Generalized Weil's reciprocity law and multiplicativity theorems

Fix a one-dimensional group variety $G$ with Euler-characteristic $\chi (G)=0$, and a quasi-projective variety $Y$, both defined over $\bold {C}$. For any $f\in Hom(Y,G)$ and constructible sheaf ${\cal F}$ on $Y$, we construct an invariant $c_{{\cal F}}(f)\in G$, which provides substantial information about the topology of the fiber-structure of $f$ and the structure of ${\cal F}$ along the fibers of $f$. Moreover, $c_{{\cal F}}:Hom(Y,G)\to G$ is a group homomorphism.