Journal of Topology

Journal of Topology Advance Access originally published online on October 28, 2007
Journal of Topology 2008 1(1):187-216; doi:10.1112/jtopol/jtm009

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© 2007 London Mathematical Society

Artin groups and the fundamental groups of some moduli spaces

Eduard Looijenga

Faculteit Wiskunde en Informatica, University of Utrecht, P.O. Box 80.010, NL-3508 TA Utrecht, The Netherlands, looijeng{at}math.ruu.nl

We define for every affine Coxeter graph, a certain factor group of the associated Artin group and prove that some of these groups appear as orbifold fundamental groups of moduli spaces. Examples are the moduli space of nonsingular cubic algebraic surfaces and the universal nonhyperelliptic smooth genus three curve. We use this to obtain a simple presentation of the mapping class group of a compact genus three topological surface with connected boundary. This leads to a modification of Wajnryb's presentation of the mapping class groups in the higher genus case that can be understood in algebro-geometric terms.

Received May 27, 2007.

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2000 Mathematics Subject Classification 14J10, 20F36 (primary), 20F34, 14H10 (secondary)

References

1. Allcock D., Carlson J. A., Toledo D. ‘The complex hyperbolic geometry for moduli of cubic surfaces’. J. Algebraic Geom. (2002) 11:659–724.
2. Birman J. ‘Braids, links, and mapping class groups’. In: Annals of Mathematics Studies (1975) 82. Princeton NJ: Princeton University Press.
3. Bourbaki N. Groupes et algèbres de Lie (1981) Paris: Masson. Ch. 4,5 et 6, 2ieme éd.
4. Brieskorn E. ‘Die Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe’. Invent. Math. (1971) 12:57–61.[CrossRef]
5. Brieskorn E., Saito K. ‘Artingruppen und Coxetergruppen’. Invent. Math. (1972) 17:245–271.[CrossRef]
6. Deligne P. ‘Les immeubles des groupes de tresses généralisées’. Invent. Math. (1972) 17:273–301.[CrossRef]
7. Demazure M. ‘Surfaces de del Pezzo’. In: in Séminaire sur les Singularités des Surfaces (1980) Berlin–New York: Springer. Lecture Notes in Mathematics 777.
8. Dolgachev I., Libgober A. ‘On the fundamental group of the complement to a discriminant variety’. In: Algebraic geometry (1981) Berlin–New York: Springer. 1–25. (Chicago, Ill, 1980), Lecture Notes in Mathematics 862.
9. Humphries S. P. ‘Generators for the mapping class group’. In: Topology of low-dimensional manifolds (Proc. Second Sussex Conf. Chelwood Gate, 1977) (1979) Berlin–New York: Springer. 44–47. Lecture Notes in Mathematics 722.
10. van der Lek H. ‘The homotopy type of complex hyperplane complements’. In: Thesis (1983) University of Nijmegen.
11. van der Lek H. ‘Extended Artin groups’. Proc. Symp. Pure Math. (1983) 40–Part 2:117–121.
12. Libgober A. ‘On the fundamental group of the space of cubic surfaces’. Math. Z. (1978) 162:63–67.[CrossRef]
13. Lönne M. ‘Fundamental Groups of Spaces of Smooth Projective Hypersurfaces’. available at arXiv:math/0608346v1[math.AG].
14. Manin Y. Cubic Forms (1986) 2. North Holland: North Holland Math. Library 4.
15. Matsumoto M. ‘A presentation of mapping class groups in terms of Artin groups and geometric monodromy groups of singularities’. Math. Ann. (2000) 316:401–418.[CrossRef]
16. Dng Nguyêñ Viêt. ‘The fundamental groups of the spaces of regular orbits of the affine Weyl groups’. Topology (1983) 22:425–435.[CrossRef][ISI]
17. Wajnryb B. ‘A simple presentation for the mapping class group of an orientable surface’. Israel J. Math. (1983) 45:157–174. See for a correction: J. S. Birman, B. Wajnryb: Presentations of the mapping class group. Errata: 3-fold branched coverings and the mapping class group of a surface in Geometry and topology (College Park, MD, 1983/84), 24–46, Lecture Notes in Math. 1167, Springer, Berlin, 1985.[CrossRef]

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This Article Abstract FREE Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Alert me to new issues of the journal Add to My Personal Archive Download to citation manager Request Permissions Google Scholar Articles by Looijenga, E. Social Bookmarking          What's this?