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.
Cite
@article{arxiv.0812.4949,
title = {Bivariant K-theory via correspondences},
author = {Heath Emerson and Ralf Meyer},
journal= {arXiv preprint arXiv:0812.4949},
year = {2012}
}
Comments
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 version