Avec cedram.org
Confluentes Mathematici
Rechercher un article
Recherche sur le site
Table des matières de ce fascicule | Article précédent | Article suivant
Daciberg Lima Gonçalves; John Guaschi; Miguel Maldonado
Embeddings and the (virtual) cohomological dimension of the braid and mapping class groups of surfaces
Confluentes Mathematici, 10 no. 1 (2018), p. 41-61
Article PDF
Class. Math.: 57N05, 20F36, 55R80, 55P20, 20F38, 57M07, 20J06
Mots clés: Mapping class groups, surface braid groups, finite coverings, embeddings, (virtual) cohomological dimension

Résumé - Abstract

We use the relations between the braid and mapping class groups of a compact, connected, non-orientable surface $N$ without boundary and those of its orientable double covering $S$ to study embeddings of these groups and their (virtual) cohomological dimensions. We first generalise results of [4, 14] to show that the mapping class group $MCG(N;k)$ of $N$ relative to a $k$-point subset embeds in the mapping class group $MCG(S;2k)$ of $S$ relative to a $2k$-point subset. We then compute the cohomological dimension of the braid groups of all compact, connected aspherical surfaces without boundary, generalising results of [15]. Finally, if the genus of $N$ is at least $2$, we deduce upper bounds for the virtual cohomological dimension of $MCG(N;k)$ that are coherent with computations of Ivanov.

Bibliographie

[1] Birman, J. S., Braids, Links and Mapping Class Groups, Ann. Math. Stud. 82, Princeton Univ. Press, 1975.
[2] Birman, J. S., On braid groups, Comm. Pure Appl. Math. 22 (1969), 41–72.
[3] Birman, J. S., Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math. 22 (1969), 213–238.
[4] Birman, J. S., Chillingworth, D. R. J., On the homeotopy of a non-orientable surface, Proc. Camb. Phil. Soc. 71 (1972), 437–448, and Erratum, Math. Proc. Cambridge Philos. Soc. 136 (2004), 441.
[5] Boldsen, S. K., Different versions of mapping class groups of surfaces, arXiv: 0908.2221 (2009).
[6] Brown, K. S., Cohomology of groups, Graduate Texts in Mathematics 87, Springer-Verlag, 1982.
[7] Fadell, E., Husseini, S., The Nielsen number on surfaces, Topological methods in nonlinear functional analysis (Toronto, Ont., 1982), Contemp. Math., Amer. Math. Soc., Providence, RI, 21 (1983), 59–98.
[8] Fadell, E., Neuwirth, L., Configuration spaces, Math. Scand. 10 (1962), 111–118.
[9] Farb, B., Some problems on mapping class groups and moduli space, In Problems on mapping class groups and related topics, Proc. Sympos. Pure Math. 74, 11–55, Amer. Math. Soc., Providence, RI, 2006.
[10] Farb, B., Margalit D., A primer on mapping class groups, Princeton Mathematical Series 49, Princeton Univ. Press, Princeton, NJ, 2012.
[11] Fox, R. H., Neuwirth, L., The braid groups, Math. Scand. 10 (1962), 119–126.
[12] Gonçalves, D. L., Guaschi, J., On the structure of surface pure braid groups, J. Pure Appl. Algebra 182 (2003), 33–64 (due to a printer’s error, this article was republished in its entirety with the reference 186 (2004) 187–218).
[13] Gonçalves, D. L., Guaschi, J., Braid groups of non-orientable surfaces and the Fadell-Neuwirth short exact sequence, J. Pure Appl. Algebra 214 (2010), 667–677.
[14] Gonçalves, D. L., Guaschi, J., Surface braid groups and coverings, J. London Math. Soc. 85 (2012), 855–868.
[15] Gonçalves, D. L., Guaschi, J., Inclusion of configuration spaces in Cartesian products, and the virtual cohomological dimension of the braid groups of $\mathbb{S}^{2}$ and $\mathbb{R}P^{2}$, Pac. J. Math. 287 (2017), 71–99.
[16] Gonçalves, D. L., Kudryavtseva, E., Zieschang, H., Roots of mappings on nonorientable surfaces and equations in free groups, Manuscripta Math. 107 (2002), 311–341.
[17] Guaschi, J., de Miranda e Pereiro, C., The lower central and derived series of the braid groups of compact surfaces, arXiv:1802.07636 (2018).
[18] Hamstrom, M.-E., The space of homeomorphisms on a torus, Illinois J. Math. 9 (1965), 59–65.
[19] Hamstrom, M.-E., Homotopy properties of the space of homeomorphisms on $P^2$ and the Klein bottle, Trans. Amer. Math. Soc. 120 (1965), 37–45.
[20] Hamstrom, M.-E., Homotopy groups of the space of homeomorphisms on a $2$-manifold, Illinois J. Math. 10 (1966), 563–573.
[21] Harer, J. L., The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math. 84 (1986), 157–176.
[22] Hope, G. and Tillmann, U., On the Farrell cohomology of the mapping class group of non-orientable surfaces, Proc. Amer. Math. Soc. 137 (2009), 393–400.
[23] Ivanov, N. V., Complexes of curves and the Teichmuller modular group, Russ. Math. Surv., 42–55, 1987.
[24] Ivanov, N. V., Mapping class groups, in Handbook of geometric topology, 523–633, North-Holland, Amsterdam, 2002.
[25] Johnson, D. L., Presentations of groups, London Mathematical Society Student Texts 15, $2$nd edition, Cambridge University Press, Cambridge, 1997.
[26] Lickorish, W. B. R., On the homeomorphisms of a non-orientable surface, Proc. Camb. Phil. Soc. 61 (1965), 61–64.
[27] Lima, E. L., Fundamental Groups and Covering Spaces, A. K. Peters, Ltd., Natick, MA, 2003.
[28] McCarty Jr., G. S., Homeotopy groups, Trans. Amer. Math. Soc. 106 (1963), 293–304.
[29] Olum, P., Mappings of manifolds and the notion of degree, Ann. Math. 58 (1953), 458–480.
[30] Paris, L., From braid groups to mapping class groups, In Problems on mapping class groups and related topics, Proc. Sympos. Pure Math. 74, 355–371, Amer. Math. Soc., Providence, RI, 2006.
[31] Paris, L. and Rolfsen D., Geometric subgroups of surface braid groups, Ann. Inst. Fourier 49 (1999), 417–472.
[32] Pereiro, C. M., Os grupos de tranças do toro e da garrafa de Klein, Ph.D thesis, Universidade Federal de São Carlos, Brazil, and Université de Caen Basse-Normandie, France, 2015.
[33] Scott, G. P., Braid groups and the group of homeomorphisms of a surface, Proc. Camb. Phil. Soc. 68 (1970), 605–617.
eISSN : 1793-7434

haut