Index theory on the Mi\v{s}\v{c}enko bundle

TL;DR

This paper provides an index-theoretic interpretation of the assembly map for principal bundles with discrete group fibers, proves Atiyah's $L^2$-index theorem in this context, and links Baum-Connes surjectivity to the Kadison-Kaplansky conjecture.

Contribution

It introduces a tensor-product presentation for the Miščenko bundle's section module and offers a new proof of Atiyah's $L^2$-index theorem without relying on geometric $K$-homology.

Findings

Index-theoretic interpretation of the assembly map

Proof of Atiyah's $L^2$-index theorem in a general setting

Surjectivity of Baum-Connes implies Kadison-Kaplansky conjecture in torsion-free case

Abstract

We consider the assembly map for principal bundles with fiber a countable discrete group. We obtain an index-theoretic interpretation of this homomorphism by providing a tensor-product presentation for the module of sections associated to the Mi\v{s}\v{c}enko line bundle. In addition, we give a proof of Atiyah's $L^2$-index theorem in the general context of principal bundles over compact Hausdorff spaces. We thereby also reestablish that the surjectivity of the Baum-Connes assembly map implies the Kadison-Kaplansky idempotent conjecture in the torsion-free case. Our approach does not rely on geometric $K$-homology but rather on an explicit construction of Alexander-Spanier cohomology classes coming from a Chern character for tracial function algebras.