Osaka Journal of Mathematics

A presentation for the mapping class group of a non-orientable surface from the action on the complex of curves

Błażej Szepietowski

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Abstract

We study the action of the mapping class group $\mathcal{M}(F)$ on the complex of curves of a non-orientable surface $F$. Following the outline of [1] we obtain, using the result of [4], a presentation for $\mathcal{M}(F)$ defined in terms of the mapping class groups of the complementary surfaces of collections of curves, provided that $F$ is not sporadic, i.e. the complex of curves of $F$ is simply connected. We also compute a finite presentation for the mapping class group of each sporadic surface.

Article information

Source
Osaka J. Math., Volume 45, Number 2 (2008), 283-326.

Dates
First available in Project Euclid: 15 July 2008

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1216151101

Mathematical Reviews number (MathSciNet)
MR2441942

Zentralblatt MATH identifier
1152.57019

Subjects
Primary: 57N05: Topology of $E^2$ , 2-manifolds
Secondary: 20F05: Generators, relations, and presentations 20F38: Other groups related to topology or analysis

Citation

Szepietowski, Błażej. A presentation for the mapping class group of a non-orientable surface from the action on the complex of curves. Osaka J. Math. 45 (2008), no. 2, 283--326. https://projecteuclid.org/euclid.ojm/1216151101


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References

  • S. Benvenuti: Finite presentations for the mapping class group via the ordered complex of curves, Adv. Geom. 1 (2001), 291--321.
  • J.S. Birman: Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math. 22 (1969), 213--238.
  • J.S. Birman and D.R.J. Chillingworth: On the homeotopy group of a non-orientable surface, Proc. Cambridge Philos. Soc. 71 (1972), 437--448.
  • K.S. Brown: Presentations for groups acting on simply-connected complexes, J. Pure Appl. Algebra 32 (1984), 1--10.
  • M. Dehn: Die Gruppe der Abbildungsklassen, Acta Math. 69 (1938), 135--206.
  • D.B.A. Epstein: Curves on $2$-manifolds and isotopies, Acta Math. 115 (1966), 83--107.
  • S. Gervais: A finite presentation of the mapping class group of a punctured surface, Topology 40 (2001), 703--725.
  • J. Harer: The second homology group of the mapping class group of an orientable surface, Invent. Math. 72 (1983), 221--239.
  • W.J. Harvey: Boundary structure of the modular group; in Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), Ann. of Math. Stud., 97, Princeton Univ. Press, Princeton, N.J., 1981, 245--251.
  • A. Hatcher and W. Thurston: A presentation for the mapping class group of a closed orientable surface, Topology 19 (1980), 221--237.
  • S. Hirose: A complex of curves and a presentation for the mapping class group of a surface, Osaka J. Math. 39 (2002), 795--820.
  • N.V. Ivanov: Complexes of curves and Teichmüller modular groups, Uspekhi Mat. Nauk 42 (1987), 49--91, English transl.: Russ. Math. Surv. 42, (1987) 55--107.
  • N.V. Ivanov: Automorphisms of Teichmüller modular groups: in Topology and Geometry---Rohlin Seminar, Lecture Notes in Math. 1346, Springer, Berlin, 1988, 199--270.
  • D.L. Johnson: Homeomorphisms of a surface which act trivially on homology, Proc. Amer. Math. Soc. 75 (1979), 119--125.
  • M. Korkmaz: Mapping class groups of nonorientable surfaces, Geom. Dedicata 89 (2002), 109--133.
  • W.B.R. Lickorish: On the homeomorphisms of a non-orientable surface, Proc. Cambridge Philos. Soc. 61 (1965), 61--64.
  • W. Magnus, A. Karrass and D. Solitar: Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations, Second revised edition, Dover, New York, 1976.
  • L. Paris and D. Rolfsen: Geometric subgroups of mapping class groups, J. Reine Angew. Math. 521 (2000), 47--83.
  • L. Paris: Actions and irreducible representations of the mapping class group, Math. Ann. 322 (2002), 301--315.
  • M. Stukow: Dehn twists on nonorientable surfaces, Fund. Math. 189 (2006), 117--147.
  • B. Szepietowski: The mapping class group of a nonorientable surface is generated by three elements and by four involutions, Geom. Dedicata 117 (2006), 1--9.
  • B. Wajnryb: A simple presentation for the mapping class group of an orientable surface, Israel J. Math. 45 (1983), 157--174.