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.