The monoid $Cb$ of name substitutions and the notion of finitely supported $Cb$-sets introduced by Pitts as a generalization of nominal sets. A simple finitely supported $Cb$-set is a one point extension of a cyclic nominal set. The support map of a simple finitely supported $Cb$-set is an injective map. Also, for every two distinct elements of a simple finitely supported $Cb$-set, there exists...

Abstract Consider a proper, isometric action by unimodular locally compact group G on Riemannian manifold M with boundary, such that / is compact. Then an equivariant Dirac-type operator D under suitable boundary condition has index $${{\,\mathrm{index}\,}}_G(D)$$ index <m...

This expository paper is an introductory text on topologicalK-theory and the Atiyah-Singer index theorem, suitable for graduate students or advanced undegraduates already possessing a background in algebraic topology. The bulk of the material presented here is distilled from Atiyah’s classic “K-Theory” text, as well as his series of seminal papers “The Index of Elliptic Operators” with Singer. ...

A significant group of problems coming from the realm of Combinatorial Geometry can only be approached through the use of Algebraic Topology. From the first such application to Kneser’s problem in 1978 by Lovász [13] through the solution of the Lovász conjecture [1], [8], many methods from Algebraic Topology have been used. Specifically, it appears that the understanding of equivariant theories...

The purpose of the paper is twofold: First, known results of the noncommutative spin geometry of the standard Podleś sphere are extended by discussing Poincaré duality and orientability. In the discussion of orientability, Hochschild homology is replaced by a twisted version which avoids the dimension drop. The twisted Hochschild cycle representing an orientation is related to the volume form o...

For a proper action by locally compact group $G$ on manifold $M$ with $G$-equivariant Spin-structure, we obtain obstructions to the existence of complete $G$-invariant Riemannian metrics uniformly positive scalar curvature. We focus case where $M/G$ is noncompact. The follow from Callias-type index theorem, and relate curvature near hypersurfaces in $M$. also deduce some other applications this...

For an algebra B with an action of a Hopf algebra H we establish the pairing between equivariant cyclic cohomology and equivariant K-theory for B. We then extend this formalism to compact quantum group actions and show that equivariant cyclic cohomology is a target space for the equivariant Chern character of equivariant summable Fredholm modules. We prove an analogue of Julg’s theorem relating...

We survey work by the author and Ralf Meyer on equivariant KKtheory. Duality plays a key role in our approach. We organize the survey around the objective of computing a certain homotopy-invariant of a space equipped with a proper action of a group or groupoid called the Lefschetz map. The latter associates an equivariant K-homology class to an equivariant Kasparov self-morphism of a space X ad...