Bivariant K-theory via correspondences

Abstract

We use correspondences to define a purely topological equivariant bivariant K-theory for spaces with a proper groupoid action. Our notion of correspondence differs slightly from that of Connes and Skandalis. We replace smooth K-oriented maps by a class of K-oriented normal maps, which are maps together with a certain factorisation. Our construction does not use any special features of equivariant K-theory. To highlight this, we construct bivariant extensions for arbitrary equivariant multiplicative cohomology theories. We formulate necessary and sufficient conditions for certain duality isomorphisms in the geometric bivariant K-theory and verify these conditions in some cases, including smooth manifolds with a smooth cocompact action of a Lie group. One of these duality isomorphisms reduces bivariant K-theory to K-theory with support conditions. Since similar duality isomorphisms exist in Kasparov theory, both bivariant K-theories agree if there is such a duality isomorphism.Comment: The article was split into two parts to make it more accessible. Some results were added and som notation is changed, notably normal maps are now called normally non-singular maps. Thus this is essentially a new article, superseding the first versio

The class of a fibre in noncommutative geometry

“…Then a result going back to Kasparov shows that there is a Poincaré duality isomorphism [22]. The group RKK Γ * (Z; A, B), explained in [33], is identical, by the definitions, to the groupoid-equivariant group KK G Γ * +d (C 0 (Z) ⊗ A, C 0 (Z) ⊗ B defined by LeGall; this point of view is convenient, because KK G is equivalent to the category of G-equivariant correspondences, by [21], for proper groupoids G.…”

“…so that we might expect a description of inflate([ Γ ⋉ X]) in the form of a G Γ -equivariant correspondence from Z × X to Z -that is, a bundle of Baum-Douglas cycles for X, parameterized by the points of Z. As it turns out, inflate([ Γ ⋉ X]) is represented by a piece of geometric data which generalizes slightly the data involved in a G Γ -equivariant correspondence in the sense of [21]. We will call it a '(smooth, G Γ -equivariant...) correspondence with singular support.'…”

Analytic and topological index maps with values in the $K$-theory of mapping cones

“…In particular, it is useful to construct variants of the Baum-Douglas model which are associated to various index problems; for example, models associated to non-integer valued index maps are of interest. We refer to the Baum-Douglas model and its variants as geometric models and assume the reader is familiar with the original (M, E, f )-model, see any of [1,2,3,8,18,22]. This paper is continuation of [6]; the setup is as follows.…”

Motivic bivariant characteristic classes

“…to the "KK-theory via correspondences" of Emerson-Meyer [14,15] (and more generally to their counterpart based on a complex oriented cohomology theory).…”