  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.
