Abstract
The notions of genus which we have studied until now for algebraic surfaces, as well as those formulated in all dimensions by Severi, are defined by algebraic means. None of them generalizes Riemann’s topological viewpoint, based on curves drawn on differentiable surfaces (see Chap. 14).
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Notes
- 1.
That is, in the algebraic differential forms without poles; see Chap. 40 for a description of the passage of the formulation in terms of integrals to the formulation in terms of forms.
- 2.
Poincaré used the term “variété”, which may be translated in English to either “manifold” or “variety”. Nowadays, the second term is generally reserved for algebraic or analytic possibly singular spaces. As Poincaré worked with differentiable objects, for which the term “variety” is much less used, we chose the first term for our translation. Nevertheless, it is very difficult to make all his ideas work without also allowing singular such “manifolds”. This is one of the reasons why “singular homology” was later developed in order to construct a completely rigorous incarnation of his ideas.
References
N. Basbois, L’émergence de la notion de groupe d’homologie. Gaz. Math. 127, 15–44 (2011)
E. Betti, Sopra gli spazi di un numero qualunque di dimensioni. Ann. Matem. Pura Appl. (2), 4, 140–158 (1871)
J. Dieudonné, Emmy Noether and algebraic topology. J. Pure Appl. Algebra 31, 5–6 (1984)
S.M. Lane, Topology becomes algebraic with Vietoris and Noether. J. Pure Appl. Algebra 39, 305–307 (1986)
H. Poincaré, Analysis situs. J. École Polytech. 1, 1–121 (1895). Republished in Œuvres de Henri Poincaré, vol. VI (Gauthier-Villars, Paris, 1953), pp. 193–288
J.-C. Pont, La topologie algébrique des origines à Poincaré (Presses Universitaires de France, Paris, 1974)
A. Weil, Riemann, Betti and the birth of Topology. Arch. Hist. Exact Sci. 20, 91–96 (1979)
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Popescu-Pampu, P. (2016). Poincaré and Analysis Situs. In: What is the Genus?. Lecture Notes in Mathematics(), vol 2162. Springer, Cham. https://doi.org/10.1007/978-3-319-42312-8_38
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