Index theory on the Miščenko bundle

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 Miščenko line bundle. In addition, we give a proof of Atiyah’s $L^2$-index theorem in the general context of flat bundles of finitely generated projective Hilbert $C^{\ast }$-modules 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.