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The Mapping Class Group of a Nonorientable Surface is Generated by Three Elements and by Four Involutions

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Abstract

We prove that the mapping class group of a closed nonorientable surface is generated by three elements and by four involutions

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References

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

    MathSciNet  MATH  Google Scholar 

  2. Brendle T.E., Farb B. (2004). Every mapping class group is generated by 6 involutions. J. Algebra 278:187–198

    Article  MATH  MathSciNet  Google Scholar 

  3. Chillingworth D.R.J. (1969). A finite set of generators for the homeotopy group of a non-orientable surface. Proc. Cambridge. Philos. Soc 65:409–430

    Article  MATH  MathSciNet  Google Scholar 

  4. Humphries S. (1979). Generators of the mapping class group. In: Fenn R (eds). Topology of Low Dimensional Manifolds, Lecture Notes in Math. 722. Springer-Verlag, Berlin, pp. 44–47

    Chapter  Google Scholar 

  5. Kassabov, M.: Generating mapping class groups by involutions, arXiv:math.GT/031 1455, v1 25 Nov 2003.

  6. Korkmaz M. (1998). First homology group of mapping class group of nonorientable surfaces. Math. Proc. Cambridge. Philos. Soc 123:487–499

    Article  MATH  MathSciNet  Google Scholar 

  7. Korkmaz M. (2002). Mapping class groups of nonorientable surfaces. Geom. Dedicata 89:109–133

    Article  MathSciNet  Google Scholar 

  8. Korkmaz M. (2005). Generating the surface mapping class group by two elements. Trans. Amer. Math. Soc 357:3299–3310

    Article  MATH  MathSciNet  Google Scholar 

  9. Lickorish W.B.R. (1963). Homeomorphisms of non-orientable two-manifolds. Proc. Cambridge. Philos. Soc 59:307–317

    MATH  MathSciNet  Google Scholar 

  10. Luo, F.: Torsion elements in the mapping class group of a surface, arXiv:math.GT/0004048, v1 8 Apr 2000.

  11. McCarthy J., Papadopoulus A. (1987). Involutions in surface mapping class groups. Enseign. Math. 33:275–290

    MATH  MathSciNet  Google Scholar 

  12. Stukow M. (2004). The extended mapping class group is generated by 3 symmetries. C.R. Acad. Sci. Paris, Ser I 338(5):403–406

    MATH  MathSciNet  Google Scholar 

  13. Stukow, M.: Dehn twists on nonorientable surfaces, preprint 2004

  14. Szepietowski B. (2002). Mapping class group of a non-orientable surface and moduli space of Klein surfaces. C.R. Acad. Sci. Paris, Ser. I 335:1053–1056

    MATH  MathSciNet  Google Scholar 

  15. Wajnryb B. (1996). Mapping class group of a surface is generated by two elements. Topology 35:377–383

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Gdańsk University, Wita Stwosza 57, 80-952, Gdańsk, Poland

    Błażej Szepietowski

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  1. Błażej Szepietowski
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Correspondence to Błażej Szepietowski.

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Mathematics Subject Classification (2000). primary: 57N05; secondary: 20F38, 20F05.

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Szepietowski, B. The Mapping Class Group of a Nonorientable Surface is Generated by Three Elements and by Four Involutions. Geom Dedicata 117, 1–9 (2006). https://doi.org/10.1007/s10711-005-9004-5

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  • DOI: https://doi.org/10.1007/s10711-005-9004-5

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