Abstract：

I will describe a complete set of numerical invariants for deciding whether a real vector bundle is stably trivial. These invariants are integers and rational angles which will be a priori given in contrast to the usual theory of secondary characteristic classes. The definition of these invariants features a duality between the reals and quaternions, moreover a quaternionic version of index theory, in particular for Dirac type operators on quaternionic spin manifolds, is invoked.