Related papers: Topological K-theory of complex noncommutative spa…

We compute the equivariant $K$-homology of the classifying space for proper actions, for compact 3-dimensional hyperbolic reflection groups. This coincides with the topological $K$-theory of the reduced $C^\ast$-algebra associated to the…

We construct the Schubert basis of the torus-equivariant K-homology of the affine Grassmannian of a simple algebraic group G, using the K-theoretic NilHecke ring of Kostant and Kumar. This is the K-theoretic analogue of a construction of…

We prove that any countable discrete and torsion free subgroup of a general linear group over an arbitrary field or a similar subgroup of an almost connected Lie group satisfies the integral algebraic K-theoretic (split) Novikov conjecture…

We associate to each unital $C^*$-algebra $A$ a geometric object---a diagram of topological spaces representing quotient spaces of the noncommutative space underlying $A$---meant to serve the role of a generalized Gel'fand spectrum. After…

The author presents a new proof of injectivity of the composition of the inverse of the rational Chern Character in homology applied to the classifying space BG of a (countable) discrete group G, restricted to dimensions less or equal than…

We show that unital simple C*-algebras with tracial topological rank zero which are locally approximated by subhomogeneous C^-algebras can be classified by their ordered $K$-theory. We apply this classification result to show that certain…

Algebraic K-theory is the stable homotopy theory of homotopy theories, and it interacts with algebraic structures accordingly. In particular, we prove the Deligne Conjecture for algebraic K-theory.

Given a henselian pair $(R, I)$ of commutative rings, we show that the relative $K$-theory and relative topological cyclic homology with finite coefficients are identified via the cyclotomic trace $K \to \mathrm{TC}$. This yields a…

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…

The classical trace map is a highly non-trivial map from algebraic K-theory to topological Hochschild homology (or topological cyclic homology) introduced by B\"okstedt, Hsiang and Madsen. It led to many computations of algebraic K-theory…

Using a canonical topology on differential K-theory induced from the Frech\'et space topology on differential forms and the discrete topology on topological K-theory, we prove that differential K-theory is uniquely determined by the…

Let V be a smooth variety defined over the real numbers. Every algebraic vector bundle on V induces a complex vector bundle on the underlying topological space V(C), and the involution coming from complex conjugation makes it a Real vector…

This paper is our first step in establishing a de Rham model for equivariant twisted $K$-theory using machinery from noncommutative geometry. Let $G$ be a compact Lie group, $M$ a compact manifold on which $G$ acts smoothly. For any $\alpha…

C. Weibel and Thomason-Trobaugh proved (under some assumptions) that algebraic K-theory with coefficients is A1-homotopy invariant. In this article we generalize this result from schemes to the broad setting of dg categories. Along the way,…

We present the construction of a Chern character in cyclic cohomology, involving an arbitrary number of associative algebras in contravariant or covariant position. This is a generalization of the bivariant Chern character for bornological…

Let $G$ be a compact connected Lie group and $K$ a closed connected subgroup. Assume that the order of any torsion element in the integral cohomology of $G$ and $K$ is invertible in a given principal ideal domain $k$. It is known that in…

Let G be a compact, simple and simply connected Lie group and $\A$ be an equivariant Dixmier-Douady bundle over G. For any fixed level k, we can define a G-C*-algebra $C_{\A^{k+h}}(G)$ as all the continuous sections of the tensor power…

In this note we give a simple, model-independent construction of Chern classes as natural transformations from differential complex K-theory to differential integral cohomology. We verify the expected behaviour of these Chern classes with…

The article gives the second part of the treatise on Regular Algebraic $K$-theory (Sections V & VI) of the author. Regular algebraic $K$-theory for groups is a homology theory for discrete groups closely connected to (but different from)…

Topological cyclic homology is a refinement of Connes--Tsygan's cyclic homology which was introduced by B\"okstedt--Hsiang--Madsen in 1993 as an approximation to algebraic $K$-theory. There is a trace map from algebraic $K$-theory to…