## Algebraic & Geometric Topology

### The center of some braid groups and the Farrell cohomology of certain pure mapping class groups

#### Abstract

In this paper we first show that many braid groups of low genus surfaces have their centers as direct factors. We then give a description of centralizers and normalizers of prime order elements in pure mapping class groups of surfaces with spherical quotients using automorphism groups of fundamental groups of the quotient surfaces. As an application, we use these to show that the $p$–primary part of the Farrell cohomology groups of certain mapping class groups are elementary abelian groups. At the end we compute the $p$–primary part of the Farrell cohomology of a few pure mapping class groups.

#### Article information

Source
Algebr. Geom. Topol., Volume 7, Number 4 (2007), 1987-2006.

Dates
Revised: 16 September 2007
Accepted: 5 October 2007
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796780

Digital Object Identifier
doi:10.2140/agt.2007.7.1987

Mathematical Reviews number (MathSciNet)
MR2366185

Zentralblatt MATH identifier
1161.20033

Subjects
Primary: 20F36: Braid groups; Artin groups 20J06: Cohomology of groups

#### Citation

Chen, Yu Qing; Glover, Henry; Jensen, Craig. The center of some braid groups and the Farrell cohomology of certain pure mapping class groups. Algebr. Geom. Topol. 7 (2007), no. 4, 1987--2006. doi:10.2140/agt.2007.7.1987. https://projecteuclid.org/euclid.agt/1513796780

#### References

• E Artin, Theorie der Zopfe, Hamburg Abh. 4 (1925) 47–72
• E Artin, Theory of braids, Ann. of Math. $(2)$ 48 (1947) 101–126
• E Artin, Braids and permutations, Ann. of Math. $(2)$ 48 (1947) 643–649
• J S Birman, Braids, links, and mapping class groups, Annals of Mathematics Studies 82, Princeton University Press, Princeton, N.J. (1974)
• J S Birman, H M Hilden, On isotopies of homeomorphisms of Riemann surfaces, Ann. of Math. $(2)$ 97 (1973) 424–439
• K S Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer, New York (1982)
• Y Q Chen, H H Glover, C A Jensen, Prime order subgroups of mapping class groups (in preparation)
• F R Cohen, Artin's braid group and the homology of certain subgroups of the mapping class group, Mem. Aemr. Math. Soc. 443 (1991) 6–28
• W Dicks, E Formanek, Automorphism subgroups of finite index in algebraic mapping class groups, J. Algebra 189 (1997) 58–89
• E Fadell, L Neuwirth, Configuration spaces, Math. Scand. 10 (1962) 111–118
• Q Lu, Cohomological properties of the punctured mapping class groups, PhD thesis, Ohio State University (1998)
• Q Lu, Periodicity of the punctured mapping class group, J. Pure Appl. Algebra 155 (2001) 211–235
• Q Lu, Farrell cohomology of low genus pure mapping class groups with punctures, Algebr. Geom. Topol. 2 (2002) 537–562
• G Mislin, Mapping class groups, characteristic classes, and Bernoulli numbers, from: “The Hilton Symposium 1993 (Montreal, PQ)”, CRM Proc. Lecture Notes 6, Amer. Math. Soc., Providence, RI (1994) 103–131
• Y Xia, The $p$–torsion of the Farrell–Tate cohomology of the mapping class group $\Gamma\sb {(p-1)/2}$, from: “Topology '90 (Columbus, OH, 1990)”, Ohio State Univ. Math. Res. Inst. Publ. 1, de Gruyter, Berlin (1992) 391–398
• H Zieschang, Finite groups of mapping classes of surfaces, Lecture Notes in Mathematics 875, Springer, Berlin (1981)