Index pairing with Alexander-Spanier cocycles
Alexander Gorokhovsky, Henri Moscovici

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…
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