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.     

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.     

Algebraic K-Theory of Two-Dimensional Crystallographic Groups

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

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.     

Elliptic Operators on Manifolds with Singularities and K-Homology

Elliptic operators on smooth compact manifolds are classified by K-homology. We prove that a similar classification is valid also for manifolds with simplest singularities: isolated conical points and edges. The main ingredients of the proof of these results are: Atiyah–Singer difference construction in the noncommutative case and Poincaré isomorphism in K-theory for (our) singular manifolds. As an application we give a formula in topological terms for the obstruction to Fredholm problems on manifolds with edges.Mathematics Subject Classification (2000): 58J05(Primary), 19K33 35S35 47L15(Secondary)(Received: June 2004)     

The K-Theory of Heegaard-Type Quantum 3-Spheres

We use a Heegaard splitting of the topological 3-sphere as a guiding principle to construct a family of its noncommutative deformations. The main technical point is an identification of the universal C^*-algebras defining our quantum 3-spheres with an appropriate fiber product of crossed-product C^*-algebras. Then we employ this result to show that the K-groups of our family of noncommutative 3-spheres coincide with their classical counterparts. Dedicated to the memory of Olaf Richter An erratum to this article is available at .     

K-Theory of Suspended Pseudo-Differential Operators

We compute the K-theory groups of Melrose's algebra of 1-suspended pseudo-differential operators. The boundary map in the six-term long exact sequence turns out to be related to both the eta invariant of Melrose and to the index of elliptic operators. The proof is based on a new identity between the formal trace and the Wodzicki residue trace on the suspended algebra.Partially supported by the European Commission RTN HPRN-CT-1999-00118 Geometric Analysis.     

K-Theory of Surfaces at the Prime 2

We prove that for smooth surfaces over real closed fields, and a class of smooth projective surfaces over a real number field, the map between mod 2 algebraic and étale K-theory is an isomorphism in sufficiently large degrees. For a class of smooth projective surfaces over a real closed field, including rational surfaces, complete intersections and K3-surfaces over the real numbers, we explicate the abutment of the mod 2 motivic cohomology to algebraic K-theory spectral sequence.     

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.     

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.     

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)     

Some mildly wild circles in Sn arising from algebraic K-Theory

We study wild embeddings of S ¹ in S ⁿ which are tame in a sense introduced by Quinn. We show that if is a finitely presented group with H ₁()=H ₂()=0, then any finiteness obstruction K ₀() can be realized on the complement of such an embedded S ¹. We also realize trivially symmetric K _–1() obstructions on the complements of such embeddings. For trivially symmetric , the embeddings constructed are shown to be isotopy homogeneous.     

The Double Coset Theorem Formula for Algebraic K-Theory of Spaces

The goal of this paper is to show that if a smooth fiber bundle has a compact Lie group as a structure group, then the transfer map for the algebraic K-theory of spaces satisfies analogs of the Mackey double coset formula and Feshbach's sum formula. We also prove a cut and paste formula for parametrized Reidemeister torsion.     

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.     

Higher K-Theory of Toric Varieties

A natural higher K-theoretic analogue of the triviality of vector bundles on affine toric varieties is the conjecture on nilpotence of the multiplicative action of the natural numbers on the K-theory of these varieties. This includes both Quillen's fundamental result on K-homotopy invariance of regular rings and the stable version of the triviality of vector bundles on affine toric varieties. Moreover, it yields a similar behavior of not necessarily affine toric varieties and, further, of their equivariant closed subsets. The conjecture is equivalent to the claim that the relevant admissible morphisms of the category of vector bundles on an affine toric variety can be supported by monomials not in a nondegenerate corner subcone of the underlying polyhedral cone. We prove that one can in fact eliminate all lattice points in such a subcone, except maybe one point. The elimination of the last point is also possible in 0 characteristic if the action of the big Witt vectors satisfies a very natural condition. A partial result of this in the arithmetic case provides first nonsimplicial examples, actually an explicit infinite series of combinatorially different affine toric varieties, simultaneously verifying the conjecture for all higher groups.Supported by the Deutsche Forschungsgemeinschaft, INTAS grant 99-00817 and TMR grant ERB FMRX CT-97-0107     

Algebraic K-Theory of Mapping Class Groups

We show that the Fibered Isomorphism Conjecture of T. Farrell and L. Jones holds for various mapping class groups. In many cases, we explicitly calculate the lower algebraic K-groups, showing that they do not always vanish.     

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.     

K-Theory of Stable Generalized Operator Algebras

It is proved that algebraic and topological K-functors are isomorphic on the category of stable generalized operator algebras which are K _i -regular for all i > 0.     

K-Theory for the Integer Heisenberg Groups

We give a closed formula for topological K-theory of the homogeneous space N/, where is the standard integer lattice in the simply connected Heisenberg Lie group N of dimension 2n+1, n . The main tools in our calculations are obtained by computing diagonal forms for certain incidence matrices that arise naturally in combinatorics.     

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.