Topic Proposal the Topology of Surface Bundles

Let M be a closed oriented manifold, and let Sg denote the closed surface of genus g; unless otherwise specified, assume g ≥ 2. A surface bundle over M is a fiber bundle with base space M and fiber Sg. The study of surface bundles in low dimensions unites ideas in the topology of manifolds of dimensions one, two, three, and four. The Virtual Fibering Conjecture (recently resolved) asserts that nearly all irreducible threemanifolds have a surface bundle as a finite cover. Work of Donaldson shows that all symplectic four-manifolds admit the structure of a Lefschetz fibration, which is a surface bundle over a surface off of a finite set of points in the base. Work of Farb shows that the family of surface bundles over a surface is large enough to have an unsolvable homeomorphism problem. The study of surface bundles also draws on results from mapping class groups and the group Homeo(S1), bringing in connections to two and one dimensions, respectively. A basic problem in the theory of fiber bundles is to understand the family of bundles over a particular base space. Let F be a closed oriented n-manifold. The theory of classifying spaces gives the existence of a space BDiff(F ) equipped with a “universal oriented F bundle” ζ and a correspondence { Isomorphism classes of oriented F bundles over M } ←→ { Homotopy classes of maps M → BDiff(F ) }