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Poincaré’s works leading to the Poincaré conjecture

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Abstract

In the last decade, the Poincaré conjecture has probably been the most famous statement among all the contributions of Poincaré to the mathematics community. There have been many papers and books that describe various attempts and the final works of Perelman leading to a positive solution to the conjecture, but the evolution of Poincaré’s works leading to this conjecture has not been carefully discussed or described, and some other historical aspects about it have not been addressed either. For example, one question is how it fits into the overall work of Poincaré in topology, and what are some other related questions that he had raised. Since Poincaré did not state the Poincaré conjecture as a conjecture but rather raised it as a question, one natural question is why he did this. In order to address these issues, in this paper, we examine Poincaré’s works in topology in the framework of classifying manifolds through numerical and algebraic invariants. Consequently, we also provide a full history of the formulation of the Poincaré conjecture which is richer than what is usually described and accepted and hence gain a better understanding of overall works of Poincaré in topology. In addition, this analysis clarifies a puzzling question on the relation between Poincaré’s stated motivations for topology and the Poincaré conjecture.

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Notes

  1. Some summaries and analysis of Poincaré’s work in topology, including the other three supplements, have been carried out in, for example, Dieudonné (1989, Chap 1), Sarkaria (1999; 1994), Novikov (2004), the translator’s introduction to Poincaré (2010), Volkert (1996), Gray (2013, Chap 8), Verhulst (2012, Chap. 10), de Saint-Gervais (2014). Our discussion is complementary to them in some sense.

  2. See Poincaré (1892, 633; 2010, 1). In this paper, we will use the English translation of these Poincaré’s papers in Poincaré (2010).

  3. See Poincaré (1892, 634; 2010, 2).

  4. See Poincaré (1895, §13, 50; 2010, §13, 41).

  5. See Poincaré (1895, 55; 2010, 44).

  6. See Poincaré (1895, 65; 2010, 52).

  7. See Poincaré (1895, 51; 2010, §13, 41).

  8. See Poincaré (1895, 66; 2010, 53).

  9. See Poincaré (1895, 51; 2010, 41).

  10. See Poincaré (1895, 66; 2010, 53).

  11. See Poincaré (1895, 65; 2010, 52).

  12. See Poincaré (1892, 635–636; 2010, 2–3).

  13. See Poincaré (1895, §14, 66; 2010, 53).

  14. See Poincaré (1900, 308; 2010, 134).

  15. See Poincaré (1900, 307; 2010, 133).

  16. See Poincaré (1904, 45; 2010, 179).

  17. See Poincaré (1904, 46; 2010, 179).

  18. See Poincaré (1904, 62; 2010, 190).

  19. See Poincaré (1904, 110; 2010, 224).

  20. See Poincaré (1895, 51; 2010, 41).

  21. See Poincaré (1895, 6; 2010, 9).

  22. This omission of the claim in Poincaré (1900, 308) about a characterization of the sphere makes one wonder whether Poincaré had already became suspicious of his statement, given that he published a counter-example to the claim a few years later in Poincaré (1904) as explained earlier.

  23. It should be emphasized that the original reason for writing this paper was entirely to understand better the history of the Poincaré conjecture. This answer to the question posed in Ji and Wang (2020, 393–394) was an accidental, or unexpected, outcome.

  24. See Poincaré (1895, 2; 2010, 6).

  25. A detailed explanation of these stated motivations and applications for topology by Poincaré was given in Ji and Wang (2020). This was also the main purpose of that paper.

  26. Jeremy Gray commented that though Pont’s statement is true, no one had the classification of compact orientable surfaces before Riemann. The problem is that Riemann’s definitions and ideas were far from rigorous.

  27. Segal wrote (Segal 1999, 849): “it was von Dyck’s work which inspired Poincaré to what we today call the Euler–Poincaré characteristic.”

  28. The influence of Dyck on Poincaré was also asserted by Dieudonné (1989, 17): “Poincaré starting point was the same as those of his predecessors (he quoted Riemann, Betti, and von Dyck)”, and by Pont (1974, 153): “Located at the boundary of the period we are studying, Dyck’s contributions act as a center. If, in fact, it is possible to see them as the culmination of a long chain of work, one can also identify one of the sources of the current of ideas which leads to Poincaré’s research and to modern topology.”

  29. See Klein (1882; 1963, 24).

  30. Furthermore, the Poincaré index theorem is more than the general topological thinking. It is a very special case of the Poincaré–Hopf index theorem in differential topology which relates the indices of the singularities of a nonzero vector field on a closed manifold of any dimension to the Euler characteristic number of the manifold. The general Poincaré–Hopf index theorem for vector fields on compact manifolds of higher dimensions was proved by Hopf in Hopf (1927).

  31. See Poincaré (1895, 18; 2010, 18). This approach is an early form of the later de Rham cohomology groups of manifolds.

  32. See Hadamard (1897, 1898); Barrow-Green (1997).

  33. Some basic references on Thurston’s geometrization program include: (1) Thurston (1982), which contains the original statements by Thurston on this program, (2) Scott (1983), which contains a fairly detailed description of the eight geometries involved in the program, and (3) McMullen (2011), which gives an accessible and historical account of the geometrization program, in particular a one line summary: “Thus a principal corollary of the geometrization conjecture is that most 3-manifolds are hyperbolic” on page 262.

    One can see from Thurston (1982, 361) and McMullen (2011, 266) that some of the most striking results of Thurston are that certain manifolds, for example, Haken manifolds, are hyperbolic. Thurston’s geometrization program is the first one on a list of 24 problems proposed by Thurston (1982, 379–380). The last ones on the list that were solved assert that certain manifolds are virtually Haken and hence virtually hyperbolic, and hyperbolic manifolds virtually have the positive first Betti number (Agol 2014). We also note that the Seifert manifolds mentioned in Sect. 4 play an important role in Thurston’s geometrization program.

  34. See also the first section of Johnson (1981).

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Acknowledgements

This paper was substantially modified from one part of the preprint in 2019 titled “Poincaré’s topology and Poincaré Conjecture” which was read and commented by Jeremy Gray. We would like to thank him for his many valuable suggestions and communications on how to understand and write the history of mathematics. An anonymous referee and Jeremy Gray also read an earlier version of the current paper. We would like to thank them for suggesting to understand the history of the Poincaré conjecture in connection with Poincaré’s earlier works related to topology as well and hence to gain a bigger historical perspective. The whole section, §6, was written based on their suggestions and comments. We are also grateful to Jeremy Gray for his many comments and suggestions on how to improve the final version of this paper. We would also like to thank Chanchan Guo and Liyun Pan for their help with the reference (Dehn and Heegaard 1907).

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  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48105, USA

    Lizhen Ji

  2. Institute for Advanced Study in History of Science, Northwest University, Xi’an, Shaanxi, 710127, China

    Chang Wang

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Lizhen Ji: Partially supported by a Grant from the Simons Foundation #353785.

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Ji, L., Wang, C. Poincaré’s works leading to the Poincaré conjecture. Arch. Hist. Exact Sci. 76, 223–260 (2022). https://doi.org/10.1007/s00407-021-00283-2

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