## Abstract

For a smooth manifold X and an integer d > dim(X) we construct and investigate a natural map sigma(d) : K-d (C-infinity (X)) -> ku C/Z(-d-1)(X). Here K-d (C-infinity(X)) is the algebraic K-theory group of the algebra of complex valued smooth functions on X, and ku C/Z* is the generalized cohomology theory called connective complex K-theory with coefficients in C/Z. If the manifold X is closed of ...

## Abstract

For a smooth manifold X and an integer d > dim(X) we construct and investigate a natural map sigma(d) : K-d (C-infinity (X)) -> ku C/Z(-d-1)(X). Here K-d (C-infinity(X)) is the algebraic K-theory group of the algebra of complex valued smooth functions on X, and ku C/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 (sic), then we state and partially prove the conjecture stating that the following two maps K-d (C-infinity(X)) -> C/Z coincide: 1. Pair the result of sigma(d) with the K-homology class of (sic). 2. Compose the Connes-Karoubi multiplicative character with the classifying map of the d-summable Fredholm module of (sic).