It follows from a theorem of Hambleton–Korzeniewski–Ranicki, published in 2007, that the signature of a closed manifold of dimension \(4m\) fibering over a surface is divisible by 4. Going further, one may ask if restricting the diffeomorphism type of the fiber gives stronger divisibility constraints. I present a synopsis of joint work with M. Krannich on the classification of manifolds which arise as total spaces of fiber bundles over a surface with fiber a highly connected \(2n\)-manifold, up to cobordism. As a corollary, we determine all signatures of such manifolds satisfying a weak condition; it turns out that signature 4 is viable if and only if \(n\) is one of 1, 3 or 7.