In this final chapter the various threads examined in the preceding chap ters are pulled together to fully justify the proof outlined in the Intro duction. Then the original application of the APS to the signature of a compact manifold with boundary is described. Some other applications of the description of 6-geometry and analysis are made; in particular it is shown how the eta invariant and the analytic torsion can be defined on manifolds with exact 6-metrics, under appropriate conditions. 9.1. The APS theorem.