Geometry Topology Student Seminar
Wednesday, April 22, 2015 - 14:05
1 hour (actually 50 minutes)
For Prof. Wickelgren's Stable Homotopy Theory class
Harer's homology stability theorem states that the homology of the mapping class group for oriented surfaces of genus g with n boundary components is independent of g for low degrees, increasing with g. Therefore the (co)homology of the mapping class group stabilizes. In this talk, we present Tillmann's result that the classifying space of the stable mapping class group is homotopic to an infinite loop space. The string category of a space X roughly consists of objects given by disjoint unions of loops in X, with morphisms given by cobordisms between collections of loops. Sending X to the loop space of the realization of the nerve of the string category of X is a homotopy functor from Top to the category of infinite loop spaces. Applying this construction for X=pt obtains the result. This result is an important component of the proof of Mumford's conjecture stating that the rational cohomology of the stable mapping class group is generated by certain tautological classes.