An Exposition of the Madsen-Weiss Theorem

The theorem of Madsen and Weiss [MW] identifies the homology of mapping class groups of surfaces, in a stable dimension range, with the homology of a certain infinite loopspace. This result is not only intrinsically interesting, showing that two objects that appear to be quite different turn out to be homologically equivalent, but it also allows explicit calculations of the stable homology, rather easily for rational coefficients and with some work for mod p coefficients. Outside the stable dimension range the homology of mapping class groups appears to be quite complicated and is still very poorly understood, so it is surprising that there is such a simple and appealing description in the stable range. The Madsen-Weiss theorem has a very classical flavor, and in retrospect it seems that it could have been proved in the 1970s or 1980s since the main ingredients were available then. However, at that time it was regarded as very unlikely that the stable homology of mapping class groups could be that of an infinite loopspace. The initial breakthrough came in a 1997 paper of Tillmann [T] where this unexpected result was proved. It remained then to determine whether the infinite loopspace was a familiar one. A conjecture in this direction was made in a 2001 paper of Madsen and Tillmann [MT], with some supporting evidence, and this conjecture became the Madsen-Weiss theorem. The original proof of the Madsen-Weiss theorem was rather lengthy, but major simplifications have been found since then. The purpose of the present paper is to present a proof that uses a number of these later simplifications, particularly some due to Galatius and Randal-Williams which make the proof really quite elementary, apart from the three main classical ingredients: