Geometry & Topology
- Geom. Topol.
- Volume 14, Number 4 (2010), 2349-2381.
Adams operations in smooth $K$–theory
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.
Geom. Topol., Volume 14, Number 4 (2010), 2349-2381.
Received: 28 April 2009
Accepted: 20 August 2010
First available in Project Euclid: 21 December 2017
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Bunke, Ulrich. Adams operations in smooth $K$–theory. Geom. Topol. 14 (2010), no. 4, 2349--2381. doi:10.2140/gt.2010.14.2349. https://projecteuclid.org/euclid.gt/1513883522