A regulator for smooth manifolds and an index theorem

TL;DR

This paper constructs a new homomorphism linking algebraic K-theory of smooth functions on a manifold to topological K-theory, extending previous work and proposing a conjecture relating it to Dirac operators and cyclic cohomology.

Contribution

It introduces a novel homomorphism from algebraic to topological K-theory for smooth manifolds and states a conjecture connecting it with Dirac operators and the Connes-Karoubi character.

Findings

Constructed a homomorphism from algebraic K-theory to topological K-theory.

Extended Suslin's map to higher-dimensional manifolds.

Partially proved a conjecture relating the map to Dirac operators and cyclic cohomology.

Abstract

For a smooth manifold X of dimension <d we construct a homomorphism from the algebraic K-theory group in degree d of the algebra of smooth functions on X to the degree -d-1 topological K-theory of X with coefficients in C/Z. This map generalizes the map used by Suslin in order to calculate the torsion subgroup of algebraic K-theory of C (the case X=*). We state and partially prove a conjecture which compares the composition of the map above with the evaluation against the K-homology class of a Dirac operator on X on the one hand, and the Connes-Karoubi multiplicative character of the associated d-summable Fredholm module on the other.