We show that the Adams operation [math] , [math] , in complex [math] –theory lifts to an operation [math] in smooth [math] –theory. If [math] is a [math] –oriented vector bundle with Thom isomorphism [math] , then there is a characteristic class [math] such that [math] in [math] for all [math] . We lift this class to a [math] –valued characteristic class for real vector bundles with geometric [math] –structures.
¶ If [math] is a [math] –oriented proper submersion, then for all [math] we have [math] in [math] , where [math] is the stable [math] –oriented normal bundle of [math] . To a smooth [math] –orientation [math] of [math] we associate a class [math] refining [math] . Our main theorem states that if [math] is compact, then [math] in [math] for all [math] . We apply this result to the [math] –invariant of bundles of framed manifolds and [math] –invariants of flat vector bundles.