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Mapping class groups of non-orientable surfaces (Nathalie Wahl)

### Abstract:

The work of Madsen and Weiss on the stable mapping class groups of surfaces could be extended to mapping class groups of non-orientable surfaces, provided that we have an appropriate homological stability theorem for these groups. The conjectural statement says that the homology of the mapping class group of a non-orientable surface agrees in a range with the (calculable) homology of the infinite loop space of a Thom spectrum associated to non-orientable plane bundles, where the range in which they agree should increase with the number of copies of RP^2 in the decomposition of the surface.
The first homology group of the mapping class groups of a closed non-orientable surface is known to stabilize by work of Korkmaz.

In my talk, I will describe properties of some curve and arc complexes in orientable and non-orientable surfaces, as part of a work in progress to prove homological stability.