Index of elliptic operators for a diffeomorphism

TL;DR

This paper develops an elliptic operator index theory for diffeomorphisms on closed manifolds, deriving an index formula involving topological invariants and K-theory pairings.

Contribution

It introduces a new elliptic theory for operators associated with diffeomorphisms and provides an explicit index formula in terms of topological and algebraic invariants.

Findings

Derived an index formula using K-theory and cyclic cohomology.

Expressed the index as a pairing between K-theory class and Todd class.

Extended elliptic theory to operators linked with diffeomorphisms.

Abstract

We develop elliptic theory of operators associated with a diffeomorphism of a closed smooth manifold. The aim of the present paper is to obtain an index formula for such operators in terms of topological invariants of the manifold and of the symbol of the operator. The symbol in this situation is an element of a certain crossed product. We express the index as the pairing of the class in K-theory defined by the symbol and the Todd class in periodic cyclic cohomology of the crossed product.