Skip to main content
SpringerLink
Log in

Abelianization and Nielsen realization problem of the mapping class group of a handlebody

  • Original Research
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Birman J.S.: Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math. 22, 213–238 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  4. Farb, B., Margalit D.: A Primer on Mapping Class Groups, to appear in Princeton Mathematical Series, Princeton University Press (2011)

  5. Franks J., Handel M.: Global fixed points for centralizers and Morita’s Theorem. Math. Geom. Topol. 13, 87–98 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fathi F., Laudenbach A.: Difféomorphisms pseudo-Anosov et décomposition de Heegaa. C. R. Acad. Sc. Paris t. Ser. A 291, 423–425 (1980)

    MathSciNet  MATH  Google Scholar 

  7. Hirose S.: Realization of the mapping class group of handlebody by diffeomorphisms. Proc. Amer. Math. Soc. 138, 4157–4159 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ishida A.: The structure of subgroup of mapping class groups generated by two Dehn twists. Proc. Jpn. Acad. Ser. A 72, 240–241 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. Johnson D.: Homeomorphisms of a surface which act trivially on homology. Proc. Amer. Math. Soc. 75, 119–125 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kerckhoff S.: The Nielsen realization problem. Ann. Math. 117, 235–265 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  11. Korkmaz M.: Low-dimensional homology groups of mapping class groups: a survey. Turkish J. Math. 26, 101–114 (2002)

    MathSciNet  MATH  Google Scholar 

  12. Markovic V.: Realization of the mapping class group by homeomorphisms. Invent Math. 168, 523–566 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Markovic, V., Saric, D.: The mapping class group cannot be realized by homeomorphisms, preprint (arXiv:0807.0182) (2008)

  14. Morita S.: Characteristic classes of surface bundles. Invent Math. 90, 551–577 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  15. Paris L., Rolfsen D.: Geometric subgroups of mapping class groups. J. Reine Angew. Math. 521, 47–83 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Suzuki S.: On homeomorphisms of a 3-dimensional handlebody. Can. J. Math. 29, 111–124 (1977)

    Article  MATH  Google Scholar 

  17. Wajnryb B.: Mapping class group of a handlebody. Fund. Math. 158, 195–228 (1998)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Author notes

  1. Susumu Hirose

    Present address: Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, Noda, Chiba, 278-8510, Japan

Authors and Affiliations

  1. Department of Mathematics, Graduate School of Science and Engineering, Saga University, Saga, 840-8502, Japan

    Susumu Hirose

Authors
  1. Susumu Hirose
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Susumu Hirose.

Additional information

This research was supported by Grant-in-Aid for Scientific Research (C) (No. 20540083), Japan Society for the Promotion of Science.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hirose, S. Abelianization and Nielsen realization problem of the mapping class group of a handlebody. Geom Dedicata 157, 217–225 (2012). https://doi.org/10.1007/s10711-011-9606-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-011-9606-z

Keywords

Mathematics Subject Classification (2000)

Search

Navigation