Elliptic complexes on manifolds with boundary

TL;DR

This paper develops a Fredholm theory for elliptic complexes on manifolds with boundary, showing how to impose boundary conditions to achieve Fredholm properties, and characterizing when boundary conditions without projections are possible.

Contribution

It introduces a Fredholm framework for elliptic complexes with boundary conditions involving global pseudodifferential projections and characterizes the topological obstructions to simpler boundary conditions.

Findings

Elliptic complexes can be complemented to a Fredholm problem with boundary projections.

Boundary conditions without projections exist iff the Atiyah-Bott obstruction vanishes.

A new Fredholm theory for Toeplitz-type pseudodifferential complexes is developed.

Abstract

We show that elliptic complexes of (pseudo)differential operators on smooth compact manifolds with boundary can always be complemented to a Fredholm problem by boundary conditions involving global pseudodifferential projections on the boundary (similarly as the spectral boundary conditions of Atiyah, Patodi and Singer for a single operator). We prove that boundary conditions without projections can be chosen if, and only if, the topological Atiyah-Bott obstruction vanishes. These results make use of a Fredholm theory for complexes of operators in algebras of generalized pseudodifferential operators of Toeplitz type which we also develop in the present paper.