MATH 7530: Algebraic Topology II (fall 2008)

Instructor: Allen Hatcher

The goal of the course will be two big theorems in algebraic topology from the last decade:

1. The theorem of Madsen-Weiss-Tillmann computing the homology of mapping class groups of surfaces in the stable dimension range by relating it to classical objects in homotopy theory. In particular this proves Mumford's conjecture on the stable rational homology, that it is a polynomial ring with one generator in each even dimension.
2. The more recent theorem of Galatius giving the analogous calculation for automorphism groups of free groups, including the surprising connection with stable homotopy groups of spheres.

A major part of the course will be devoted to developing the necessary background material, such as spectral sequences. There will not be time to go through everything in detail, so some things will only be sketched. The main prerequisite is basic algebraic topology as in 651, but there may also be a few other topics we need that can be picked up by reading sections of my book.