Skip to main content
SpringerLink
Log in

Cohomology of the hyperelliptic Torelli group

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

Let SI(S g ) denote the hyperelliptic Torelli group of a closed surface S g of genus g. This is the subgroup of the mapping class group of S g consisting of elements that act trivially on H 1(S g ; ℤ) and that commute with some fixed hyperelliptic involution of S g . We prove that the cohomological dimension of SI(S g ) is g − 1 when g ≥ 1. We also show that H g−1(SI(S g ); ℤ) is infinitely generated when g ≥ 2. In particular, SI(S 3) is not finitely presentable. Finally, we apply our main results to show that the kernel of the Burau representation of the braid group B n at t = −1 has cohomological dimension equal to the integer part of n/2, and it has infinitely generated homology in this top dimension.

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. V. I. Arnol’d, A remark on the branching of hyperelliptic integrals as functions of the parameters, Akademija Nauk SSSR. Funkcional’nyi Analiz i ego Priloženija 2 (1968), 1–3.

    Google Scholar 

  2. M. Bestvina, K.-U. Bux and D. Margalit, The dimension of the Torelli group, Journal of the American Mathematical Society 23 (2010), 61–105.

    Article  MathSciNet  MATH  Google Scholar 

  3. J. S. Birman, Mapping class groups and their relationship to braid groups, Communications on Pure and Applied Mathematics 22 (1969), 213–238.

    Article  MathSciNet  MATH  Google Scholar 

  4. J. S. Birman and H. M. Hilden, On the mapping class groups of closed surfaces as covering spaces, in Advances in the Theory of Riemann surfaces (Proc. Conf., Stony Brook, NY, 1969), Annals of Mathematics Studies, Vol. 66. Princeton University Press, Princeton, NJ, 1971, pp. 81–115.

    Google Scholar 

  5. J. S. Birman and H. M. Hilden, On isotopies of homeomorphisms of Riemann surfaces, Annals of Mathematics 97 (1973), 424–439.

    Article  MathSciNet  MATH  Google Scholar 

  6. T. Brendle and D. Margalit, Point pushing, homology, and the hyperelliptic involution, Preprint.

  7. K. S. Brown, Cohomology of Groups, Graduate Texts in Mathematics, Vol. 87, Springer-Verlag, New York, 1982.

    Book  MATH  Google Scholar 

  8. M. Culler and K. Vogtmann, Moduli of graphs and automorphisms of free groups, Inventiones Mathematicae 84 (1986), 91–119.

    Article  MathSciNet  MATH  Google Scholar 

  9. J. S. Ellenberg, On the Euler characteristic of Torelli groups, http://quomodocumque.files.wordpress.com/2011/09/virtual-dimension-eulercharacteristic-torelli.pdf.

  10. B. Farb, Some problems on mapping class groups and moduli space, in Problems on Mapping Class Groups and Related Topics, Proceedings of Symposia in Pure Mathematics, Vol. 74, American Mathematical Society, Providence, RI, 2006, pp. 11–55.

    Chapter  Google Scholar 

  11. B. Farb and D. Margalit, A Primer on Mapping Class Groups, Princeton University Press, Princeton, NJ, 2011.

    MATH  Google Scholar 

  12. R. Hain, Finiteness and Torelli spaces, in Problems on Mapping Class Groups and Related Topics, Proceedings of Symposia in Pure Mathematics, Vol. 74, American Mathematical Society, Providence, RI, 2006, pp. 57–70.

    Chapter  Google Scholar 

  13. J. L. Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Inventiones Mathematicae 84 (1986), 157–176.

    Article  MathSciNet  MATH  Google Scholar 

  14. A. Hatcher and D. Margalit, Generating the Torelli group, in preparation.

  15. G. Hochschild and J.-P. Serre, Cohomology of group extensions, Transactions of the American Mathematical Society 74 (1953), 110–134.

    Article  MathSciNet  MATH  Google Scholar 

  16. N. V. Ivanov, Complexes of curves and Teichmüller modular groups, Uspekhi Matematicheskih Nauk 42 (1987), 49–91.

    MATH  Google Scholar 

  17. N. V. Ivanov, Subgroups of Teichmüller modular groups, Translations of Mathematical Monographs, Vol. 115, American Mathematical Society, Providence, RI, 1992, Translated from the Russian by E. J. F. Primrose and revised by the author.

    MATH  Google Scholar 

  18. D. Johnson, An abelian quotient of the mapping class group Ig, Mathematische Annalen 249 (1980), 225–242.

    Article  MathSciNet  MATH  Google Scholar 

  19. D. Johnson, The structure of the Torelli group. II. A characterization of the group generated by twists on bounding curves, Topology 24 (1985), 113–126.

    Article  MathSciNet  MATH  Google Scholar 

  20. W. Magnus and A. Peluso, On a theorem of V. I. Arnol’d, Communications on Pure and Applied Mathematics 22 (1969), 683–692.

    Article  MathSciNet  MATH  Google Scholar 

  21. G. Mess, Unit tangent bundle subgroups of the mapping class groups, Preprint IHES/M/90/30 (1990).

  22. G. Mess, The Torelli groups for genus 2 and 3 surfaces, Topology 31 (1992), 775–790.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. J. Powell, Two theorems on the mapping class group of a surface, Proceedings of the American Mathematical Society 68 (1978), 347–350.

    Article  MathSciNet  MATH  Google Scholar 

  25. J.-P. Serre, Cohomologie des groupes discrets, in Prospects in Mathematics (Proc. Sympos., Princeton Univ., Princeton, NJ, 1970), Annals of Mathematics Studies, Vol. 70, Princeton University Press, Princeton, NJ, 1971, pp. 77–169.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics & Statistics University Gardens, University of Glasgow, G12 8QW, Scotland, UK

    Tara Brendle

  2. Pittsburg State University, 224 Yates Hall, Pittsburg, KS, 66762-7502, USA

    Leah Childers

  3. School of Mathematics, Georgia Institute of Technology, 686 Cherry St., Atlanta, GA, 30332, USA

    Dan Margalit

Authors
  1. Tara Brendle
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Leah Childers
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dan Margalit
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tara Brendle.

Additional information

The third author gratefully acknowledges support from the National Science Foundation and the Sloan Foundation.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Brendle, T., Childers, L. & Margalit, D. Cohomology of the hyperelliptic Torelli group. Isr. J. Math. 195, 613–630 (2013). https://doi.org/10.1007/s11856-012-0075-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-012-0075-3

Keywords

Search

Navigation