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.

Belongs to: Stockholm Mathematics Centre
Last changed: Oct 11, 2019