Poincaré and the early history of 3-manifolds
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Abstract:
Recent developments in the theory of 3-manifolds, centered around the Poincaré conjecture, use methods that were not envisioned by Poincaré and his contemporaries. Nevertheless, the main themes of 3-manifold topology originated in Poincaré’s time. The purpose of this article is to reveal the origins of the subject by revisiting the world of the early topologists.References
- S. I. Adyan, Unsolvability of some algorithmic problems in the theory of groups. , