The mod 8 signature of a surface bundle

The talk will report on the current status of a joint project with Dave Benson, Caterina Campagnolo and Carmen Rovi. Werner Meyer (Bonn thesis 1972) proved that the signature $\sigma(E) \in \mathbb{Z}$ of a surface bundle $\Sigma_g \to E \to \Sigma_h$ is divisible by 4, and can be computed from a  cohomology class $\tau \in H^2(\operatorname{Sp}(2g,\mathbb{Z});\mathbb{Z})$ and the monodromy $\pi_1(\Sigma_h) \to \operatorname{Sp}(2g,\mathbb{Z})$. In our project we prove that  $\sigma(E)/4 \in \mathbb{Z}_2$ can be computed from a  cohomology class $\tau_8 \in H^2(\operatorname{Sp}(2g,\mathbb{Z}_4);\mathbb{Z}_2)$ and the monodromy $\pi_1(\Sigma_h) \to \operatorname{Sp}(2g,\mathbb{Z}_4)$. The symmetric $L$-theory of $\mathbb{Z}_4$ is used to construct an explicit cocycle for $\tau_8$.