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

Abstract

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