Secondary Characteristic Classes of Surface Bundles

The Miller-Morita-Mumford classes associate to an oriented surface bundle $E\to B$ a class ${\kappa }_i(E)\in H^{2i}(B;\mathbb{Z})$. In this note we define for each prime $p$ and each integer $i\ge 1$ a secondary characteristic class ${\lambda }_i(E)\in H^{2i(p-1)-2}(B;\mathbb{Z})\mathrm{/}\mathbb{Z}{\kappa }_{i(p-1)-1}$. The mod $p$ reduction ${\lambda }_i(E)\in H^{\ast }(B;{\text{F}}_p)$ has zero indeterminacy and satisfies $p{\lambda }_i(E)={\kappa }_{i(p-1)-1}(E)\in H^{\ast }(B;\mathbb{Z}\mathrm{/}{p}^2)$.