# Topology of moduli spaces

### Aims

To prove Mumford's conjecture about the stable cohomology of the moduli space of Riemann surfaces.

### Contents

The course is about the cohomological structure of the "space of all" complex manifolds that are diffeomorphic to a smooth surface of genus g, the so-called moduli space M(g). For g=0,1, the moduli space is discrete, namely a single point (Riemann's mapping theorem), and $\mathbb{R}^2$, respectively. For g larger than or equal to 2, M(g) has the dimension 6g-6, and is not contractible.

I will introduce the space Emb(g) of all differentiable surfaces of genus g embedded in some (high-dimensional) euclidian space, define a map from Emb(g) to M(g), and show that this map induces an isomorphism in cohomology (with rational coefficients).

Then I will construct 2i-dimensional cohomology classes in Emb(g), and formulate Mumford's conjecture.

The final part of the course is a description of Emb(g) up to homotopy that allows us to calculate its entire rational cohomology ring. It turns out to be a polynomial algebra in even dimensional cohomology classes.

### Prerequisites

Topology (E2005), Riemann Surfaces (F2006), Introduction to Algebraic Topology (F2006), or similar.

### Types of teaching

2 hours of lectures per week and 2 hours of (student) seminars.