Abelianization and Nielsen realization problem of the mapping class group of a handlebody
For the oriented 3-dimensional handlebody constructed from a 3-ball by attaching g 1-handles, it is shown that the natural surjection from the group of orientation preserving diffeomorphisms to the mapping class group has no section when g is at least 5. In order to prove the above result, we show the vanishing of the first homology group with the real coefficient of the mapping class group of the handlebody with genus g at least 3 and a distinguished disk on its boundary.
KeywordsMapping class group 3-Dimensional handlebody Abelianization Nielsen realization
Mathematics Subject Classification (2000)57M60 57N10
Unable to display preview. Download preview PDF.
- 2.Casson A.J., Bleiler S.A.: Automorphisms of Surfaces after Nielsen and Thurston London Mathematical Society Student Texts 9. Cambridge University Press, Cambridge (1988)Google Scholar
- 3.Farb, B.: Some problems on mapping class groups and moduli space, Problems on mapping class groups and related topics. In: Proc. Sympos. Pure Math. 74, Amer. Math. Soc., pp. 11–55. Providence, RI (2006)Google Scholar
- 4.Farb, B., Margalit D.: A Primer on Mapping Class Groups, to appear in Princeton Mathematical Series, Princeton University Press (2011)Google Scholar
- 13.Markovic, V., Saric, D.: The mapping class group cannot be realized by homeomorphisms, preprint (arXiv:0807.0182) (2008)Google Scholar