Delta-structures on mapping class groups and braid groups

Berrick, A.; Hanbury, E.; Wu, J.

doi:10.1090/S0002-9947-2013-05889-8

Delta-structures on mapping class groups and braid groups
HTML articles powered by AMS MathViewer

by A. J. Berrick, E. Hanbury and J. Wu PDF

Trans. Amer. Math. Soc. 366 (2014), 1879-1903 Request permission

Abstract:

We describe a Delta-group structure on the mapping class groups of surfaces, and show that it is compatible with the Delta-group structures of the braid groups of surfaces given by Berrick-Cohen-Wong-Wu. We then prove an isomorphism theorem relating these two Delta-groups. This is the first of a pair of papers on this topic.

References

A. J. Berrick, F. R. Cohen, Y. L. Wong, and J. Wu, Configurations, braids, and homotopy groups, J. Amer. Math. Soc. 19 (2006), no. 2, 265–326. MR 2188127, DOI 10.1090/S0894-0347-05-00507-2
A. J. Berrick and E. Hanbury: Simplicial structures and normal forms for mapping class groups and braid groups, preprint (2012).
A. J. Berrick, E. Hanbury and J. Wu: Brunnian subgroups of mapping class groups and braid groups, Proc. London Math. Soc. 107 (2013), 875–906.
Frederick R. Cohen, Introduction to configuration spaces and their applications, Braids, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 19, World Sci. Publ., Hackensack, NJ, 2010, pp. 183–261. MR 2605307, DOI 10.1142/9789814291415_{0}003
F. R. Cohen and J. Wu, On braid groups, free groups, and the loop space of the 2-sphere, Categorical decomposition techniques in algebraic topology (Isle of Skye, 2001) Progr. Math., vol. 215, Birkhäuser, Basel, 2004, pp. 93–105. MR 2039761
F. R. Cohen and J. Wu, On braid groups and homotopy groups, Groups, homotopy and configuration spaces, Geom. Topol. Monogr., vol. 13, Geom. Topol. Publ., Coventry, 2008, pp. 169–193. MR 2508205, DOI 10.2140/gtm.2008.13.169
Alain Connes, Cohomologie cyclique et foncteurs $\textrm {Ext}^n$, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 23, 953–958 (French, with English summary). MR 777584
Clifford J. Earle and James Eells, A fibre bundle description of Teichmüller theory, J. Differential Geometry 3 (1969), 19–43. MR 276999
C. J. Earle and A. Schatz, Teichmüller theory for surfaces with boundary, J. Differential Geometry 4 (1970), 169–185. MR 277000, DOI 10.4310/jdg/1214429381
Edward Fadell and Lee Neuwirth, Configuration spaces, Math. Scand. 10 (1962), 111–118. MR 141126, DOI 10.7146/math.scand.a-10517
Zbigniew Fiedorowicz and Jean-Louis Loday, Crossed simplicial groups and their associated homology, Trans. Amer. Math. Soc. 326 (1991), no. 1, 57–87. MR 998125, DOI 10.1090/S0002-9947-1991-0998125-4
Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976. MR 0448362, DOI 10.1007/978-1-4684-9449-5
Christian Kassel and Vladimir Turaev, Braid groups, Graduate Texts in Mathematics, vol. 247, Springer, New York, 2008. With the graphical assistance of Olivier Dodane. MR 2435235, DOI 10.1007/978-0-387-68548-9
Jingyan Li and Jie Wu, Artin braid groups and homotopy groups, Proc. Lond. Math. Soc. (3) 99 (2009), no. 3, 521–556. MR 2551462, DOI 10.1112/plms/pdp005
Jean-Louis Loday, Homologies diédrale et quaternionique, Adv. in Math. 66 (1987), no. 2, 119–148 (French). MR 917736, DOI 10.1016/0001-8708(87)90032-6
Richard S. Palais, Natural operations on differential forms, Trans. Amer. Math. Soc. 92 (1959), 125–141. MR 116352, DOI 10.1090/S0002-9947-1959-0116352-7
Richard S. Palais, Local triviality of the restriction map for embeddings, Comment. Math. Helv. 34 (1960), 305–312. MR 123338, DOI 10.1007/BF02565942
Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
Norman Steenrod, The Topology of Fibre Bundles, Princeton Mathematical Series, vol. 14, Princeton University Press, Princeton, N. J., 1951. MR 0039258, DOI 10.1515/9781400883875
Jie Wu, A braided simplicial group, Proc. London Math. Soc. (3) 84 (2002), no. 3, 645–662. MR 1888426, DOI 10.1112/S0024611502013370
J. Wu, Combinatorial descriptions of homotopy groups of certain spaces, Math. Proc. Cambridge Philos. Soc. 130 (2001), no. 3, 489–513. MR 1816806, DOI 10.1017/S030500410100487X
W. Zhang: Group operads and homotopy theory, preprint, 2011. arxiv.org/pdf/1111.7090.