New sheaf theoretic methods in differential topology

Michael Weiss

Address: Department of Mathematical Sciences, University of Aberdeen Aberdeen, AB24 3UE, Scotland, UK


Abstract: The Mumford conjecture predicts the ring of rational characteristic classes for surface bundles with oriented connected fibers of large genus. The first proof in [11] relied on a number of well known but difficult theorems in differential topology. Most of these difficult ingredients have been eliminated in the years since then. This can be seen particularly in [7] which has a second proof of the Mumford conjecture, and in the work of Galatius [5] which is concerned mainly with a “graph” analogue of the Mumford conjecture. The newer proofs emphasize Tillmann’s theorem [23] as well as some sheaf-theoretic concepts and their relations with classifying spaces of categories. These notes are an overview of the shortest known proof, or more precisely, the shortest known reduction of the Mumford conjecture to the Harer-Ivanov stability theorems for the homology of mapping class groups. Some digressions on the theme of classifying spaces and sheaf theory are included for motivation.

AMSclassification: primary 57R19; secondary 57R20, 57R22.

Keywords: surface bundle, sheaf, classifying space, homological stability.