Periodicity of the pure mapping class group of non-orientable surfaces

Abstract

We show that the pure mapping class group ${\mathcal{N}}_g^k$ of a non-orientable closed surface of genus $g⩾2$ with $k⩾1$ marked points has $p$-periodic cohomology for each odd prime $p$ for which ${\mathcal{N}}_g^k$ has $p$-torsion. Using the Yagita invariant and the cohomology classes obtained by the representation of subgroups of order $p$, we obtain that the $p$-period is less than or equal to $4$ when $g⩾3$ and $k⩾1$. Moreover, combining the Nielsen realization theorem and a characterization of the $p$-period given in terms of normalizers and centralizers of cyclic subgroups of order $p$, we show that the $p$-period of ${\mathcal{N}}_g^k$ is bounded below by $4$, whenever ${\mathcal{N}}_g^k$ has $p$-periodic cohomology, $g⩾3$ and $k⩾0$. These results provide partial answers to questions proposed by G. Hope and U. Tillmann.

16 pages