## ABSTRACT

In this chapter three applications of the construction of parametrices will be made. The first result is the relative index formula, Theorem 6.5, re lating the index of an elliptic differential operator for different values of the weighting of the 6-Sobolev spaces. This was proved for pseudodiffer ential operators in [64] and allows the proof of the APS theorem to be reduced to the Fredholm case. To illustrate the relative index theorem, an idea from [8], extended recently by Gromov and Shubin [40], is used to deduce the Riemann-Roch theorem, for Riemann surfaces, from it. Sec ondly the cohomology of a compact manifold with boundary is represented in terms of harmonic forms. That is, the Hodge theory associated to an exact 6-metric is developed. Finally the resolvent of a second-order elliptic and self-adjoint family of 6-differential operators is analyzed. In particular the resolvent kernel is shown to have an analytic extension to an infinitely branched covering of the complex plane and this is used to give a detailed description of the spectrum. The relationship between the extended L2 null space of a Dirac operator and its adjoint is also investigated. This applies in particular to the Dirac Laplacian, ђ2Е. 6.1. Boundary pairing.