The problem of the invariance of dimension in the growth of modern topology. II
This book-size paper continues the study of the development of the topological dimension concept, initiated in the earlier papers ibid. 17, 261-295 (1977; Zbl 0372.01002) and in particular ibid. 20, 97-188 (1979; Zbl 0424.54020), to which the present paper is the second part. It starts with Chapter 5 (42 pages), mainly devoted to the ideas of Henri Poincaré. The author discusses two theories of dimension by Poincaré, the first and older one based on group theory, the second one of a topological nature. In order to explain Poincaré's approach, his philosophical ideas (described as a ''philosophy of conventionalism'') are very lucidly discussed. On the basis of this discussion the contributions of the great Frenchman (and their limitations) are made clear. The last part of Chapter 5 deals with the work of F. Riesz, R. Baire and M. Fréchet (insofar as it touches dimension). In particular, Baire showed (1907) how an n-dimensional Jordan separation theorem entails n- dimensional domain invariance, and how domain invariance entails invariance of dimension (these interrelations were later taken up by Brouwer); however, he never succeeded in proving the n-dimensional Jordan theorem. Chapters 6 and 7 (together with 3 appendices; they cover 95 pages) form the main part of the paper. They are devoted to the contributions to dimension theory of L. E. J. Brouwer, and his controversy with Henri Lebesgue. The historical development of Brouwer's ideas is carefully reconstructed, using not only the material in the edition of the ''Collected Works'' of \textit{L. E. J. Brouwer} (1976; Zbl 0328.01020) and the important contributions of Hans Freudenthal, the editor of Vol. II of these ''Collected Works'', to the understanding of Brouwer's work, but also additional material, which was unavailable at the time the ''Collected Works'' were prepared for publication (much of this additional material is reproduced in the appendices). The author presents a detailed account of the evolution of Brouwer's ideas, from his early involvement with the work of A. Schoenflies, his contacts with J. Hadamard (whom Brouwer met personally in the first days of 1910, just when his great new ideas were taking shape), through the writing (early 1910) and publishing in Math. Ann. 70, 161-165 (February 1911) of his key paper ''Beweis der Invarianz der Dimensionenzahl'', and the ensuing conflict with Henri Lebesgue (the editor of Math. Ann., Otto Blumenthal, having placed immediately after Brouwer's paper a note of Lebesgue, sketching a different proof of invariance of dimension, by means of the tiling principle [ibid. 70, 166-168 (1911)]). Brouwer quickly constructed his own rigorous proofs for the Lebesgue approach; he also picked up Baire's ideas; and already in 1913 his comprehensive paper ''Über den natürlichen Dimensionsbegriff'' appeared in J. Reine Angew. Math. 142, 146-152 (1913), in which all the main problems of dimension and invariance were solved. Everyone interested in topology should read this! After these central chapters on Brouwer and his contemporaries (in which, I should add, the contributions of Lebesgue are also discussed on their own merits, and not only in relation to the activities of Brouwer) the final chapter (30 pages) on ''Glimpses of the development of dimension theory after Brouwer'' comes almost as an anticlimax. Nevertheless, it brings a nice description of the way P. S. Uryson and K. Menger independently developed their inductive definition of dimension. The contributions of E. Sperner and W. Hurewicz in providing elegant and simple proofs to Brouwer's main result (making them finally accessible) are also mentioned, as is the work of the early Polish School. At about 1930 the historical chronicle is broken off. The last pages of this chapter contain some final remarks and conclusions. A 15-page bibliography to this second part of the paper, and a page of errata for Part I, conclude the paper. In this and his previous papers, the author has done a great service to all those interested in topology and in its history, but also to those interested in the history of ideas and in the development of theories. Readers not well versed in geometry and topology will at times have difficulties in following the description of technical work; even those who consider themselves topologists will find it necessary at times to consult the original papers in order to fully understand some of the technical passages. So much the better. This paper tells us about the evolution of the concept of dimension, so important both philosophically and mathematically. The story is exciting, and it is well told.
scientific article; zbMATH DE number 3844618
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