To prove Mumford's conjecture about the stable cohomology of the moduli space of Riemann surfaces.
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.
Topology (E2005), Riemann Surfaces (F2006), Introduction to Algebraic Topology (F2006), or similar.
2 hours of lectures per week and 2 hours of (student) seminars.
The original journal papers supplemented by my manuscripts for the individual lectures.
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Revised 19.12.2006
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