Characteristic Classes of Surface Bundles Discussed with Professor Farb

We understand a lot about fiber bundle F → E → B, when F is a vector space and the structure group is a linear group. We can compute their cohomology completely. However, if we consider other bundles, such as bundles with fiber F a manifold, where the structure group is the diffeomorphism group, we know nearly nothing at all. In this article, we discuss surface bundles, which is the first step to understand manifold bundles. Even more importantly, this relates to many fundamental objects in 3-manifold theory, symplectic 4-manifold theory and algebraic geometry. It is a well-known fact proved by Earle-Eells that Diff0 (Sg), g > 1 is contractible. Then Diff (Sg) is homotopy equivalence to Diff(Sg)/Diff + 0 (Sg), denoted Mod(Sg). By homotopy theory, we know that the BDiff(Sg) = K(Mod(Sg),1). Since the classifying space of of surface bundle is a K(π,1) space, from the theory of K(π,1) space, we have the following correspondence: