Invertible Dirac operators and handle attachments on manifolds with boundary

TL;DR

This paper studies how to extend metrics with invertible Dirac operators on spin manifolds with boundary over handle attachments, enabling new constructions and classifications of such metrics.

Contribution

It proves that invertibility properties of Dirac operators can be preserved under handle attachments of codimension at least two, advancing geometric analysis on manifolds with boundary.

Findings

Metrics with invertible Dirac operators can be extended over certain handle attachments.

Construction of non-isotopic metrics with invertible Dirac operators.

A theorem on the existence and classification of concordances for these metrics.

Abstract

For spin manifolds with boundary we consider Riemannian metrics which are product near the boundary and are such that the corresponding Dirac operator is invertible when half-infinite cylinders are attached at the boundary. The main result of this paper is that these properties of a metric can be preserved when the metric is extended over a handle of codimension at least two attached at the boundary. Applications of this result include the construction of non-isotopic metrics with invertible Dirac operator, and a concordance existence and classification theorem.