This is a survey of the relationship between C *-algebraic deformation quan-tization and the tangent groupoid in noncommutative geometry, emphasizing the role of index theory. We first explain how C *-algebraic versions of deformation quantization are related to the bivariant E-theory of Connes and Higson. With this background, we review how Weyl–Moyal quantization may be described using the ta...

In this paper we consider a family of Dirac-type operators on fibration P → B equivariant with respect to an action of an étale groupoid. Such a family defines an element in the bivariant K theory. We compute the action of the bivariant Chern character of this element on the image of Connes’ map Φ in the cyclic cohomology. A particular case of this result is Connes’ index theorem for étale grou...

In this paper we consider a family of Dirac-type operators on fibration P → B equivariant with respect to an action of an étale groupoid. Such a family defines an element in the bivariant K theory. We compute the action of the bivariant Chern character of this element on the image of Connes’ map Φ in the cyclic cohomology. A particular case of this result is Connes’ index theorem for étale grou...

We compute the Fadell-Husseini index of the dihedral group D8 = (Z2)2 ⋊ Z2 on S × S, that is, the kernel of the map in equivariant cohomology H ∗ D8 (pt,F2) −→ H ∗ D8 (Sd × Sd,F2). This establishes the complete cohomological lower bound for the two hyperplane case of Grünbaum’s 1960 mass partition problem: For which d and j can two hyperplanes be found that cut j arbitrary measures on R into fo...

Let Γ < SLn(Z) be a subgroup of finite index, where n≥5. Suppose Γ acts continuously on a manifold M , where π1(M) = Z n, preserving a measure that is positive on open sets. Further assume that the induced Γ action on H(M) is non-trivial. We show there exists a finite index subgroup Γ′ < Γ and a Γ′ equivariant continuous map ψ :M→Tn that induces an isomorphism on fundamental group. We prove mor...

We prove a version of the L2-index Theorem of Atiyah, which uses the universal center-valued trace instead of the standard trace. We construct for G-equivariant K-homology an equivariant Chern character, which is an isomorphism and lives over the ring Z ⊂ ΛG ⊂ Q obtained from the integers by inverting the orders of all finite subgroups of G. We use these two results to show that the Baum-Connes...

Abstract. We study the Z2-equivariant K-theory of M(A), where M(A) is the complement of the complexification of a real hyperplane arrangement, and Z2 acts on M(A) by complex conjugation. We compute the rational equivariant Kand KO-rings of M(A), and we give two different combinatorial descriptions of a subring Line(A) of the integral equivariant KO-ring, where Line(A) is defined to be the subri...

Abstract. We study the Z2-equivariant K-theory of M(A), where M(A) is the complement of the complexification of a real hyperplane arrangement, and Z2 acts on M(A) by complex conjugation. We compute the rational equivariant K and KO rings of M(A), and we give two different combinatorial descriptions of a subring Line(A) of the integral equivariant KO ring, where Line(A) is defined to be the subr...

We characterize all equivariant odd spectral triples on the quantum SU(2) group having a nontrivial Chern character. It is shown that the dimension of an equivariant spectral triple is at least three, and there does exist a 3-summable equivariant spectral triple. We also show that given any odd spectral triple, there is an odd equivariant spectral triple that induces the same element in K. AMS ...

Borsuk-Ulam type theorems for arbitrary compact Lie group actions are proven. The transfer plays a major role in this approach. We present Borsuk-Ulam type theorems for arbitrary compact Lie group actions. The essence of our approach is a generalization of the ideal-valued index of FadellHusseini [FH88] using transfer [Boa66], [BG75], [Dol76], [KP72], [Rou71]. Once an appropriate concept (Defin...