Skip to main content
SpringerLink
Log in

A small normal generating set for the handlebody subgroup of the Torelli group

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

Abstract

We prove that the handlebody subgroup of the Torelli group of an orientable surface is generated by genus one BP-maps . As an application, we give a normal generating set for the handlebody subgroup of the level d mapping class group of an orientable surface.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Bass, H., Milnor, J., Serre, J.-P.: Solution of the congruence subgroup problem for \({\rm SL}_n\) \((n\ge 3)\) and \({\rm Sp}_{2n}\) \((n\ge 2)\). Inst. Hautes Etudes Sci. Publ. Math. No. 33, 59–137 (1967)

    Article  Google Scholar 

  2. Birman, J.S.: On Siegel’s modular group. Math. Ann. 191, 59–68 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  3. Birman, J.S.: On the Equivalence of Heegaard Splittings of Closed, Orientable 3-Manifolds. Knots, Groups, and 3-Manifolds (Papers Dedicated to the Memory of R. H. Fox). Annals of Mathematics Studies, pp. 137–164. Princeton University Press, Princeton (1975)

    Google Scholar 

  4. Birman, J.S., Craggs, R.: The \(\mu \)-invariant of 3-manifolds and certain structural properties of the group of homeomorphisms of a closed, oriented 2-manifold. Trans. Am. Math. Soc. 237, 283–309 (1978)

    MathSciNet  MATH  Google Scholar 

  5. Birman, J.S.: The topology of 3-manifolds, Heegaard distance and the mapping class group of a 2-manifold, Problems on mapping class groups and related topics. In: Proc. Sympos. Pure Math., 74, Amer. Math. Soc., Providence, RI, pp. 133–149 (2006)

  6. Birman, J.S., Brendle, T.E., Broaddus, N.: Calculating the Image of the Second Johnson-Morita Representation. Groups of Diffeomorphisms, Advanced Studies in Pure Mathematics, pp. 119–134. Mathematical Society, Tokyo (2008)

    MATH  Google Scholar 

  7. Day, M., Putman, A.: The complex of partial bases for \(F_n\) and finite generation of the Torelli subgroup of \({\rm Aut}(F_n)\). Geom. Dedic. 164, 139–153 (2013)

    Article  MATH  Google Scholar 

  8. Griffiths, H.B.: Automorphisms of a 3-dimensional handlebody. Abh. Math. Semin. Univ. Hamburg 26, 191–210 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hirose, S.: The action of the handlebody group on the first homology group of the surface. Kyungpook Math. J. 46(3), 399–408 (2006)

    MathSciNet  MATH  Google Scholar 

  10. Johnson, D.L.: Homeomorphisms of a surface which act trivially on homology. Proc. Am. Math. Soc. 75(1), 119–125 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  11. Johnson, D.L.: An abelian quotient of the mapping class group \({\cal{I}}_g\). Math. Ann. 249(3), 225–242 (1980)

    Article  MathSciNet  Google Scholar 

  12. Johnson, D.L.: The structure of the Torelli group. I. A finite set of generators for \({\cal{I}}\). Ann. Math. (2) 118(3), 423–442 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  13. Johnson, D.L.: The structure of the Torelli group. III. The abelianization of \({\cal{I}}\). Topology 24(2), 127–144 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  14. Luft, E.: Actions of the homeotopy group of an orientable 3-dimensional handlebody. Math. Ann. 234(3), 279–292 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  15. Magnus, W.: Über \(n\)-dimensionale Gittertransformationen (German). Acta Math. 64(1), 353–367 (1935)

    Article  MathSciNet  MATH  Google Scholar 

  16. McCarthy, J.D., Pinkall, U.: Representing homology automorphisms of nonorientable surfaces, Max Planck Inst. Preprint MPI/SFB 85-11, revised version written on 26 Feb 2004. http://www.math.msu.edu/~mccarthy/publications/selected.papers.html

  17. Morita, S.: Casson’s invariant for homology 3-spheres and characteristic classes of surface bundles. I. Topology 28(3), 305–323 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  18. Morita, S.: The extension of Johnson’s homomorphism from the Torelli group to the mapping class group. Invent. Math. 111(1), 197–224 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  19. Pitsch, W.: Trivial cocycles and invariants of homology 3-spheres. Adv. Math. 220(1), 278–302 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Powell, J.: Two theorems on the mapping class group of a surface. Proc. Am. Math. Soc. 68(3), 347–350 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  21. Putman, A.: Small generating sets for the Torelli group. Geom. Topol. 16(1), 111–125 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Putman, A.: The congruence subgroup problem for \({\rm SL}_n({\mathbb{Z}})\). http://www.math.rice.edu/~andyp/notes/

Download references

Acknowledgements

The author would like to express his gratitude to Hisaaki Endo, for his encouragement and helpful advices. The author also wishes to thank Susumu Hirose and Wolfgang Pitsch for their comments and helpful advices. This work was supported by JST CREST Grant Number JPMJCR17J4, Japan.

Author information

Authors and Affiliations

  1. (Genki Omori) Course of Mathematics, Programs in Mathematics, Electronics and Informatics, Graduate School of Science and Engineering, Saitama University, Shimo-Okubo 255, Sakura-ku, Saitama-shi, 338-8570, Japan

    Genki Omori

Authors
  1. Genki Omori
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Genki Omori.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Omori, G. A small normal generating set for the handlebody subgroup of the Torelli group. Geom Dedicata 201, 353–367 (2019). https://doi.org/10.1007/s10711-018-0396-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-018-0396-4

Keywords

Search

Navigation