Abstract
In this paper, the authors systematically discuss orbit braids in M × I with regards to orbit configuration space FG(M, n), where M is a connected topological manifold of dimension at least 2 with an effective action of a finite group G. These orbit braids form a group, named orbit braid group, which enriches the theory of ordinary braids.
The authors analyze the substantial relations among various braid groups associated to those configuration spaces FG(M, n), F(M/G, n) and F(M, n). They also consider the presentations of orbit braid groups in terms of orbit braids as generators by choosing M = ℂ with typical actions of ℤp and (ℤ2)2.
This is a preview of subscription content, log in via an institution to check access.
References
Allcock, D. and Basak, T., Geometric generators for braid-like groups, Geom. Topol., 20, 2016, 747–778.
Allcock, D. and Basak, T., Generators for a complex hyperbolic braid group, Geom. Topol., 22(6), 2018, 3435–3500.
Artin, E., Theorie der Zöpfe, Ahb. Math. Sem. Univ. Hamburg, 4, 1925, 47–72.
Artin, E., Theory of braids, Ann. of Math., 48, 1947, 101–126.
Bibby, C. and Gadish, N., Combinatorics of orbit configuration spaces, 2018, arXiv:1804.06863.
Birman, J. S., Braids, Links, and Mapping Class Groups, Princeton Univ. Press, Princeton, NJ, 1974.
Bredon, G. E., Intrduction to compact transformation groups, Pure and Applied Mathematics, 46, Academic Press, New York, London, 1972.
Brieskorn, E., Die Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe, Invent. Math., 12, 1971, 57–61.
Brieskorn, E., Sur les groupes de tresses (d’apräs V.I. Arnol’d), Séminaire Bourbaki, 24ème année (1971/1972), Lecture Notes in Math., 317, 1973, 21–44.
Chen, J. D., Lä, Z. and Wu, J., Orbit configuration spaces of small covers and quasi-toric manifolds, Science China Mathematics., 64(1), 2021, 167–196.
Chow, W. L., On the algebraical braid group, Ann. of Math., 49, 1948, 654–658.
Davis, M. and Januszkiewicz, T., Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J., 61, 1991, 417–451.
Deligne, P., Les immeubles des groupes de tresses généralisés, Invent. Math., 17, 1972, 273–302.
Fadell, E. and Neuwirth, L., Configuration spaces, Math. Scand., 10, 1962, 111–118.
Fox, R. and Neuwirth, L., The braid groups, Math. Scand., 10, 1962, 119–126.
Fricke, R. and Klein, F., Vorlesungen über die Theorie der automorphen Funktionen, Bd. I. Gruppentheoretischen Grundlagen, Teubner, Leipzig, 1897, Johnson, New York, 1965.
Goryunov, V. V., The cohomology of braid groups of series C and D, Trudy Moskov. Mat. Obshch., 42, 1981, 234–242.
Hurwitz, A., Über Riemannsche Flächen mit gegebenen Verzweigungspunkten, Math. Ann., 39(1), 1891, 1–60.
Looijenga, E., Artin groups and the fundamental groups of some moduli spaces, J. Topol., 1(1), 2008, 187–216.
Randell, R., The fundamental group of the complement of a union of complex hyperplanes: Correction, Invent. math., 80, 1985, 467–468.
Rhodes, F., On the fundamental group of a transformation group, Proceedings of the London Mathematical Society, 3(1), 1966, 635–650.
Switzer, R. M., Algebraic Topology—Homotopy and Homology, Springer-Verlag, New York, Heidelberg, 1975.
Vershinin, V. V., Braid groups and loop spaces, Russian Mathematical Surveys, 54, 1999, 273–350.
Xicoténcatl, M. A., m Orbit configuration spaces, infinitesimal braid relations in homology and equivariant loop spaces, Thesis (Ph.D.), University of Rochester, Rochester, 1997.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the National Natural Science Foundation of China (No. 11971112).
Rights and permissions
About this article
Cite this article
Li, F., Li, H. & Lü, Z. A Theory of Orbit Braids. Chin. Ann. Math. Ser. B 44, 165–192 (2023). https://doi.org/10.1007/s11401-023-0009-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11401-023-0009-x