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 $$ \sigma\_{d}\colon K\_{d}(C^{\infty}(X))\to \mathbf {ku}\mathbb C/\mathbb Z^{-d-1}(X)\ . $$ Here $K\_{d}(C^{\infty}(X))$ is the algebraic $K$-theory group of the algebra of complex valued smooth functions on $X$, and $\mathbf {ku}\mathbb C/\mathbb Z^{\*}$ 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 $$ K\_{d}(C^{\infty}(X))\to \mathbb C/\mathbb Z $$ 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.