Skip to main content
SpringerLink
Log in

The unstable integral homology of the mapping class groups of a surface with boundary

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We construct a graph complex calculating the integral homology of the bordered mapping class groups. We compute the homology of the bordered mapping class groups of the surfaces S 1,1, S 1,2 and S 2,1. Using the circle action on this graph complex, we build a double complex and a spectral sequence converging to the homology of the unbordered mapping class groups. We compute the homology of the punctured mapping class groups associated to the surfaces S 1,1 and S 2,1. Finally, we use Miller’s operad to get the first Kudo–Araki and Browder operations on our graph complex. We also consider an unstable version of the higher Kudo–Araki–Dyer–Lashoff operations.

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. Bödigheimer, C.-F. Moduli spaces of Riemann surfaces with boundary. http://www.math.uni-bonn.de/people/cfb/PUBLICATIONS/moduli-spaces-of-riemann-surfaces-with-boundary.pdf (2003)

  2. Bödigheimer, C.-F., Tillmann, U. Stripping and splitting decorated mapping class groups. In: Cohomological methods in homotopy theory (Bellaterra, 1998). Progr. Math., vol. 196, pp. 47–57. Birkhäuser, Basel (2001)

  3. Bowditch B.H., Epstein D.B.A. (1988) Natural triangulations associated to a surface. Topology 27(1): 91–117

    Article  MATH  MathSciNet  Google Scholar 

  4. Chas, M., Sullivan, D. String topology. Ann. Math. Preprint: math.GT/9911159 (1999)

  5. Cohen, F.R., Tillmann, U. Toward homology operations for mapping class groups. In Homotopy theory via algebraic geometry and group representations (Evanston, IL, 1997). Contemp. Math., vol. 220, pp. 35–46. Amer. Math. Soc., Providence, RI (1998)

  6. Cohen, R., Godin, V. A polarized view of string topology. In: Proceedings of 2002 Conference on Top. Geo. and Quant. Field theory in honor of G. Segal, (2003)

  7. Conant J., Vogtmann K. (2003) On a theorem of Kontsevich. Algebr. Geom. Topol. 3, 1167–1224 (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  8. Culler M., Vogtmann K. (1986) Moduli of graphs and automorphisms of free groups. Invent. Math. 84(1): 91–119

    Article  MATH  MathSciNet  Google Scholar 

  9. Ehrenfried, R. Die Homologie der Modulräume berandeter Riemannscher Flächen von kleinem Geschlecht. Bonner Mathematische Schriften [Bonn Mathematical Publications], 306. Universität Bonn Mathematisches Institut, Bonn, 1998. Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, 1997

  10. Fiedorowicz, Z., Song, Y. The braid structure of mapping class groups. Sci. Bull. Josai Univ., (Special issue 2), 21–29, 1997. Surgery and geometric topology (Sakado, 1996)

  11. Galatius S. (2004) Mod p homology of the stable mapping class group. Topology 43(5): 1105–1132

    Article  MATH  MathSciNet  Google Scholar 

  12. Harer J., Zagier D. (1986) The Euler characteristic of the moduli space of curves. Invent. Math. 85(3): 457–485

    Article  MATH  MathSciNet  Google Scholar 

  13. Harer J.L. (1986) The virtual cohomological dimension of the mapping class group. Invent. Math. 84(1): 157–176

    Article  MATH  MathSciNet  Google Scholar 

  14. Harer J.L. (1990) Stability of the homology of the moduli spaces of Riemann surfaces with spin structure. Math. Ann. 287(2): 323–334

    Article  MATH  MathSciNet  Google Scholar 

  15. Igusa, K. Higher Franz-Reidemeister torsion. In: AMS/IP Studies in Advanced Mathematics vol. 31. Chap. 8 American Mathematical Society, Providence, RI, 2002.

  16. Jones J., D.S. Cyclic homology and equivariant homology. Invent. Math. 87(2), 403–423 (1987)

  17. Kaufmann R.M., Livernet M., Penner R.C. (2003) Arc operads and arc algebras (electronic). Geom. Topol. 7, 511–568

    Article  MATH  MathSciNet  Google Scholar 

  18. Kontsevich M. (1992) Intersection theory on the moduli space of curves and the matrix Airy function. Comm. Math. Phys. 147(1): 1–23

    Article  MATH  MathSciNet  Google Scholar 

  19. Madsen, I., Weiss, M. The stable moduli space of Riemann surfaces : Mumford’s conjecture. preprint (2002)

  20. Miller E.Y. (1986) The homology of the mapping class group. J. Differ. Geom. 24(1): 1–14

    MATH  Google Scholar 

  21. Milnor J. Introduction to algebraic K-theory. Princeton University Press, Princeton, N.J. Annals of Mathematics Studies, No. 72 (1971)

  22. Mumford, D. Towards an enumerative geometry of the moduli space of curves. In: Arithmetic and geometry, vol. II. Progr. Math. vol. 36 pp. 271–328. Birkhäuser Boston, Boston, MA (1983)

  23. Nielsen J. (1924) Die isomorphismengruppe der frien gruppen. Ann. Math. 91, 169–209

    Article  MATH  Google Scholar 

  24. Penner R.C. (1987) The decorated Teichmüller space of punctured surfaces. Comm. Math. Phys. 113(2): 299–339

    Article  MATH  MathSciNet  Google Scholar 

  25. Penner R.C. (1988) Perturbative series and the moduli space of Riemann surfaces. J. Diff. Geom. 27(1): 35–53

    MATH  MathSciNet  Google Scholar 

  26. Stasheff, J.D. Homotopy associativity of H-spaces. I, II. Trans. Amer. Math. Soc. 108 (1963), 275-292; ibid. 108, 293–312 (1963)

  27. Strebel, K. Quadratic differentials. In: Ergebnisse der Mathematik und ihrer Grenzgebiete (3), [Results in Mathematics and Related Areas (3)] vol. 5. Springer, Berlin Heidelberg New York (1984)

  28. Tillmann U. (1997) On the homotopy of the stable mapping class group. Invent. Math. 130(2): 257–275

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

    Véronique Godin

Authors
  1. Véronique Godin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Véronique Godin.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Godin, V. The unstable integral homology of the mapping class groups of a surface with boundary. Math. Ann. 337, 15–60 (2007). https://doi.org/10.1007/s00208-006-0025-7

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-006-0025-7

Keywords

Search

Navigation