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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.

Lecturer

Ib Madsen

Teaching materials/Text-books

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

Capacity limit

20

Revised 19.12.2006

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