Abstract
We show that the Adams operation Psi(k), k is an element of {-1, 0, 1, 2, ... }, in complex K-theory lifts to an operation (Psi) over cap (k) in smooth K-theory. If V -> X is a K-oriented vector bundle with Thom isomorphism Thom(V), then there is a characteristic class rho(k) (V) is an element of K[1/k](0) (X) such that Psi(k) (Thom(V) (x)) = Thom(V)(rho(k)(V) boolean OR Psi(k) (x)) in K[1/k](X) ...
Abstract
We show that the Adams operation Psi(k), k is an element of {-1, 0, 1, 2, ... }, in complex K-theory lifts to an operation (Psi) over cap (k) in smooth K-theory. If V -> X is a K-oriented vector bundle with Thom isomorphism Thom(V), then there is a characteristic class rho(k) (V) is an element of K[1/k](0) (X) such that Psi(k) (Thom(V) (x)) = Thom(V)(rho(k)(V) boolean OR Psi(k) (x)) in K[1/k](X) for all x is an element of K(X). We lift this class to a (K) over cap (0)(...)[1/k]-valued characteristic class for real vector bundles with geometric Spin(c)-structures. If pi : E -> B is a K-oriented proper submersion, then for all x is an element of K (X) we have Psi(k) (pi(!)(x)) = pi(!) (rho(k) (N) boolean OR Psi(k) (x)) in K[1/k](B), where N -> E is the stable K-oriented normal bundle of pi. To a smooth K-orientation 0(pi) of pi we associate a class (rho) over cap (k) (0(pi)) is an element of(K) over cap (0) (E)[1/k] refining rho(k.)(N). Our main theorem states that if B is compact, then (Psi) over cap (k) ((pi) over cap (!)((x) over cap)) = (pi) over cap((rho) over cap (k)(0(pi)) boolean OR (Psi) over cap (k)(x) over cap) in (K) over cap (B)[1/k] for all (x) over cap is an element of (K) over cap (E). We apply this result to the e-invariant of bundles of framed manifolds and rho-invariants of flat vector bundles.
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