Adams operations in smooth $K$–theory

Preprint, Other literature type English OPEN
Bunke, Ulrich (2010)
  • Publisher: MSP
  • Journal: (issn: 1465-3060)
  • Related identifiers: doi: 10.2140/gt.2010.14.2349
  • Subject: Adams operations | differential $K$–theory | 19L20 | Mathematics - K-Theory and Homology
    acm: MathematicsofComputing_GENERAL

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.
  • References (2)

    R. J. Szabo and A. Valentino. Ramond-Ramond fields, fractional branes and orbifold differential K-theory, arXiv:0710.2773.

    Zen-ichi Yosimura. Universal coefficient sequences for cohomology theories of CW-spectra. Osaka J. Math., 12(2):305-323, 1975.

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