A short exposition of the Madsen-Weiss theorem

Abstract

This is an exposition of a proof of the Madsen-Weiss Theorem, which asserts that the homology of mapping class groups of surfaces, in a stable dimension range, is isomorphic to the homology of a certain infinite loopspace that arises naturally when one applies the "scanning method". The proof given here utilizes simplifications introduced by Galatius and Randal-Williams.Comment: Version 2 adds three appendices containing background material: (1) Gramain's proof of the Earle-Eells theorem on contractibility of the components of diffeomorphism groups of surfaces, (2) the calculation of the stable rational homology, and (3) a proof of the Group Completion Theorem following an argument of Galatius. The exposition of the paper has also been reorganized significantl