Relative Chern character, boundaries and index formulae

TL;DR

This paper derives explicit formulas for the Chern character of the index bundle for elliptic pseudodifferential operators with boundary conditions, using geometric K-theory and symbol analysis.

Contribution

It provides new explicit index formulas for three classes of elliptic operators with boundary, connecting K-theory, symbols, and cohomology.

Findings

Explicit index formulas for transmission algebra operators

Explicit index formulas for zero algebra operators

Explicit index formulas for scattering algebra operators

Abstract

For three classes of elliptic pseudodifferential operators on a compact manifold with boundary which have `geometric K-theory', namely the `transmission algebra' introduced by Boutet de Monvel, the `zero algebra' introduced by Mazzeo and the `scattering algebra' from [MR95k:58168] we give explicit formulae for the Chern character of the index bundle in terms of the symbols (including normal operators at the boundary) of a Fredholm family of fibre operators. This involves appropriate descriptions, in each case, of the cohomology with compact supports in the interior of the total space of a vector bundle over a manifold with boundary in which the Chern character, mapping from the corresponding realization of K-theory, naturally takes values.