The total space of a closed oriented surface bundle over a surface is a $4$ manifold. Such a $4$ manifold can have non-trivial signature. In fact, Meyer showed that the signature of surface bundle is divisible by 4, and can be computed using an element of $H^2(Sp(2g,\mathbb{Z});\mathbb{Z})$.

In this talk I will report on joint work with Dave Benson, Caterina Campagnolo and Andrew Ranicki about how a class in the second cohomology of a finite quotient of $Sp(2g,\mathbb{Z})$ enables us to compute the signature of a surface bundle modulo $8$. I will also present geometric constructions which are relevant to the signature cocycle.