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

- Lecturer: JProf. Dr. Gabriela Weitze-Schmithüsen
- Classes: Seminar (0171900)
- Weekly hours: 2
- Audience: Mathematics, Computer Science (from 4. semester)

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).**

Schedule | ||
---|---|---|

Seminar: | Tuesday 14:00-15:30 |

Lecturers | ||
---|---|---|

Lecturer | JProf. Dr. Gabriela Weitze-Schmithüsen | |

Office hours: no office hours in this semester | ||

Room 1.033 Kollegiengebäude Mathematik (20.30) | ||

Email: weitze-schmithuesen@kit.edu | ||

Lecturer | Dr. Tobias Columbus | |

Office hours: Friday, 10:30 - 12:00 | ||

Room 1.024 Kollegiengebäude Mathematik (20.30) | ||

Email: tobias dot columbus at posteo de |

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

of Teichmüller space by the operation of the mapping class group . 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, is just an open ball and hence easy to understand. Important topological invariants of , e.g. its cohomology, may thus be computed from corresponding invariants of the group .

We will introduce the cohomology groups and and will study their dependence on and . 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 .

**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

# References

As main source we use the following two articles:

- Nathalie Wahl: The Mumford conjecture, Madsen-Weiss and homological stability for mapping class groups of surfaces, PCMI lectures, UTAH 2011
- Nathalie Wahl: Homological stability for mapping class groups of surfaces, in: Handbook of Moduli, Vol. III, 547-583. Advanced Lectures in Mathematics 26 (2012)