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History of topology. (English) Zbl 0922.54003
Amsterdam: Elsevier. ix, 1056 p. (1999).
[The articles of this volume will be reviewed individually.]
From the ‘Preface’: “Topology, for many years, has been one of the most exciting and influential fields of research in modern mathematics. Although its origins may be traced back several hundred years it was Poincaré who, to borrow an expression used of Möbius, “gave topology wings” in a classic series of articles published around the turn of the century. While the earlier history, sometimes called the prehistory, is also considered, this volume is mainly concerned with the more recent history of topology, from Poincaré onwards. As will be seen from the list of contents the articles cover a wide range of topics. Some are more technical than others, but the reader without a great deal of technical knowledge should still find most of the articles accessible. Some are written by professional historians of mathematics, others by historically-minded mathematicians, who tend to have a different viewpoint. Most of the material has not been published before. Topology is a large subject, with many branches, and it proved quite impossible to cover everything. The emphasis is on what might be called classical topology rather than on general (or point-set) topology: a separate history of general topology is in the process of publication under a different aegis. However, the editor believes the articles will be found to cover most of the major topics. The order in which they are arranged is partly chronological and partly according to subject matter. The last part of the book is more concerned with the people who were important in the development of the subject. In particular short biographies of a number of them are given, based on material already in the literature, and rather longer biographies of some others, which contain material not previously published.”
List of contents: T. Crilly and D. Johnson: The emergence of topological dimension theory (pp. 1-24); E. Scholz: The concept of manifold, 1850-1950 (pp. 25-64); R. Vanden Eynde: Development of the concept of homotopy (pp. 65-102); G. Burde and H. Zieschang: Development of the concept of a complex (pp. 103-110); V. J. Katz: Differential forms (pp. 111-122); K. S. Sarkaria: The topological work of Henri Poincaré (pp. 123-167); T. Hawkins: Weyl and the topology of continuous groups (pp. 169-198); T. Koetsier and J. van Mill: By their fruits ye shall know them: Some remarks on the interaction of general topology with other areas of mathematics (pp. 199-239); S. Mardešić: Absolute neighborhood retracts and shape theory (pp. 241-269); R. F. Brown: Fixed point theory (pp. 271-299); M. Epple: Geometric aspects in the development of knot theory (pp. 301-357); C. Nash: Topology and physics – a historical essay (pp. 359-415); A. H. Durfee: Singularities (pp. 417-434); S. K. Donaldson: One hundred years of manifold topology (pp. 435-447); C. McA. Gordon: 3-dimensional topology up to 1960 (pp. 449-489); N. H. Kuiper: A short history of triangulation and related matters (pp. 491-502); R. J. Wilson: Graph theory (pp. 503-529); S. Lefschetz: The early development of algebraic topology (pp. 531-559); I. M. James: From combinatorial topology to algebraic topology (pp. 561-573); H. Samelson: $$\pi_3 (S^2)$$, H. Hopf, W. K. Clifford, F. Klein (pp. 575-578); W. S. Massey: A history of cohomology theory (pp. 579-603); M. Zisman: Fibre bundles, fibre maps (pp. 605-629); J. McCleary: A history of spectral sequences: Origins to 1953 (pp. 631-663); J. P. May: Stable algebraic topology, 1945-1966 (pp. 665-723); J. C. Becker and D. H. Gottlieb: A history of duality in algebraic topology (pp. 725-745); J. R. Hubbuck: A short history of $$H$$-spaces (pp. 747-755); K. Hess: A history of rational homotopy theory (pp. 757-796); C. A. Weibel: History of homological algebra (pp. 797-836); I. M. James: Topologists at conferences (pp. 837-848); S. L. Segal: Topologists in Hitler’s Germany (pp. 849-861); M. Mimura: The Japanese school of topology (pp. 863-882); I. M. James: Some topologists (pp. 883-908); E. Breitenberger: Johann Benedikt Listing (pp. 909-924); E. S. Munkholm and H. J. Munkholm: Poul Heegaard (pp. 925-946); D. van Dalen: Luitzen Egbertus Jan Brouwer (pp. 947-964); J. Stillwell: Max Dehn (pp. 965-978); V. L. Hansen: Jakob Nielsen and his contributions to topology (pp. 979-989); G. Frei and U. Stammbach: Heinz Hopf (pp. 991-1008); W. T. van Est: Hans Freudenthal (pp. 1009-1019); D. Puppe: Herbert Seifert (1907-1996) (pp. 1021-1027).
