Dirac operators, boundary value problems, and the b-calculus

It is well-known that the index of a Dirac operator with augmented Atiyah-Patodi-Singer (= APS) boundary conditions on a compact manifold with boundary can be identified with the L-2 index of a corresponding operator on a manifold with cylindrical ends. The augmented APS condition is a specific example of an "ideal boundary condition," which is a boundary condition that differs from the APS condition by a projection on the kernel of the boundary Dirac operator. Following Melrose and Piazza [42] we show that the index and eta invariants of a Dirac operator on a compact manifold with boundary with any ideal boundary condition can be identified with parallel invariants of a perturbation of the corresponding Dirac operator on the manifold with cylindrical ends with L-2 domain by a b-smoothing operator constructed from the ideal boundary condition. In this sense, the "b-category" of objects is able to give a complete description of index and eta invariants for all ideal boundary conditions, and not just the augmented APS condition.