Mapping class group of a non-orientable surface and moduli space of Klein surfaces

Błażej Szepietowski

doi:10.1016/S1631-073X(02)02617-1

Mapping class group of a non-orientable surface and moduli space of Klein surfaces
[Groupe modulaire d'une surface non-orientable et l'espace des modules des surfaces de Klein]

Błażej Szepietowski ¹

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

Comptes Rendus. Mathématique, Volume 335 (2002) no. 12, pp. 1053-1056.

Abstract (VO)
Résumé

As for Riemann surfaces, the moduli space of closed non-orientable Klein surfaces of genus g can be defined as the orbit space of the Teichmüller space $𝒯_{g}$ by the mapping class group Mod_g of a closed non-orientable surface. Using the set of generators given by Birman and Chillingworth, we prove that the latter group is generated by involutions. We conclude, using the Armstrong's result, that the moduli space is simply-connected.

Comme pour les surfaces de Riemann, l'espace des modules des surfaces de Klein fermées, non-orientable et de genre g peut être défini comme l'espace des orbites de l'espace de Teichmüller $𝒯_{g}$ sous l'action du groupe modulaire Mod_g d'une surface fermée, non-orientable. Utilisant l'ensemble de générateurs donné par Birman et Chillingworth nous prouvons que le dernier groupe est engendré par des involutions. On en déduit, utilisant le résultat d'Armstrong, que l'espace des modules est simplement-connexe.

Reçu le : 2002-10-15
Accepté le : 2002-10-23
Publié le : 2002-11-21

DOI : 10.1016/S1631-073X(02)02617-1

Affiliations des auteurs :

Błażej Szepietowski ¹

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

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     title = {Mapping class group of a non-orientable surface and moduli space of {Klein} surfaces},
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Błażej Szepietowski. Mapping class group of a non-orientable surface and moduli space of Klein surfaces. Comptes Rendus. Mathématique, Volume 335 (2002) no. 12, pp. 1053-1056. doi : 10.1016/S1631-073X(02)02617-1. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02617-1/

Bibliographie
Cité par

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[2] M.A. Armstrong The fundamental group of the orbit space of a discontinous group, Proc. Cambridge Philos. Soc., Volume 64 (1968), pp. 299-301

[3] J.S. Birman; D.R. Chillingworth On the homeotopy group of a non-orientable surface, Proc. Cambridge Philos. Soc., Volume 71 (1972), pp. 437-448

[4] W.B.R. Lickorish Homeomorphisms of non-orientable two-manifolds, Proc. Cambridge Philos. Soc., Volume 59 (1963), pp. 307-317

[5] A.M. Macbeth; D. Singerman Spaces of subgroups and Teichmüller space, Proc. London Math. Soc., Volume 31 (1975), pp. 211-256

[6] C. Maclachlan Modulus space is simply-connected, Proc. Amer. Math. Soc., Volume 29 (1971), pp. 185-186

[7] J. McCarthy; A. Papadopoulus Involutions in surface mapping class groups, Enseign. Math., Volume 33 (1987), pp. 275-290

Marta Leśniak; Błażej Szepietowski Generating the mapping class group of a nonorientable surface by three torsions, Geometriae Dedicata, Volume 216 (2022) no. 4, p. 14 (Id/No 40) | DOI:10.1007/s10711-022-00698-3 | Zbl:1493.57009
Marta Leśniak Torsion normal generators of the mapping class group of a nonorientable surface, Journal of Pure and Applied Algebra, Volume 226 (2022) no. 6, p. 24 (Id/No 106964) | DOI:10.1016/j.jpaa.2021.106964 | Zbl:1489.57010
Błażej Szepietowski Low-dimensional linear representations of the mapping class group of a nonorientable surface, Algebraic Geometric Topology, Volume 14 (2014) no. 4, pp. 2445-2474 | DOI:10.2140/agt.2014.14.565 | Zbl:1304.57031
Błażej Szepietowski Low-dimensional linear representations of the mapping class group of a nonorientable surface, Algebraic Geometric Topology, Volume 14 (2014) no. 4, p. 2445 | DOI:10.2140/agt.2014.14.2445
Michał Stukow Generating mapping class groups of nonorientable surfaces with boundary, advg, Volume 10 (2010) no. 2, p. 249 | DOI:10.1515/advgeom.2010.010
Błażej Szepietowski The mapping class group of a nonorientable surface is generated by three elements and by four involutions., Geometriae Dedicata, Volume 117 (2006), pp. 1-9 | DOI:10.1007/s10711-005-9004-5 | Zbl:1091.57014

Cité par 6 documents. Sources : Crossref, zbMATH

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