Geometry & Topology

Adams operations in smooth $K$–theory

Ulrich Bunke

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 $ρk(V)∈K[1∕k]0(X)$ such that $Ψk(ThomV(x))= ThomV(ρk(V)∪Ψk(x))$ in $K[1∕k](X)$ for all $x∈K(X)$. We lift this class to a $K̂0(⋯)[1∕k]$–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 $K[1∕k](B)$, where $N→E$ is the stable $K$–oriented normal bundle of $π$. To a smooth $K$–orientation $oπ$ of $π$ we associate a class $ρ̂k(oπ)∈K̂0(E)[1∕k]$ refining $ρk(N)$. Our main theorem states that if $B$ is compact, then $Ψ̂k(π̂!(x̂))=π̂(ρ̂k(oπ)∪Ψ̂k(x̂))$ in $K̂(B)[1∕k]$ 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.

Article information

Source
Geom. Topol., Volume 14, Number 4 (2010), 2349-2381.

Dates
Accepted: 20 August 2010
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.gt/1513883522

Digital Object Identifier
doi:10.2140/gt.2010.14.2349

Mathematical Reviews number (MathSciNet)
MR2740650

Zentralblatt MATH identifier
1209.19004

Citation

Bunke, Ulrich. Adams operations in smooth $K$–theory. Geom. Topol. 14 (2010), no. 4, 2349--2381. doi:10.2140/gt.2010.14.2349. https://projecteuclid.org/euclid.gt/1513883522

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