# A regulator for smooth manifolds and an index theorem

### Ulrich Bunke

Universität Regensburg, Germany

## Abstract

For a smooth manifold $X$ and an integer $d$ > dim$(X)$ we construct and investigate a natural map

Here $K_{d}(C_{∞}(X))$ is the algebraic $K$-theory group of the algebra of complex valued smooth functions on $X$, and $kuC/Z_{∗}$ is the generalized cohomology theory called connective complex $K$-theory with coefficients in $C/Z$. If the manifold $X$ is closed of odd dimension $d−1$ 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 $σ_{d}$ with the $K$-homology class of the Dirac operator. 2. Compose the Connes–Karoubi multiplicative character with the classifying map of the $d$-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