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Clifford and the Number of Holes

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Part of the Lecture Notes in Mathematics book series (HISTORYMS,volume 2162)

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.

Keywords

  • Riemann Surface
  • Stereographical Projection
  • Orientable Surface
  • Rational Curf
  • Smooth Point

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Fig. 19.1
Fig. 19.2
Fig. 19.3

Notes

  1. 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
    figure 1

    Clifford’s circuits around and through the holes

References

  1. 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

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  2. H. Kneser, Geschlossen Flächen in dreidimensionalen Mannigfaltigkeiten. Jahresb. Deutschen Math. Ver. 38, 248–260 (1929)

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  3. J. Milnor, A unique decomposition theorem for 3-manifolds. Am. J. Math. 84, 1–7 (1962)

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© 2016 Springer International Publishing Switzerland

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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

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