Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-28T13:53:16.744Z Has data issue: false hasContentIssue false
  • English
  • Français

Braid groups and the group of homeomorphisms of a surface

Published online by Cambridge University Press:  24 October 2008

G. P. Scott
G. P. Scott
Affiliation:
University of Liverpool
Get access Rights & Permissions [Opens in a new window]

Extract

Braid groups on the plane were denned by Artin(1) in 1925. More recently Fox(8) defined braid groups on arbitrary topological spaces, the situation being particularly interesting if the space is a 2-manifold. Presentations of the braid groups on R2, S2 and P2 can be found in (2), (6) and (13) respectively, and some general results on braid groups of compact 2-manifolds can be found in (7). In section 1 of this paper, we give finite presentations for the braid groups on all closed 2-manifolds except S2 and P2.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 68 , Issue 3 , November 1970 , pp. 605 - 617
Copyright
Copyright © Cambridge Philosophical Society 1970

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Artin, E.Theorie der Zïpfe. Abh. Math. Sem. Univ. Hamburg 4 (1925), 47–72.CrossRefGoogle Scholar
(2)Artin, E.Theory of braids. Ann. of Math. 48 (1947), 101–126.CrossRefGoogle Scholar
(3)Artin, E.Braids and permutations. Ann. of Math. 48 (1947), 643–649.CrossRefGoogle Scholar
(4)Earle, C. J. and Eells, J.The diffeomorphism group of a compact Riemann surface. Bull. Amer. Math. Soc. 73 (1967), 557–559.Google Scholar
(5)Epstein, D. B. A.Curves on 2-manifolds and isotopies. Acta Math. 115 (1966), 83–107.CrossRefGoogle Scholar
(6)Fadell, E. and Van Buskirk, J.The braid groups of E 2 and S 2. Duke Math. J. 29 (1962), 243–258.Google Scholar
(7)Fadell, E. and Neuwirth, L.Configuration Spaces. Math. Scand. 10 (1962), 111–118.CrossRefGoogle Scholar
(8)Fox, R. H. and Neuwirth, L.The braid groups. Math. Scand. 10 (1962), 119–126.Google Scholar
(9)Hamstrom, M. E.Homotopy groups of the space of homeomorphiems on a 2-manifold. Illinois J. Math. 10 (1966), 563–573.Google Scholar
(10)McCarty, G. S.Homeotopy groups. Trans. Amer. Math. Soc. 106 (1963), 293–304.CrossRefGoogle Scholar
(11)Nielsen, J.Die struktur periodischer transformationen von Flächen. Danske Vid. Selsk. Mat.-Fis. (1937), pp. 1–37.Google Scholar
(12)Scott, G. P.The space of homeomorphisms of a 2-manifold. Topology 9 (1970), 97–109.CrossRefGoogle Scholar
(13)Van Buskirk, J.Braid groups of compact 2-manifolds with elements of finite order. Trans. Amer. Math. Soc. 122 (1966), 81–97.Google Scholar
(14)Magnus, W.Das Identitätsproblem fur Gruppen mit einer definierenden Relation. Math. Ann. 106 (1932), 295–307.CrossRefGoogle Scholar