Volume 14, issue 4 (2010)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 29
Issue 2, 549–1114
Issue 1, 1–548

Volume 28, 9 issues

Volume 27, 9 issues

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Procedure
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN 1364-0380 (online)
ISSN 1465-3060 (print)
Author Index
To Appear
 
Other MSP Journals
Adams operations in smooth $K$–theory

Ulrich Bunke

Geometry & Topology 14 (2010) 2349–2381
Abstract

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.

Keywords
Adams operations, differential $K$–theory
References
Publication
Received: 28 April 2009
Accepted: 20 August 2010
Published: 20 November 2010
Proposed: Peter Teichner
Seconded: Ralph Cohen, Steven Ferry
Authors
Ulrich Bunke
Fakultät für Mathematik
Universität Regensburg
93040 Regensburg
Germany
http://www.mathematik.uni-regensburg.de/Bunke/