Fredholm realizations of elliptic symbols on manifolds with boundary II: fibered boundary

TL;DR

This paper develops an adiabatic calculus interpolating between two pseudodifferential operator calculi on manifolds with fibered boundary, enabling computation of K-theory groups, Fredholm quantizations, and a families index theorem.

Contribution

It introduces an adiabatic calculus bridging Mazzeo's edge and phi calculi, advancing the analysis of elliptic operators on fibered boundary manifolds.

Findings

Computed smooth K-theory groups of the edge calculus

Established conditions for Fredholm quantizations of elliptic symbols

Proved a families index theorem in K-theory

Abstract

We consider two calculi of pseudodifferential operators on manifolds with fibered boundary: Mazzeo's edge calculus, which has as local model the operators associated to products of closed manifolds with asymptotically hyperbolic spaces, and the phi calculus of Mazzeo and the second author, which is similarly modeled on products of closed manifolds with asymptotically Euclidean spaces. We construct an adiabatic calculus of operators interpolating between them, and use this to compute the `smooth' K-theory groups of the edge calculus, determine the existence of Fredholm quantizations of elliptic symbols, and establish a families index theorem in K-theory.