Pseudodifferential operators on manifolds with fibred corners

TL;DR

This paper develops a pseudodifferential calculus on manifolds with fibred corners, generalizing existing frameworks, to analyze singularities of stratified pseudomanifolds and establish duality results in K-theory.

Contribution

It introduces a new pseudodifferential calculus on manifolds with fibred corners, extending the alculus, and connects it to K-theory duality for stratified pseudomanifolds.

Findings

Defined a calculus of pseudodifferential operators on manifolds with fibred corners

Established criteria for Fredholmness and compactness of operators

Proved a Poincare9 duality between K-homology and K-theory groups

Abstract

One way to geometrically encode the singularities of a stratified pseudomanifold is to endow its interior with an iterated fibred cusp metric. For such a metric, we develop and study a pseudodifferential calculus generalizing the \Phi-calculus of Mazzeo and Melrose. Our starting point is the observation, going back to Melrose, that a stratified pseudomanifold can be `resolved' into a manifold with fibred corners. This allows us to define pseudodifferential operators as conormal distributions on a suitably blown-up double space. Various symbol maps are introduced, leading to the notion of full ellipticity. This is used to construct refined parametrices and to provide criteria for the mapping properties of operators such as Fredholmness or compactness. We also introduce a semiclassical version of the calculus and use it to establish a Poincar\'e duality between the K-homology of the stratified pseudomanifold and the K-group of fully elliptic operators.