Abstract
We saw that Riemann denoted the genus by “p”, a notation which is still frequently used today, in particular for the generalizations of this notion in higher dimensions. On the other hand, Riemann did not give a name to this notion, and his definition was not the one we saw in the Introduction, in terms of holes. There is a good reason for this, namely, that the surfaces imagined by Riemann consisted of sheets which thinly cover the plane, and therefore do not admit visible holes.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The curves around each hole are for instance those represented in Fig. 4 of the Introduction. In Fig. 19.1 is also represented a curve through each hole.
Fig. 19.1 
Clifford’s circuits around and through the holes
References
W.K. Clifford, On the canonical form and dissection of a Riemann’s surface. Proc. Lond. Math. Soc. 8 (122), 292–304 (1877). Republished in Mathematical Papers (Macmillan, London, 1882). Reprinted by Chelsea, New York, 1968
H. Kneser, Geschlossen Flächen in dreidimensionalen Mannigfaltigkeiten. Jahresb. Deutschen Math. Ver. 38, 248–260 (1929)
J. Milnor, A unique decomposition theorem for 3-manifolds. Am. J. Math. 84, 1–7 (1962)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Popescu-Pampu, P. (2016). Clifford and the Number of Holes. In: What is the Genus?. Lecture Notes in Mathematics(), vol 2162. Springer, Cham. https://doi.org/10.1007/978-3-319-42312-8_19
Download citation
DOI: https://doi.org/10.1007/978-3-319-42312-8_19
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-42311-1
Online ISBN: 978-3-319-42312-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)