The Topology of Surface Bundles: Cohomology and Enumerations of Fiberings

This thesis undertakes a study of surface bundles, especially $4$-manifolds equipped with one or more surface bundle structures. A central theme is the interplay between the number of surface bundle structures on a manifold, the properties of the associated monodromy representations, and the algebro-topological invariants of the manifold. In Chapter 1, we show that any non-trivial surface bundle with monodromy in the Johnson kernel has a unique fibering. In Chapter 2, we provide the first examples of $4$-manifolds admitting $3$ or more surface bundle structures. In Chapter 3, we study how the cohomology algebra of a surface bundle can be computed from the monodromy representation, and relate this problem to the cohomology of the mapping class group and the Torelli subgroup.