Adams operations in smooth K–theory

Bunke, Ulrich

doi:10.2140/gt.2010.14.2349

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Adams operations in smooth $K$–theory

Ulrich Bunke

Geometry & Topology 14 (2010) 2349–2381

DOI: 10.2140/gt.2010.14.2349

Abstract

We show that the Adams operation $Ψ^{k}$ , $k \in {- 1, 0, 1, 2, \dots}$ , in complex $K$ –theory lifts to an operation ${\hat{Ψ}}^{k}$ in smooth $K$ –theory. If $V \to X$ 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 $Ψ^{k} ({Thom}_{V} (x)) = {Thom}_{V} (ρ^{k} (V) \cup Ψ^{k} (x))$ in $K [1 ∕ k] (X)$ for all $x \in K (X)$ . We lift this class to a ${\hat{K}}^{0} (\dots) [1 ∕ k]$ –valued characteristic class for real vector bundles with geometric ${Spin}^{c}$ –structures.

If $π : E \to B$ is a $K$ –oriented proper submersion, then for all $x \in K (X)$ we have $Ψ^{k} (π_{!} (x)) = π_{!} (ρ^{k} (N) \cup Ψ^{k} (x))$ in $K [1 ∕ k] (B)$ , where $N \to E$ is the stable $K$ –oriented normal bundle of $π$ . To a smooth $K$ –orientation $o_{π}$ of $π$ we associate a class ${\hat{ρ}}^{k} (o_{π}) \in {\hat{K}}^{0} (E) [1 ∕ k]$ refining $ρ^{k} (N)$ . Our main theorem states that if $B$ is compact, then ${\hat{Ψ}}^{k} ({\hat{π}}_{!} (\hat{x})) = \hat{π} ({\hat{ρ}}^{k} (o_{π}) \cup {\hat{Ψ}}^{k} (\hat{x}))$ in $\hat{K} (B) [1 ∕ k]$ for all $\hat{x} \in \hat{K} (E)$ . 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

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/

Volume 14, issue 4 (2010)