adjoint case, the corresponding Calderón projector is a b-pseudodifferential operator of order 0, characterize Fredholmness, prove relative index formulæ, and solve the Bojarski conjecture. Mathematics Subject Classifications (2000): 58J28, 58J52.     

K-Theory of Stratified Vector Bundles

We show that the Atiyah–Hirzebruch K-theory of spaces admits a canonical generalization for stratified spaces. For this we study algebraic constructions on stratified vector bundles. In particular the tangent bundle of a stratified manifold is such a stratified vector bundle.     

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.     

Polynomial operations on complex K-theory

We adapt here the results of the author concerning polynomial operations on the 0th stable cohomotopy to the case of the 0th complex K-theory and consider polynomial operations : Kh, where h is a ring-valued contravariant functor, defined on finite CW-complexes, satisfying some properties. We construct a family of generating operations for the ring Pol(K,h) of all polynomial operations : Kh and doing so, we describe the additive structure of this ring in terms of the h(BU(n)'s. As an illustration of how polynomiality could be used to study operations in the setting of algebraic K-theory, we consider, from our point of view, the well known situation operations : KK on complex K-theory.     

Algebraic K-Theory of Two-Dimensional Crystallographic Groups

Abstract. We explicitly compute the lower algebraicK-groups of the two-dimensional crystallographic 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.     

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.     

Milnor K-Groups and Finite Field Extensions

Let E/F be a finite separable field extension and let m denote the integral part of log₂ [E : F]. David Leep recently showed that if char(F) 2, then for n m the nth power of the fundamental ideal in the Witt ring of E satisfies the equality I ⁿ E = I ^n–m F · I ^m E. The aim of this note is to prove the analogous equality for the Milnor K-groups, that is K _n E = K _n–m F · K _m E for n m. In either of these equalities one may not replace m by m – 1, as examples of certain m-quadratic extensions indicate.     

On K2 of One-Dimensional Local Rings

We study K₂ of one-dimensional local domains which are essentially of finite type over a field of characteristic 0. In particular, we show that Berger’s conjecture implies Geller’s conjecture for such rings. This verifies Geller’s conjecture in many new cases of interest. Received: September 2003     

The Higher K-Theory of Real Curves

If X is a smooth curve defined over the real numbers , we show that K _n (X) is the sum of a divisible group and a finite elementary Abelian 2-group when n 2. We determine the torsion subgroup of K _n (X), which is a finite sum of copies of and 2, only depending on the topological invariants of X() and X(), and show that (for n 2) these torsion subgroups are periodic of order 8.     

N-dimensional rings with an isolated singular point having nonzero K?n

We construct examples of normal affine k-algebras of dimension N with an isolated singular point and nonzero K _–N , giving counter-examples to a conjecture of Weibel.