Cobordism, Relative Indices and Stein Fillings

TL;DR

This paper extends the understanding of index gluing properties for SpinC-Dirac operators with sub-elliptic boundary conditions, applying these results to Stein fillability of 3D contact manifolds.

Contribution

It advances the analytic framework for boundary value problems of the SpinC-Dirac operator and applies it to Stein fillability in contact geometry.

Findings

Extended analytic results for sub-elliptic boundary value problems.

Established gluing formulas for indices on manifolds with multiple boundary components.

Applied index theory to problems in Stein fillability of contact manifolds.

Abstract

In this paper we build on the framework developed in "Subelliptic Boundary Value Problems for the Spin_C Dirac Operator, I, II, III" to obtain a more complete understanding of the gluing properties for indices of boundary value problems for the SpinC-Dirac operator with sub-elliptic boundary conditions. We extend our analytic results for sub-elliptic boundary value problems for the SpinC-Dirac operator, and gluing results for the indices of these boundary problems to SpinC-manifolds with several pseudoconvex (pseudoconcave) boundary components. These results are applied to study Stein fillability for compact, 3-dimensional, contact manifolds.