Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-03T02:42:33.599Z Has data issue: false hasContentIssue false

On the homeotopy group of a non-orientable surface

Published online by Cambridge University Press:  24 October 2008

Joan S. Birman
Affiliation:
Stevens Institute of Technology
D. R. J. Chillingworth
Affiliation:
University of Warwick

Extract

Let X be a closed, compact connected 2-manifold (a surface), which we will denote by O or N if we wish to stress that X is orientable or non-orientable. Let G(X) denote the group of all homeomorphisms XX, D(X) the normal subgroup of homeomorphisms isotopic to the identity, and H(X) the factor group G(X)/D(X), i.e. the homeotopy group of X. The problem of determining generators for H(O) was considered by Lickorish in (7, 8), and the second of these papers specifies a finite set of generators of a particularly simple type. In (10) and (11) Lickorish considered the analogous problem for non-orientable surfaces, and, using Lickorish's partial results, Chilling-worth (4) determined a finite set of generators for H(N). While the generators obtained for H(O) and H(N) were strikingly similar, it was noteworthy that the techniques used in the two cases were different, and in particular that little use was made in the non-orientable case of the earlier results obtained on the orientable case. The purpose of this note is to show that the results of Lickorish and Chillingworth on non-orientable surfaces follow rather easily from the work in (7, 8) by an application of some ideas from the theory of covering spaces (2). Moreover, while Lickorish and Chillingworth sought only to find generators, we are able to show (Theorem 1) how in fact the entire structure of the group H(N) is determined by H(O), where O is an orientable double cover of N. Finally, we are able to determine defining relations for H(N) for the case where N is the connected sum of 3 projective planes (Theorem 3).

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1972

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)Birman, J.Mapping class groups and their relationship to braid groups. Comm. Pure Appl. Math. 22 (1969), 213238.CrossRefGoogle Scholar
(2)Birman, J. and Hilden, H. On lifting and projecting homeomorphisms (to appear).Google Scholar
(3)Birman, J. On Siegel's modular group, to appear in Math. Ann.Google Scholar
(4)Chillingworth, D. R. J.A finite set of generators for the homeotopy group of a non-orientable surface. Proc. Cambridge Philos. Soc. 65 (1969), 409430.CrossRefGoogle Scholar
(5)Dehn, M.Die Gruppe der Abbildungsklassen, Acta Math. 69 (1938), 135206.CrossRefGoogle Scholar
(6)Karass, A., Magnus, W. and Solitar, D.Elements of finite order in a group with a single defining relation. Comm. Pure Appl. Math. 13 (1960), 5766.CrossRefGoogle Scholar
(7)Lickorish, W. B. R.A representation of orientable combinatorial 3-manifolds. Ann. of Math. 76 (1962), 531540.CrossRefGoogle Scholar
(8)Lickorish, W. B. R.A finite set of generators for the homeotopy group of a 2-manifold. Proc. Cambridge Philos. Soc. 60 (1964), 769778.CrossRefGoogle Scholar
(9)Lickorish, W. B. R.On the homeotopy group of a 2-manifold (corrigendum). Proc. Cambridge Philos. Soc. 62 (1966), 679681.CrossRefGoogle Scholar
(10)Lickorish, W. B. R.Homeomorphisms of non-orientable 2-manifolds. Proc. Cambridge Philos. Soc. 59 (1963), 307317.CrossRefGoogle Scholar
(11)Lickorish, W. B. R.On the homeomorphisms of a non-orientable surface. Proc. Cambridge Philos. Soc. 61 (1965), 6164.CrossRefGoogle Scholar
(12)Magnus, W., Karass, A. and Solitar, D.Combinatorial Group Theory, John Wiley and Son, 1966.Google Scholar
(13)Nielsen, J.Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen, Acta Math. 50 (1927), 189358.CrossRefGoogle Scholar