Secondary characteristic classes of surface bundles

Abstract

The Miller–Morita–Mumford classes associate to an oriented surface bundle $E\to B$ a class ${\kappa }_i\left(E\right)\in H^{2i}\left(B;\mathbb{Z}\right)$. It was proved by the author, Madsen and Tillman [J. Amer. Math. Soc. 19 (2006) 759-779] that the mod $p$ reduction ${\kappa }_i\left(E\right)\in H^{2i}\left(B;\mathbb{Z}∕p\right)$ vanishes when $i+1$ is divisible by $\left(p-1\right)$. In this note we prove that the $p^2$ reduction ${\kappa }_i\left(E\right)\in H^{2i}\left(B;\mathbb{Z}∕p^2\right)$ vanishes when $i+1$ is divisible by $p\left(p-1\right)$. We also define for each integer $i\ge 1$ a characteristic class ${\lambda }_i\left(E\right)\in H^{2i\left(p-1\right)-2}\left(B;\mathbb{Z}∕p\right)$ which satisfies $p{\lambda }_i\left(E\right)={\kappa }_{i\left(p-1\right)-1}\left(E\right)\in H^{\ast }\left(B;\mathbb{Z}∕p^2\right)$.