Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-20T03:39:09.072Z Has data issue: false hasContentIssue false
  • English
  • Français

A Note on the Construction of a Set of Fundamental Circuits on a Surface of Genus p

Published online by Cambridge University Press:  20 November 2018

R. G. de Buda
R. G. de Buda*
Affiliation:
Canadian General Electric Co. Ltd.,Toronto, Ont.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

On an orientable surface of genus p, a set of 2p fundamental circuits can be selected which all pass through a single point A. After cutting along the 2p circuits, the surface can be unfolded into a flat region bounded by a 4p-gon so that: the set of vertices corresponds to the one point A; and the 2p pairs of edges to the 2p fundamental circuits; and the interior of the polygon to the remainder of the surface. If the edges of the polygon are directed, the 2 edges which correspond to one fundamental circuit will be directed in opposite sense, since the surface is orientable [1]. The sequence and direction of the edges is the same as the sequence of the fundamental circuits.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 4 , Issue 1 , January 1961 , pp. 13 - 21
Copyright
Copyright © Canadian Mathematical Society 1961

References

1. Brahana, H. R., Systems of circuits on two dimensional manifolds, Ann, of Math. 23(1921), 144-168.Google Scholar
2. Coxeter, H. S. M. and Moser, W. O. J., Generators and relations for discrete groups, Ergebnisse der Mathematik 14, (Berlin, 1957), equation 3.55.Google Scholar
3. Burnside, W., Theory of groups of finite order, (London, 1911), fig. 8, p. 396.Google Scholar
4. Threlfall, W., Gruppenbilder, Abh. Sachs. Akad. Wiss. Math. - Phys. Kl. 41, no. 6, 1-59, particularly p. 35.Google Scholar
5. Hilbert, D. and Cohn Vossen, S., Geometry and the Imagination, second edition, (New York, 1952), p. 296 and figures on pp. 300-301.Google Scholar
6. Nielsen, J., Untersuchungen zur Topologie der geschlossenen zweiseitigen Flaechen, Acta Math. 50 (1927) 189-358, particularly p. 195.Google Scholar
7. James, R. G., Combinatorial topology of surfaces, Math. Mag. 29 (1955), 1-39.Google Scholar