Geometry & Topology

Adams operations in smooth $K$–theory

Ulrich Bunke

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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 VX 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 xK(X). We lift this class to a K̂0()[1k]–valued characteristic class for real vector bundles with geometric Spinc–structures.

If π:EB is a K–oriented proper submersion, then for all xK(X) we have Ψk(π!(x))=π!(ρk(N)Ψk(x)) in K[1k](B), where NE 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 x̂K̂(E). We apply this result to the e–invariant of bundles of framed manifolds and ρ–invariants of flat vector bundles.

Article information

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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Adams operations differential $K$–theory


Bunke, Ulrich. Adams operations in smooth $K$–theory. Geom. Topol. 14 (2010), no. 4, 2349--2381. doi:10.2140/gt.2010.14.2349.

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