#### Volume 14, issue 4 (2010)

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Adams operations in smooth $K$–theory

### Ulrich Bunke

Geometry & Topology 14 (2010) 2349–2381
##### Abstract

We show that the Adams operation ${\Psi }^{k}$, $k\in \left\{-1,0,1,2,\dots \phantom{\rule{0.3em}{0ex}}\right\}$, in complex $K$–theory lifts to an operation ${\stackrel{̂}{\Psi }}^{k}$ 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 ${\rho }^{k}\left(V\right)\in K{\left[1∕k\right]}^{0}\left(X\right)$ such that ${\Psi }^{k}\left({Thom}_{V}\left(x\right)\right)={Thom}_{V}\left({\rho }^{k}\left(V\right)\cup {\Psi }^{k}\left(x\right)\right)$ in $K\left[1∕k\right]\left(X\right)$ for all $x\in K\left(X\right)$. We lift this class to a ${\stackrel{̂}{K}}^{0}\left(\cdots \phantom{\rule{0.3em}{0ex}}\right)\left[1∕k\right]$–valued characteristic class for real vector bundles with geometric ${Spin}^{c}$–structures.

If $\pi :E\to B$ is a $K$–oriented proper submersion, then for all $x\in K\left(X\right)$ we have ${\Psi }^{k}\left({\pi }_{!}\left(x\right)\right)={\pi }_{!}\left({\rho }^{k}\left(N\right)\cup {\Psi }^{k}\left(x\right)\right)$ in $K\left[1∕k\right]\left(B\right)$, where $N\to E$ is the stable $K$–oriented normal bundle of $\pi$. To a smooth $K$–orientation ${o}_{\pi }$ of $\pi$ we associate a class ${\stackrel{̂}{\rho }}^{k}\left({o}_{\pi }\right)\in {\stackrel{̂}{K}}^{0}\left(E\right)\left[1∕k\right]$ refining ${\rho }^{k}\left(N\right)$. Our main theorem states that if $B$ is compact, then ${\stackrel{̂}{\Psi }}^{k}\left({\stackrel{̂}{\pi }}_{!}\left(\stackrel{̂}{x}\right)\right)=\stackrel{̂}{\pi }\left({\stackrel{̂}{\rho }}^{k}\left({o}_{\pi }\right)\cup {\stackrel{̂}{\Psi }}^{k}\left(\stackrel{̂}{x}\right)\right)$ in $\stackrel{̂}{K}\left(B\right)\left[1∕k\right]$ for all $\stackrel{̂}{x}\in \stackrel{̂}{K}\left(E\right)$. We apply this result to the $e$–invariant of bundles of framed manifolds and $\rho$–invariants of flat vector bundles.

##### Keywords
Adams operations, differential $K$–theory