Pseudodifferential extension and Todd class

TL;DR

This paper develops a formalism linking cyclic cohomology of pseudodifferential symbols to the Todd class, providing an algebraic proof of the Atiyah-Singer index theorem for closed manifolds.

Contribution

It introduces a formalism connecting the Radul cocycle to the Todd class, enabling an algebraic proof of the Atiyah-Singer index theorem.

Findings

Radul cocycle corresponds to the Poincaré dual of the Todd class

Provides an algebraic proof of the Atiyah-Singer index theorem

Establishes isomorphism between cyclic cohomology and homology of the cosphere bundle

Abstract

Let M be a closed manifold. Wodzicki shows that, in the stable range, the cyclic cohomology of the associative algebra of pseudodifferential symbols of order \leq 0 is isomorphic to the homology of the cosphere bundle of M. In this article we develop a formalism which allows to calculate that, under this isomorphism, the Radul cocycle corresponds to the Poincar\'e dual of the Todd class. As an immediate corollary we obtain a purely algebraic proof of the Atiyah-Singer index theorem for elliptic pseudodifferential operators on closed manifolds.