A regulator for smooth manifolds and an index theorem

  • Ulrich Bunke

    Universität Regensburg, Germany


For a smooth manifold and an integer > dim we construct and investigate a natural map

Here is the algebraic -theory group of the algebra of complex valued smooth functions on , and is the generalized cohomology theory called connective complex -theory with coefficients in . If the manifold is closed of odd dimension and equipped with a Dirac operator, then we state and partially prove the conjecture stating that the following two maps

coincide: 1. Pair the result of with the -homology class of the Dirac operator. 2. Compose the Connes–Karoubi multiplicative character with the classifying map of the -summable Fredholm module of the Dirac operator.

Cite this article

Ulrich Bunke, A regulator for smooth manifolds and an index theorem. J. Noncommut. Geom. 12 (2018), no. 4, pp. 1293–1340

DOI 10.4171/JNCG/309