Foliated manifolds, algebraic K-theory, and a secondary invariant

We introduce a $\mathbb{C}/\mathbb{Z}$-valued invariant of a foliated manifold with a stable framing and with a partially flat vector bundle. This invariant can be expressed in terms of integration in differential $K$-theory, or alternatively, in terms of $\eta$-invariants of Dirac operators and local correction terms. Initially, the construction of the element in $\mathbb{C}/\mathbb{Z}$ involves additional choices. But if the codimension of the foliation is sufficiently small, then this element is independent of these choices and therefore an invariant of the data listed above. We show that the invariant comprises various classical invariants like Adams' $e$-invariant, the $\rho$-invariant of twisted Dirac operators, or the Godbillon-Vey invariant from foliation theory. Using methods from differential cohomology theory we construct a regulator map from the algebraic $K$-theory of smooth functions on a manifold to its connective $K$-theory with $\mathbb{C}/\mathbb{Z}$ coefficients. Our main result is a formula for the invariant in terms of this regulator and integration in algebraic and topological $K$-theory.