Homology of pseudodifferential operators I. Manifolds with boundary

TL;DR

This paper computes Hochschild and cyclic homology for cusp pseudodifferential operators on manifolds with boundary, extending classical index formulas with boundary-specific invariants like the eta invariant.

Contribution

It introduces a Hochschild 1-cocycle interpretation of the index functional and derives a new index formula incorporating boundary effects and the eta invariant.

Findings

Hochschild and cyclic homology groups are explicitly computed.

An index formula extending Atiyah-Patodi-Singer results is established.

The eta invariant plays a key role in the boundary contribution.

Abstract

The Hochschild and cyclic homology groups are computed for the algebra of `cusp' pseudodifferential operators on any compact manifold with boundary. The index functional for this algebra is interpreted as a Hochschild 1-cocycle and evaluated in terms of extensions of the trace functionals on the two natural ideals, corresponding to the two filtrations by interior order and vanishing degree at the boundary, together with the exterior derivations of the algebra. This leads to an index formula which is a pseudodifferential extension of that of Atiyah, Patodi and Singer for Dirac operators; together with a symbolic term it involves the `eta' invariant on the suspended algebra over the boundary previously introduced by the first author.