In Fall 2010 I gave a series of 10 lectures at the University of Chicago on the "scanning" approach to computing stable homology, recently used to great effect by Galatius and others. The main theorem was the proof of the Mumford conjecture on the stable homology of moduli space (originally proved by Madsen-Weiss). But the real goal of the course was to internalize scanning as a tool and expose the geometric ideas behind it. Toward this end, we computed the stable homology of braid groups, symmetric groups, and automorphism groups of free groups. We concluded with a proof of Bott periodicity by scanning.
My approach in Lectures 6, 7, and 9 is based on the exposition of Allen Hatcher. I am very grateful to Soren Galatius and to Allen Hatcher for numerous helpful conversations and explanations.
Lecture 1: an outline of scanning in the concrete case of braid groups (10-4-10)
Lecture 2: classifying spaces of topological groups (10-11-10)
Lecture 3: classifying spaces of topological monoids (10-13-10)
Lecture 4: the group completion theorem (11-1-10, John Lind)
Lecture 5: braid groups and the Barratt–Priddy–Quillen theorem (11-3-10)
Lecture 6: stable homology of the moduli space of curves, part I (11-8-10)
Lecture 7: stable homology of the moduli space of curves, part II (11-10-10)
Lecture 8: characteristic classes of surface bundles and the Mumford conjecture (11-15-10)
Lecture 9: stable homology of automorphism groups of free groups (11-17-10)
Lecture 10: Bott periodicity and the relaxation principle (11-29-10)