K-Theory and Homology

Together with F. Morel, we have constructed in \cite{CR, Cobord1, Cobord2} a theory of {\em algebraic cobordism}, an algebro-geometric version of the topological theory of complex cobordism. In this paper, we give a survey of the…

Let a compact group G act on real or complex C*-algebras A and B, with A separable and B sigma-unital. We express the G-equivariant Kasparov groups KK_n(A,B) by algebraic K-groups of a certain additive category.

For an algebra B with an action of a Hopf algebra H we establish the pairing between even equivariant cyclic cohomology and equivariant K-theory for B. We then extend this formalism to compact quantum group actions and show that equivariant…

We consider Hopf crossed products of the the type $A#_\sigma \mathcal{H}$, where $\mathcal{H}$ is a cocommutative Hopf algebra, $A$ is an $\mathcal{H}$-module algebra and $\sigma$ is a "numerical" convolution invertible 2-cocycle on…

In 1988, Brown and Ellis published [3] a generalised Hopf formula for the higher homology of a group. Although substantially correct, their result lacks one necessary condition. We give here a counterexample to the result without that…

This paper contains a long summary of the basic properties of higher FR torsion. An attempt is made to simplify the constructions from my book Higher Franz-Reidemeister Torsion (IP/AMS Studies in Advanced Math 31). Some new basic theorems…

We describe a variant of K-theory for spaces with involution, built from vector bundles which are sent to their negative under the involution.

This paper will be concerned with proving that certain Whitehead groups of torsion-free elementary amenable groups are torsion groups and related results, and then applying these results to the Bass conjecture. In particular we shall…

We establish the Hasse principle (local-global principle) in the context of the Baum-Connes conjecture with coefficients. We illustrate this principle with the discrete group $GL(2,F)$ where $F$ is any global field.

A generalization of Connes-Thom isomorphism is given for stable, homotopy invariant, and split exact functors on separable $C^*$-algebras. As examples of these functors, we concentrate on asymptotic and local cyclic cohomology and the…

The main result of the work ``The nilpotence conjecture in K-theory of toric varieties'' is extended to all coefficient fields of characteristic 0, thus covering the class of genuine toric varieties.

Crossed squares and 2-crossed modules are both algebraic models for 3-types. This paper explores the interrelationships between these two models.

Given a finite--dimensional monomial algebra $A$ we consider the trivial extension $TA$ and provide formulae, depending on the characteristic of the field, for the dimensions of the summands $HH_1(A)$ and $\Alt(DA)$ of the first Hochschild…

It is now well known that the K-theory of a Waldhausen category depends on more than just its (triangulated) homotopy category (see [Schlichting]). The purpose of this note is to show that the K-theory spectrum of a (good) Waldhausen…

We develop methods for computing Hochschild cohomology groups and deformations of crossed product rings. We use these methods to find deformations of a ring associated to a particular orbifold with discrete torsion, and give a presentation…

Let k be a field and let A be a Frobenius algebra over k. Assume that the Nakayama automorphism of A associated to a Frobenius homomorphism of A has finite order m, and k has a m-th primitive root of unity. Then, A has a natural…

We compute the Whitehead groups of the associative rings in a class which includes (twisted) formal power series rings and the augmentation localizations of group rings and polynomial rings. For any associative ring A, we obtain an…

It is shown that all nontrivial elements in higher $K$-groups of toric varieties over a class of regular rings are annihilated by iterations of the natural Frobenius type endomorphisms. This is a higher analog of the triviality of vector…

We show that if two rings have equivalent derived categories then they have the same algebraic K-theory. Similar results are given for G-theory, and for a large class of abelian categories.

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…