Abstract
We show that the Adams operation , , in complex –theory lifts to an operation in smooth –theory. If is a –oriented vector bundle with Thom isomorphism , then there is a characteristic class such that in for all . We lift this class to a –valued characteristic class for real vector bundles with geometric –structures.
If is a –oriented proper submersion, then for all we have in , where is the stable –oriented normal bundle of . To a smooth –orientation of we associate a class refining . Our main theorem states that if is compact, then in for all . We apply this result to the –invariant of bundles of framed manifolds and –invariants of flat vector bundles.
Citation
Ulrich Bunke. "Adams operations in smooth $K$–theory." Geom. Topol. 14 (4) 2349 - 2381, 2010. https://doi.org/10.2140/gt.2010.14.2349
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