Advertisement

Mathematische Zeitschrift

, Volume 270, Issue 1–2, pp 297–329 | Cite as

Improved homological stability for the mapping class group with integral or twisted coefficients

  • Søren K. Boldsen
Article

Abstract

In this paper we prove stability results for the homology of the mapping class group of a surface. We get a stability range that is near optimal, and extend the result to twisted coefficients.

Keywords

Modulus Space Stability Result Spectral Sequence Simplicial Complex Marked Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Brown, K.: Cohomology of groups. In: Graduate Texts in Mathematics, vol. 87. Springer-Verlag, Berlin (1982)Google Scholar
  2. 2.
    Cerf J.: Topologie de certains espaces de plongements. Bull. Soc. Math. Fr. 89, 227–380 (1961)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Cohen R., Madsen I.: Surfaces in a background space and the homology of the mapping class groups. In: Proc. Symp. Pure Math. 80(1), 43–76 (2009) Preprint: math.GT/0601750MathSciNetGoogle Scholar
  4. 4.
    Cohen, R., Madsen, I.: Stability for closed surfaces in a background space. Preprint: arXiv:1002.2498v1 (2010)Google Scholar
  5. 5.
    Faber, V.: A conjectural description of the tautological ring of the moduli space of curves. In: Faber, C., Looijenga, E. (eds.) Moduli of Curves and Abelian Varieties. Aspects of Mathematics, vol. 33, pp. 109–129. Vieweg, Weisbaden (1999)Google Scholar
  6. 6.
    Gramain A.: Le type d’homotopie du groupe des difféomorphismes d’une surface compacte. Ann. Sci. École Norm. Sup. (4) 6, 53–66 (1973)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Harer J.L.: Stability of the homology of the mapping class group of orientable surfaces. Ann. Math. 121, 215–249 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Harer J.L.: The third homology group of the moduli space of curves. Duke Math. J. 63(1), 25–55 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Harer, J.L.: Improved homology stability for the homology of the mapping class groups of surfaces. Preprint, DUK-M-93-09 (1993)Google Scholar
  10. 10.
    Harer, J.L.: The fourth homology group of the moduli space of curves. Preprint, DUK-M-93-08 (1993)Google Scholar
  11. 11.
    Hirsch, M.: Differential topology. In: Graduate Texts in Mathematics, vol. 33. Springer-Verlag, Berlin (1976)Google Scholar
  12. 12.
    Ivanov, N.: Complexes of curves and the Teichmüller modular group. Uspekhi Mat. Nauk 42(3), 49–91 (1987). English translation: Russ. Math. Surv. 42(3), 55–107 (1987)Google Scholar
  13. 13.
    Ivanov N.: On the homology stability for Teichmüller modular groups: closed surfaces and twisted coefficients. Contemp. Math. 150, 149–193 (1993)Google Scholar
  14. 14.
    Madsen I., Weiss M.: The stable moduli space of Riemann surfaces: Mumford’s conjecture. Ann. Math. 165(3), 843–941 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Morita S.: Generators for the tautological algebra of the moduli space pf curves. Topology 42, 787–819 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Morita S.: Characteristic classes of surface bundles. Invent. Math. 90(3), 551–577 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Wahl N.: Homological stability for the mapping class groups of non-orientable surfaces. Invent. Math. 171, 389–424 (2008)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.University of BonnBonnGermany

Personalised recommendations