# Søren Galatius: Periodicity and stability in mapping class groups and other E_2 algebras

**Time: **
Wed 2019-10-16 13.15 - 15.00

**Location: **
Sal 306, hus 6, Kräftriket, SU

**Participating: **
Søren Galatius, København

### Abstract

Let \(R_g\) be a chain complex calculating the group homology of the mapping class group of a genus g surface with one parametrized boundary component, and let \(R\) be the direct sum of \(R_g\) over all \(g\). Boundary connected sum may be used to define a bilinear map \(R_g \times R_h \rightarrow R_{g+h}\) making \(R\) into a DGA, which is in fact bigraded: one grading is by homological degree and one is by genus. There is a special cycle in genus 1 and homological degree 0 which plays a special role — multiplying by it defines a chain map\(s\colon R_g \rightarrow R_{g+1}\) often called the "stabilization map". Two influential theorems may be phrased in terms of this bigraded DGA and this element: Harer's homological stability theorem asserts that the quotient \(R/s\) is acyclic in a range of bidegrees, while Madsen-Weiss' theorem calculates the localization \(R[s^{-1}]\). I will argue that a similar point of view may be taken on \(R/s\) and a new graded self-map \(k\colon R/s \rightarrow R/s\) of bidegree \((3,2)\), replacing \(R\) and \(s\). We prove that the quotient \(R/(s,k)\) is acyclic in a larger range of bidegrees, and make preliminary calculations concerning the localization \((R/s)[k^{-1}]\). Time permitting, I will also report on a similar approach to general linear groups of fields and local rings. All is joint work with Alexander Kupers and Oscar Randal-Williams.