Georgia Institute of Technology College of Sciences

The mapping class group of a surface is a quotient of the group of orientation preserving diffeomorphisms. However the mapping class group generally can't be lifted to the group of diffeomorphisms, and even many subgroups can't be lifted. Given a surface S of genus at least 2 and a marked point z, the fundamental group of S naturally injects to a subgroup of MCG(S,z). I will present a result of Bestvina-Church-Souto that this subgroup can't be lifted to the diffeomorphisms fixing z.

Given a vector bundle over a smooth manifold, one can give an alternate definition of characteristic classes in terms of geometric data, namely connection and curvature. We will see how to define Chern classes and Euler class for the a vector bundle using this theory developed in mid 20th century.

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.