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

TL;DR

This paper investigates the periodic cohomology of the pure mapping class group of non-orientable surfaces, establishing bounds on the p-period and providing insights into the group's algebraic structure.

Contribution

It determines bounds on the p-period of the pure mapping class group of non-orientable surfaces and relates these bounds to the group's torsion properties and subgroup structures.

Findings

p-period is at most 4 for genus ≥ 3 with marked points

p-period is at least 4 when the group has p-periodic cohomology

Results partially answer questions by Hope and Tillmann

Abstract

We show that the pure mapping class group $\mathcal{N}_{g}^{k}$ of a non-orientable closed surface of genus $g\geqslant 2$ with $k\geqslant 1$ 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 $g\geqslant 3$ and $k\geqslant 1$. 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, $g\geqslant 3$ and $k\geqslant 0$. These results provide partial answers to questions proposed by G. Hope and U. Tillmann.