2022/2023

Research Seminar "Homological Stability and the Topology of Moduli Spaces"

Type: Optional course (faculty)

Delivered by: Faculty of Mathematics

When: 3, 4 module

Open to: students of all HSE University campuses

Instructors: Alexei Gorinov

Language: English

ECTS credits: 6

Contact hours: 72

Course Syllabus

Abstract

"In the past 20 years there have been several results which state that the cohomology of a certain series of geometric objects stabilises. Moreover, the stable cohomology is often more accessible than the cohomology of the individual objects, and can be explicitly described in some cases. Our goal is to understand the general ideas behind these results, and also to discuss some of the examples in detail. Here is one of these. Given an orientable genus g surface with one boundary component, consider its mapping class group Gamma_g,1, consisting of isotopy classes of diffeomorphisms acting trivially near the boundary. Gluing along one component of a genus 1 surface with two boundary components, we obtain a genus g+1 surface with one boundary component, and a corresponding map of mapping class groups Gamma_g,1 —> Gamma_g+1,1 by extending diffeomorphisms trivially beyond the boundary. Harer stability concerns the stabilization of homology for the sequence of classifying spaces BGamma_g,1 —> BGamma_g+1,1, while the Madsen-Weiss theorem determines the homotopy type of the homotopy colimit of this system of classifying spaces. The goal of this seminar is to understand Harer stability, the Madsen-Weiss theorem, and generalizations to higher dimensional manifolds due to Galatius-Madsen-Tilmann-Weiss and Galatius-Randal-Williams. If time allows, we will also discuss several related or similar phenomena."