What is the low degree cohomology of the mapping class group of a non-orientable surface? More specifically, what is the universal central extension of the mapping class group of a non-orientable surface?


(I've done some googling, but so far have only found presentations of the MCG.)


(I would be happy to have the answer in just the negative Euler characteristic case.)



Cohomology of the mapping class group of a non-orientable surface?
Let me write $\mathcal{N}\_g$ for the mapping class group of the connect sum of $g$ projective planes. Nathalie Wahl proved that these groups enjoy homological stability, and in 



> 
> O. Randal-Williams, The homology of the stable non-orientable mapping
>  class group, Algebraic & Geometric Topology 8 (2008) 1811-1832.
> 
> 
> 


I calculated the stable (co)homology. Using the best currently available general homological stability ranges, one has
$$H\_1(\mathcal{N}\_g ; \mathbb{Z}) = \mathbb{Z}/2 \quad \text{for} \quad g \geq 7$$
and
$$H\_2(\mathcal{N}\_g ; \mathbb{Z}) = (\mathbb{Z}/2)^2 \quad \text{for} \quad g \geq 10.$$


In fact, by



> 
> M. Korkmaz, First homology group of mapping class groups of
>  nonorientable surfaces, Math. Proc. Cambridge Philos. Soc. 123 (1998),
>  487-499.
> 
> 
> 


the abelianisations of all $\mathcal{N}\_g$ are known: starting at $g=1$ they are
$$0, (\mathbb{Z}/2)^2, (\mathbb{Z}/2)^2, (\mathbb{Z}/2)^3, (\mathbb{Z}/2)^2, (\mathbb{Z}/2)^2, \mathbb{Z}/2, \mathbb{Z}/2, \ldots.$$


As the abelianisation does not vanish for $g > 1$ these groups are not perfect, so they do not have a universal central extension.


