Families Index for Pseudodifferential Operators on Manifolds with Boundary

TL;DR

This paper defines an analytic index for families of cusp pseudodifferential operators on manifolds with boundary, demonstrating cobordism invariance and the role of K-theory in classifying these operators.

Contribution

It introduces a new index theory for cusp pseudodifferential operators on manifolds with boundary, including perturbation techniques to achieve invertibility and insights into cobordism invariance.

Findings

Existence of perturbations making families invertible

Cobordism invariance of the index for cusp operators

Connection to K-theory classifying spaces

Abstract

An analytic index is defined for a family of cusp pseudodifferential operators, $P_b,$ on a fibration with fibres which are compact manifolds with boundaries, provided the family is elliptic and has invertible indicial family at the boundary. In fact there is always a perturbation $Q_b$ by a family of cusp operators of order $-\infty$ such that each $P_b+Q_b$ is invertible. Thus any elliptic family of symbols has a realization as an invertible family of cusp pseudodifferential operators, which is a form of the cobordism invariance of the index. A crucial role is played by the weak contractibility of the group of cusp smoothing operators on a compact manifold with non-trivial boundary and the associated exact sequence of classifying spaces of odd and even K-theory.