Adams operations in smooth $K$–theory

Abstract

We show that the Adams operation ${\Psi }^k$, $k\in \left\{-1,0,1,2,\dots \right\}$, in complex $K$–theory lifts to an operation ${\widehat{\Psi }}^k$ 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 ${\widehat{K}}^0\left(\cdots \right)\left[1∕k\right]$–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 ${\widehat{\rho }}^k\left(o_{\pi }\right)\in {\widehat{K}}^0\left(E\right)\left[1∕k\right]$ refining ${\rho }^k\left(N\right)$. Our main theorem states that if $B$ is compact, then ${\widehat{\Psi }}^k\left({\widehat{\pi }}_{!}\left(\widehat{x}\right)\right)=\widehat{\pi }\left({\widehat{\rho }}^k\left(o_{\pi }\right)\cup {\widehat{\Psi }}^k\left(\widehat{x}\right)\right)$ in $\widehat{K}\left(B\right)\left[1∕k\right]$ for all $\widehat{x}\in \widehat{K}\left(E\right)$. We apply this result to the $e$–invariant of bundles of framed manifolds and $\rho$–invariants of flat vector bundles.