Indexed articles:
Crilly, Tony, The emergence of topological dimension theory, 1-24 [Zbl 0995.54031]
Scholz, Erhard, The concept of manifold, 1850-1950, 25-64 [Zbl 0956.57002]
Vanden Eynde, Ria, Development of the concept of homotopy, 65-102 [Zbl 0944.55001]
Burde, Gerhard; Zieschang, Heiner, Development of the concept of a complex, 103-110 [Zbl 0944.57001]
Katz, Victor J., Differential forms, 111-122 [Zbl 0967.58002]
Sarkaria, K. S., The topological work of Henri Poincaré, 123-167 [Zbl 0959.54002]
Hawkins, Thomas, Weyl and the topology of continuous groups, 169-198 [Zbl 0949.22001]
Koetsier, Teun; van Mill, Jan, By their fruits ye shall know them: Some remarks on the interaction of general topology with other areas of mathematics, 199-239 [Zbl 0949.54001]
Mardešić, Sibe, Absolute neighborhood retracts and shape theory, 241-269 [Zbl 0973.54002]
Brown, Robert F., Fixed point theory, 271-299 [Zbl 0947.54021]
Epple, Moritz, Geometric aspects in the development of knot theory, 301-357 [Zbl 0957.57002]
Nash, Charles, Topology and physics – a historical essay, 359-415 [Zbl 0963.57001]
Durfee, Alan H., Singularities, 417-434 [Zbl 0956.57003]
Donaldson, S. K., One hundred years of manifold topology, 435-447 [Zbl 0956.57004]
Gordon, C. McA., 3-dimensional topology up to 1960, 449-489 [Zbl 0956.57005]
Kuiper, N. H., A short history of triangulation and related matters, 491-502 [Zbl 0966.57003]
Wilson, Robin J., Graph theory, 503-529 [Zbl 0948.05002]
Lefschetz, Solomon, The early development of algebraic topology, 531-559 [Zbl 0945.55001]
James, I. M., From combinatorial topology to algebraic topology, 561-573 [Zbl 0956.55001]
Samelson, H., $$\pi_3 (S^2)$$, H. Hopf, W. K. Clifford, F. Klein, 575-578 [Zbl 0944.55002]
Massey, William S., A history of cohomology theory, 579-603 [Zbl 1001.55002]
Zisman, M., Fibre bundles, fibre maps, 605-629 [Zbl 0956.55002]
McCleary, John, A history of spectral sequences: Origins to 1953, 631-663 [Zbl 0956.55003]
May, J. P., Stable algebraic topology, 1945-1966, 665-723 [Zbl 0956.55004]
Becker, James C.; Gottlieb, Daniel Henry, A history of duality in algebraic topology, 725-745 [Zbl 0957.55001]
Hubbuck, J. R., A short history of $$H$$-spaces, 747-755 [Zbl 1001.55003]
Hess, Kathryn, A history of rational homotopy theory, 757-796 [Zbl 0957.55002]
Weibel, Charles A., History of homological algebra, 797-836 [Zbl 0966.55002]
James, I. M., Topologists at conferences, 837-848 [Zbl 0959.54004]
Segal, S. L., Topologists in Hitler’s Germany, 849-861 [Zbl 0998.01019]
Mimura, Mamoru, The Japanese school of topology, 863-882 [Zbl 0946.01002]
James, I. M., Some topologists, 883-908 [Zbl 0945.55002]
Breitenberger, E., Johann Benedikt Listing, 909-924 [Zbl 1003.01012]
Munkholm, Ellen S.; Munkholm, Hans J., Poul Heegaard, 925-946 [Zbl 0948.01031]
van Dalen, Dirk, Luitzen Egbertus Jan Brouwer, 947-964 [Zbl 0948.01037]
Stillwell, John, Max Dehn, 965-978 [Zbl 0948.01034]
Hansen, Vagn Lundsgaard, Jakob Nielsen and his contributions to topology, 979-989 [Zbl 0948.01024]
Frei, Günther; Stammbach, Urs, Heinz Hopf, 991-1008 [Zbl 0948.01023]
van Est, W. T., Hans Freudenthal, 1009-1019 [Zbl 0948.01038]
Puppe, Dieter, Herbert Seifert, 1021-1027 [Zbl 0949.01016]

MSC:
 54-03 History of general topology 55-03 History of algebraic topology 57-03 History of manifolds and cell complexes 00B15 Collections of articles of miscellaneous specific interest
Keywords:
History; Topology