Skip to main content
SpringerLink
Log in

Embeddings of braid groups into mapping class groups and their homology

  • Conference paper
Configuration Spaces

Part of the book series: CRM Series ((CRMSNS))

Abstract

We construct several families of embeddings of braid groups into mapping class groups of orientable and non-orientable surfaces and prove that they induce the trivial map in stable homology in the orientable case, but not so in the non-orientable case. We show that these embeddings are non-geometric in the sense that the standard generators of the braid group are not mapped to Dehn twists.

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

Access this chapter

Log in via an institution
eBook
USD 24.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 34.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. E. Artin, Theorie der Zöpfe, Abh. Math. Semin. Univ. Hamburg 4 (1925), 47–72.

    Article  MATH  MathSciNet  Google Scholar 

  2. J. S. Birman and D. R. J. Chillingworth, On the homeotopy group of a non-orientable surface, In: Proc. Cambridge Philos. Soc. 71 (1972), 437–448.

    Article  MATH  MathSciNet  Google Scholar 

  3. J. S. Birman and D. R. J. Chillingworth, Erratum: ”On the homeotopy group of a non-orientable surface” [Proc. Cambridge Philos. Soc. 71 (1972), 437-448; MR0300288], Math. Proc. Cambridge Philos. Soc. 136 (2004), no. 2, 441.

    Article  MathSciNet  Google Scholar 

  4. J. S. Birman and H. M. Hilden, Lifting and projecting homeomorphisms, Arch. Math. (Basel) 23 (1972), 428–434.

    Article  MATH  MathSciNet  Google Scholar 

  5. J. S. Birman and H. M. Hilden, On isotopies of homeomorphisms of Riemann surfaces, Ann. of Math. (2) 97 (1973), 424–439.

    Article  MATH  MathSciNet  Google Scholar 

  6. S. K. Boldsen, Improved homological stability for the mapping class group with integral or twisted coefficients, arXiv:0904.3269

    Google Scholar 

  7. C.-F. Bödigheimer, On the topology of moduli spaces of Riemann surfaces, Part II: homology operations, Math. Gottingensis, Heft 23 (1990).

    Google Scholar 

  8. F. R. Cohen, Homology of mapping class groups for surfaces of low genus, In: The Lefschetz Centennial Conference”, Part II (Mexico City, 1984), Contemp. Math., 58, II, Amer. Math. Soc., Providence, RI, 1987, 21–30.

    Google Scholar 

  9. F. R. Cohen, T. J. Lada and J. P. May, “The Homology of Iterated Loop Spaces”, Lecture Notes in Mathematics, Vol. 533. Springer-Verlag, Berlin-New York, 1976.

    Google Scholar 

  10. N. V. Ivanov, Complexes of curves and the Teichmüller modular group, Uspekhi Mat. Nauk 42 (1987), 49–91.

    MATH  MathSciNet  Google Scholar 

  11. M. Korkmaz, First homology group of mapping class groups of nonorientable surfaces. Math. Proc. Cambridge Philos. Soc. 123 (1998), no. 3, 487–499.

    Article  MATH  MathSciNet  Google Scholar 

  12. E. Y. Miller, The homology of the mapping class group, J. Differential Geom. 24 (1986), 1–14.

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  14. J. Powell, Two theorems on the mapping class group of a surface, Proc. Amer. Math. Soc. 68 (1978), no. 3, 347–350.

    Article  MATH  MathSciNet  Google Scholar 

  15. O. Randal-Williams, Resolutions of moduli spaces and homological stability. arXiv:0909.4278

    Google Scholar 

  16. G. Segal and U. Tillmann, Mapping configuration spaces to moduli spaces, In: “Groups of Diffeomorphisms”, Adv. Stud. Pure Math, Vol. 52, Math. Soc. Japan, Tokyo, 2008, 469–477.

    Google Scholar 

  17. Y. Song and U. Tillmann, Braids, mapping class groups, and categorical delooping, Math. Ann. 339 (2007), no. 2, 377–393.

    Article  MATH  MathSciNet  Google Scholar 

  18. B. Szepietowski, Embedding the braid group in mapping class groups, Publ. Mat. 54 (2010), no. 2, 359–368.

    Article  MATH  MathSciNet  Google Scholar 

  19. U. Tillmann, Higher genus surface operad detects infinite loop spaces, Math. Ann. 317 (2000), no. 3, 613–628.

    Article  MATH  MathSciNet  Google Scholar 

  20. U. Tillmann, The representation of the mapping class group of a surface on its fundamental group in stable homology, Q. J. Math. 61 (2010), no. 3, 373–380.

    Article  MATH  MathSciNet  Google Scholar 

  21. N. Wahl, Infinite loop space structure(s) on the stable mapping class group, Topology 43 (2004), no. 2, 343–368.

    Article  MATH  MathSciNet  Google Scholar 

  22. N. Wahl, Homological stability for the mapping class groups of non-orientable surfaces, Invent. Math. 171 (2008), no. 2, 389–424.

    Article  MATH  MathSciNet  Google Scholar 

  23. B. Wajnryb, Artin groups and geometric monodromy, Invent. Math. 138 (1999), no. 3, 563–571.

    Article  MATH  MathSciNet  Google Scholar 

  24. B. Wajnryb, Relations in the mapping class group. In: “Problems on Mapping Class Groups and Related Topics”, Proc. Sympos. Pure Math., Vol. 74, Amer. Math. Soc., Providence, RI, 2006, 115–120.

    Google Scholar 

Download references

Authors
  1. Carl-Friedrich Bödigheimer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ulrike Tillmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

A. Bjorner F. Cohen C. De Concini C. Procesi M. Salvetti

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Scuola Normale Superiore Pisa

About this paper

Cite this paper

Bödigheimer, CF., Tillmann, U. (2012). Embeddings of braid groups into mapping class groups and their homology. In: Bjorner, A., Cohen, F., De Concini, C., Procesi, C., Salvetti, M. (eds) Configuration Spaces. CRM Series. Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-431-1_7

Download citation

Publish with us

Policies and ethics

Search

Navigation