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Adams operations in smooth $K$–theory

Ulrich Bunke

Geometry & Topology 14 (2010) 2349–2381

We show that the Adams operation Ψk, k {1,0,1,2,}, in complex K–theory lifts to an operation Ψ̂k in smooth K–theory. If V X is a K–oriented vector bundle with Thom isomorphism ThomV , then there is a characteristic class ρk(V ) K[1k]0(X) such that Ψk(ThomV (x)) = ThomV (ρk(V ) Ψk(x)) in K[1k](X) for all x K(X). We lift this class to a K̂0()[1k]–valued characteristic class for real vector bundles with geometric Spinc–structures.

If π: E B is a K–oriented proper submersion, then for all x K(X) we have Ψk(π!(x)) = π!(ρk(N) Ψk(x)) in K[1k](B), where N E is the stable K–oriented normal bundle of π. To a smooth K–orientation oπ of π we associate a class ρ̂k(oπ) K̂0(E)[1k] refining ρk(N). Our main theorem states that if B is compact, then Ψ̂k(π̂!(x̂)) = π̂(ρ̂k(oπ) Ψ̂k(x̂)) in K̂(B)[1k] for all x̂ K̂(E). We apply this result to the e–invariant of bundles of framed manifolds and ρ–invariants of flat vector bundles.

Adams operations, differential $K$–theory
Received: 28 April 2009
Accepted: 20 August 2010
Published: 20 November 2010
Proposed: Peter Teichner
Seconded: Ralph Cohen, Steven Ferry
Ulrich Bunke
Fakultät für Mathematik
Universität Regensburg
93040 Regensburg