Geometry & Topology 14 (2010) 2349–2381

DOI: 10.2140/gt.2010.14.2349

Abstract

We show that the Adams operation Ψ^k, k in {−1,0,1,2,…}, in complex K–theory lifts to an operation ˆ Ψk in smooth K–theory. If V → X is a K–oriented vector bundle with Thom isomorphism Thom_V , then there is a characteristic class ρ^k(V ) in K[1 ∕ k]⁰(X) such that Ψ^k(Thom_V (x)) = Thom_V (ρ^k(V ) ∪ Ψ^k(x)) in K[1 ∕ k](X) for all x in K(X). We lift this class to a K⁰(⋯)[1 ∕ k]–valued characteristic class for real vector bundles with geometric Spin^c–structures.

If π: E → B is a K–oriented proper submersion, then for all x in K(X) we have Ψ^k(π_!(x)) = π_!(ρ^k(N) ∪ Ψ^k(x)) in K[1 ∕ k](B), where N → E is the stable K–oriented normal bundle of π. To a smooth K–orientation o_π of π we associate a class ρ^k(o_π) in K⁰(E)[1 ∕ k] refining ρ^k(N). Our main theorem states that if B is compact, then Ψ^k(π_!(x)) = π(ρ^k(o_π) ∪Ψ^k(x)) in K(B)[1 ∕ k] for all x in K(E). We apply this result to the e–invariant of bundles of framed manifolds and ρ–invariants of flat vector bundles.

Keywords

Adams operations, differential K–theory

Mathematical Subject Classification

References

Publication

Received: 28 April 2009
Accepted: 20 August 2010
Published: 20 November 2010
Proposed: Peter Teichner
Seconded: Ralph Cohen, Steven Ferry

Authors