On the Farrell cohomology of the mapping class group of non-orientable surfaces

We study the unstable cohomology of the mapping class groups $\mathcal{N}_g$ of non-orientable surfaces of genus $g$. In particular, we determine for all genus $g$ and all primes $p$ when the group $\mathcal{N}_g$ is $p$-periodic.

To this purpose we show that $\mathcal{N}_g$ is a subgroup of the mapping class group $\Gamma_{g-1}$ of an orientable surface of genus $g-1$ and deduce that $\mathcal{N}_g$ has finite virtual cohomological dimension. Furthermore, we describe precisely which finite groups of odd order are subgroups of $\mathcal{N}_g$.