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The New Topology

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L.E.J. Brouwer – Topologist, Intuitionist, Philosopher

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Abstract

The next step in Brouwer’s topological research was the study of continuous maps on manifolds. The program opened with a bang: in a brief note Brouwer proved the invariance of dimension under homeorphisms. This publication led to an unpleasant altercation with Lebesgue, who claimed to have already found a proof. In fact he had deduced the invariance from the paving principle, but failed to prove the paving principle. In the end Brouwer’s priority and superior insight was fully vindicated. In subsequent papers Brouwer enriched the arsenal of basic notions of topology with simplicial approximation and the mapping degree. The contacts with Baire, Hadamard, Blumenthal, and Hilbert, are described. Brouwer’s name became lastingly attached to his fixed point theorem. Brouwer also proved the invariance of domain theorem, which he subsequently used to salvage Klein’s continuity method for proving uniformisation. This brought him into a conflict with Paul Koebe, who was the uncrowned king of uniformisation and complex function theory. This first topological period closed with a significant feat: Brouwer defined, following Poincaré’s first approach, the general notion of dimension, and proved its ‘correctness’, i.e. showed that ℝn is n-dimensional.

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Notes

  1. 1.

    This chapter makes essential use of Freudenthal’s comments in Volume II of the Collected Works and of the paper The Problem of the Invariance of Dimension in the Growth of Modern Topology I, II of Dale Johnson.

  2. 2.

    Ein Beitrag zur Mannigfaltigkeitslehre.

  3. 3.

    The history of this particular theorem is published by Emmy Noether and Jean Cavaillès, Noether and Cavaillès (1937).

  4. 4.

    There is a fairly simple geometric representation of the above mapping, Johnson (1979), p. 171 and Young and Young (1906), pp. 165, 291.

  5. 5.

    Among others Hilbert and Moore, cf. Young and Young (1906).

  6. 6.

    Brouwer (1909b), p. 297.

  7. 7.

    Het wezen der meetkunde. The translation of ‘wezen’ by ‘nature’ is somewhat flat. Wezen expresses something more, like ‘essence’.

  8. 8.

    Brouwer (1909a), CW I, p. 116.

  9. 9.

    Brouwer (1909a), CW I. p. 117.

  10. 10.

    Not to be confused with Methodenreinheit (purity of methods) of older generations. Brouwer was quite prepared to use whatever means were available.

  11. 11.

    The first term was introduced by Poincaré (1885), the second one by Brouwer, Brouwer (1919f, 1919g).

  12. 12.

    Freudenthal (1975), cf. CW II, p. 422 ff.

  13. 13.

    CW II, p. 421, see also Brouwer to Scheltema 3 December 1909.

  14. 14.

    Concerning the ‘Analysis Situs’-paper, cf. p. 144.

  15. 15.

    Freudenthal in CW II, p. 425.

  16. 16.

    Hadamard (1910).

  17. 17.

    Freudenthal (1979).

  18. 18.

    Cf. p. 213.

  19. 19.

    Freudenthal (CW II, p. 435 ff.) has given a thorough historic and mathematical analysis of the invariance of dimension episode. The reader is referred to Freudenthal’s comments for more technical details.

  20. 20.

    Sur la non-applicabilité de deux domaines appartenant respectivement à des espaces à n et n+p dimensions (Extrait d’une lettre à M. O. Blumenthal), Lebesgue (1911a).

  21. 21.

    Pflaster Satz.

  22. 22.

    By Baire.

  23. 23.

    Cf. Johnson (1981), p. 191.

  24. 24.

    CW II, p. 440.

  25. 25.

    Schoenflies (1908).

  26. 26.

    Zoretti (1911).

  27. 27.

    Sur la non-applicabilité de deux continus à n et n+p dimensions.

  28. 28.

    The dimension paper, cf. p. 174.

  29. 29.

    Alexander (1922) and Alexandroff and Hopf (1935), Chap. XI elaborated the ideas involved.

  30. 30.

    Sur l’invariance du nombre de dimensions d’un espace et sur le théorème de M.Jordan relatif aux variété fermées, Lebesgue (1911b).

  31. 31.

    Johnson (1981), pp. 198, 199.

  32. 32.

    Blumenthal to Brouwer, 16.6.1911.

  33. 33.

    Johnson (1981), p. 203.

  34. 34.

    CW II, p. 440.

  35. 35.

    Über den natürlichen Dimensionsbegriff, Brouwer (1913a).

  36. 36.

    Lebesgue (1911b).

  37. 37.

    Brouwer (1913a).

  38. 38.

    Sur l’invariance du nombre de dimensions d’un espace et sur le théorème de M. Jordan relatif aux variété fermées. Lebesgue (1911b).

  39. 39.

    Baire (1907a).

  40. 40.

    Lebesgue (1911a), p. 168.

  41. 41.

    CW II, p. 439.

  42. 42.

    Baire to Brouwer, 5 December 1911.

  43. 43.

    The n-dimensional Jordan theorem, Brouwer (1911e).

  44. 44.

    Cf. Lebesgue (1911a).

  45. 45.

    Cf. Lebesgue (1921).

  46. 46.

    Basics of Set Theory.

  47. 47.

    Hopf (1966).

  48. 48.

    Über Abbildung von Mannigfaltigkeiten, Brouwer (1911c).

