# 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
, 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.

### Lecturer

Ib Madsen

### Teaching materials/Text-books

The original journal papers supplemented by my manuscripts for the individual lectures.

### Capacity limit

20