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Seminar: Stable Homology of the Mapping Class Group (Summer Semester 2013)

We want to give a proof of Harer's stability theorem concerning the cohomology of the mapping class group.

The seminar takes place on Tuesday at 14:00. Startin from Tuesday, July2, we are in the seminar room 1C-02 (Allianzgebäude).

Seminar: Tuesday 14:00-15:30

The moduli space \mathcal{M}_g classifies compact Riemannian surfaces of genus g and is an interesting object of algebraic geometry. On the other hand, \mathcal{M}_g is a kind of manifold with "harmless" singularities only that may be obtained as a quotient

$\mathcal{M}_g = \mathcal{T}_g / \Gamma_g $

of Teichmüller space \mathcal{T}_g by the operation of the mapping class group \Gamma_g. We want to exploit this very fact in order to shed some light on the topology of moduli space.

From a topological point of view, \mathcal{T}_g is just an open ball and hence easy to understand. Important topological invariants of \mathcal{M}_g, e.g. its cohomology, may thus be computed from corresponding invariants of the group \Gamma_g.

We will introduce the cohomology groups \mathrm{H}^i(\mathcal{M}_g;\mathbb{Q}) and \mathrm{H}^i(\Gamma_g;\mathbb{Q}) and will study their dependence on i and g. For this purpose, we are going to introduce and make use of simplicial complexes, classifying spaces and algebraic tools such as spectral sequences.

Our aim is to give a proof of Harer's stability theorem - a deep result about the topology of \mathcal{M}_g.

Prerequisites: Basic topology and algebra as taught in the courses Einführung in Geometrie und Topologie and Einführung in Algebra und Zahlentheorie

Briefing: Friday, 8.2.2013, 13:15 in room 1C-02

You can also download this announcement and the list of topics as a (german) PDF document.

Comments on the Challenges

Challenge 1

The minimal number of triangles is 14. Did you manage to construct it? ... Not so easy as you would think. It can be realised as the Császár polyhedron. For surfaces of genus g there exists a formula which determines the smallest possible number. The corresponding question for higher dimensional manifolds is however hard and widely unsolved, see e.g. a survey article of Frank Lutz.

Challenge 12


As main source we use the following two articles: