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

Hope, Graham; Tillmann, Ulrike

doi:10.1090/S0002-9939-08-09507-5

On the Farrell cohomology of the mapping class group of non-orientable surfaces
HTML articles powered by AMS MathViewer

by Graham Hope and Ulrike Tillmann PDF

Proc. Amer. Math. Soc. 137 (2009), 393-400 Request permission

Abstract:

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 $\mathcal {N}_g$.

References

Joan S. Birman and D. R. J. Chillingworth, On the homeotopy group of a non-orientable surface, Proc. Cambridge Philos. Soc. 71 (1972), 437–448. MR 300288, DOI 10.1017/s0305004100050714
Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956
H. H. Glover, G. Mislin, and Y. Xia, On the Farrell cohomology of mapping class groups, Invent. Math. 109 (1992), no. 3, 535–545. MR 1176203, DOI 10.1007/BF01232038
John L. Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math. 84 (1986), no. 1, 157–176. MR 830043, DOI 10.1007/BF01388737
Steven P. Kerckhoff, The Nielsen realization problem, Ann. of Math. (2) 117 (1983), no. 2, 235–265. MR 690845, DOI 10.2307/2007076
Qin Lu, Periodicity of the punctured mapping class group, J. Pure Appl. Algebra 155 (2001), no. 2-3, 211–235. MR 1801416, DOI 10.1016/S0022-4049(99)00112-7
Qin Lu, Farrell cohomology of low genus pure mapping class groups with punctures, Algebr. Geom. Topol. 2 (2002), 537–562. MR 1917066, DOI 10.2140/agt.2002.2.537
Thomas W. Tucker, Finite groups acting on surfaces and the genus of a group, J. Combin. Theory Ser. B 34 (1983), no. 1, 82–98. MR 701174, DOI 10.1016/0095-8956(83)90009-6
Nathalie Wahl, Homological stability for the mapping class groups of non-orientable surfaces, Invent. Math. 171 (2008), no. 2, 389–424. MR 2367024, DOI 10.1007/s00222-007-0085-7
Yining Xia, The $p$-periodicity of the mapping class group and the estimate of its $p$-period, Proc. Amer. Math. Soc. 116 (1992), no. 4, 1161–1169. MR 1104406, DOI 10.1090/S0002-9939-1992-1104406-2