Pseudodifferential operators on manifolds with foliated boundaries

TL;DR

This paper develops a new pseudodifferential calculus for manifolds with foliated boundaries, extending existing frameworks and providing index formulas and eta invariant limits for Dirac-type operators.

Contribution

It introduces a generalized calculus for foliated boundaries, including symbols for Fredholm operators and an index formula for Dirac-type operators with compact leaves.

Findings

Constructed a pseudodifferential calculus for foliated boundary manifolds.

Derived an index formula for Fredholm perturbations of Dirac operators.

Obtained a formula for the adiabatic limit of the eta invariant.

Abstract

Let X be a smooth compact manifold with boundary. For smooth foliations on the boundary of X admitting a `resolution' in terms of a fibration, we construct a pseudodifferential calculus generalizing the fibred cusp calculus of Mazzeo and Melrose. In particular, we introduce certain symbols leading to a simple description of the Fredholm operators inside the calculus. When the leaves of the fibration `resolving' the foliation are compact, we also obtain an index formula for Fredholm perturbations of Dirac-type operators. Along the way, we obtain a formula for the adiabatic limit of the eta invariant for invertible perturbations of Dirac-type operators, a result of independent interest generalizing the well-known formula of Bismut and Cheeger.