  49. 49.

    Poincaré had already defined the notion in Poincaré (1899). Brouwer does not quote Poincaré, so presumably he was not aware of the paper.

  50. 50.

    Note that in the letter to Hilbert of 1 January 1910, Brouwer still uses polynomial approximations.

  51. 51.

    The reader need not worry, nobility has not been created in the Netherlands for more then a century.

  52. 52.

    Dieudonné (1989), p. 161.

  53. 53.

    Simplicial topology.

  54. 54.

    Dieudonné (1989), p. 168.

  55. 55.

    After the ascent of homology theory, a much simpler definition of the degree of a mapping became available: let f be a continuous mapping from M to M′, where M and M′ are compact, connected, oriented (pseudo-) manifolds, then for f ∗:H n (M;Z)→H n (M′;Z), we have f ∗([cM])=c[M′] where c is the mapping degree.

  56. 56.

    Beweis der Invarianz des n-dimensionalen Gebiets. Brouwer (1911d).

  57. 57.

    Brouwer (1911b).

  58. 58.

    CW II, p. 443.

  59. 59.

    Baire (1907a, 1907b).

  60. 60.

    Zur Invarianz des n-dimensionalen Gebiets, Brouwer (1912c).

  61. 61.

    See the letter to Baire of 5.11.1911, Johnson (1981), p. 218.

  62. 62.

    Brouwer (1912j).

  63. 63.

    Brouwer (1910a).

  64. 64.

    Sur l’invariance de la courbe fermée, Beweis der Invarianz der geschlossenen Kurve, Brouwer (1912e, 1912i).

  65. 65.

    Brouwer (1912h).

  66. 66.

    Über den natürlichen Dimensionsbegriff. Brouwer (1913a).

  67. 67.

    Alexander (1922).

  68. 68.

    Cf. p. 190, Wiessing (1960), p. 143 ff.

  69. 69.

    Brouwer (1928e).

  70. 70.

    That is, Brouwer showed that ℝ n has dimension n.

  71. 71.

    For a precise and general definition of uniformisation the reader is referred to the literature, for example Nevanlinna (1953), Beardon (1984).

  72. 72.

    Klein (1927).

  73. 73.

    Weyl to Mulder, 29 July 1910.

  74. 74.

    Cf. Kühnau (1981).

  75. 75.

    Blumenthal to Brouwer, 26 August 1911.

  76. 76.

    The Invariance of domain-paper was sent to the Mathematische Annalen on 14 June 1911, published November 1911.

  77. 77.

    JDMV 1912.

  78. 78.

    Cf. Behnke and Sommer (1955).

  79. 79.

    …, Koebe aber nur eine gewisse Ahnung, dass sich etwas mit seinem Verzerrungssatz in der Kontinuitäts Methode lasse, mitbrachte.

  80. 80.

    He quoted the author: ‘The simplest and most natural of all these proofs [of the uniformisation theorem] is the first one given by Koebe.’ Modesty was not one of Koebe’s defects.

  81. 81.

    Brouwer (1912d).

  82. 82.

    Of which only a few have survived.

  83. 83.

    Brouwer (1912f).

  84. 84.

    I.e. the notes for the Göttinger Gesellschaft submitted by Klein and Hilbert on 13 January 1912.

  85. 85.

    Undated letter, mentioned above.

  86. 86.

    Cf. Alexandrov (1972), Zorin (1972).

  87. 87.

    Cf. Freudenthal’s commentary; CW II, p. 581 ff.

  88. 88.

    One of the objections of Koebe to Brouwer’s note.

  89. 89.

    Fricke and Klein (1897, 1912)

  90. 90.

    Lebenspendend.

  91. 91.

    14 January 1912, see CW II, p. 585.

  92. 92.

    Brouwer to Hilbert, 9 March 1912.

  93. 93.

    There goes the great function theorist.

  94. 94.

    The greatest function theorist of Luckenwalde.

  95. 95.

    Freudenthal (1984).

  96. 96.

    All of Europe talks about it: Koebe is mailing reprints.

  97. 97.

    Van de Waerden to Van Dalen, 25 February 1992.

  98. 98.

    Freudenthal in CW II, p. 575.

  99. 99.

    See CW II, p. 571. The sticker summed up Brouwer’s grievances mentioned above.

  100. 100.

    Koebe to Hilbert, 29 February 1912.

  101. 101.

    Koebe (1914).

  102. 102.

    On the absence of singularities of the module manifold, Brouwer (1912f).

  103. 103.

    rühmlichst bekannt.

  104. 104.

    Brouwer (1919i).

  105. 105.

    von einer mir unbekannten Hand.

  106. 106.

    Klein (1923).

  107. 107.

    Wesen der Kontinuitätsmethode, ‘after lectures held at the meetings of the German Mathematical Society in September 1913 in Vienna and September 1935 in Stuttgart’, Koebe (1936).

  108. 108.

    On the continuity proof of the fundamental theorem, Klein (1923).

  109. 109.

    Wiessing (1960), p. 143 ff.

  110. 110.

    High school, see p. 4.

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    Dirk van Dalen

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van Dalen, D. (2013). The New Topology. In: L.E.J. Brouwer – Topologist, Intuitionist, Philosopher. Springer, London. https://doi.org/10.1007/978-1-4471-4616-2_5

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