Georgia Institute of Technology College of Sciences

Let MCG(g) be the mapping class group of a surface of genus g. For sufficiently large g, the nth homology (and cohomology) group of MCG(g) is independent of g. Hence we say that the family of mapping class groups satisfies homological stability. Symmetric groups and braid groups also satisfy homological stability, as does the family of moduli spaces of certain higher dimensional manifolds. The proofs of homological stability for most families of groups and spaces follow the same basic structure, and we will sketch the structure of the proof in the case of the mapping class group.

To any compact Hausdorff space we can assign the ring of (classes of) vector bundles under the operations of direct sum and tensor product. This assignment allows the construction of an extraordinary cohomology theory for which the long exact sequence of a pair is 6-periodic.

The aim of this talk is to give fairly self contained proof of the following result due to Eliashberg. There is exactly one holomorphically fillable contact structure on $T^3$. If time permits we will try to indicate different notions of fillability of contact manifolds in dimension 3.