A regulator for smooth manifolds and an index theorem

Abstract

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

Here $K_d(C^{\infty }(X))$ is the algebraic $K$-theory group of the algebra of complex valued smooth functions on $X$, and $\mathbf{ku}\mathbb{C}/{\mathbb{Z}}^{\ast }$ is the generalized cohomology theory called connective complex $K$-theory with coefficients in $\mathbb{C}/\mathbb{Z}$. 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 ${\sigma }_d$ 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.