Index pairing with Alexander-Spanier cocycles

TL;DR

This paper develops a uniform approach to higher indices of elliptic operators using Alexander-Spanier cocycles, extending classical index theorems and formulas to arbitrary manifolds and dimensions through cyclic cohomology and K-theory pairings.

Contribution

It introduces a new uniform construction for higher indices of elliptic operators via Alexander-Spanier cocycles, generalizing classical index formulas to broader settings.

Findings

Reproduces Atiyah-Singer index theorem for the lowest index.

Extends Helton-Howe trace formula to higher indices on arbitrary manifolds.

Provides a representation of the Connes-Chern character in terms of cyclic homology expressions.

Abstract

We give a uniform construction of the higher indices of elliptic operators associated to Alexander-Spanier cocycles of either parity in terms of a pairing a la Connes between the K-theory and the cyclic cohomology of the algebra of complete symbols of pseudodifferential operators, implemented by means of a relative form of the Chern character in cyclic homology. While the formula for the lowest index of an elliptic operator D on a closed manifold M (which coincides with its Fredholm index) reproduces the Atiyah-Singer index theorem, our formula for the highest index of D (associated to a volume cocycle) yields an extension to arbitrary manifolds of any dimension of the Helton-Howe formula for the trace of multicommutators of classical Toeplitz operators on odd-dimensional spheres. In fact, the totality of higher analytic indices for an elliptic operator D amount to a representation of the Connes-Chern character of the K-homology cycle determined by D in terms of expressions which extrapolate the Helton-Howe formula below the dimension of M.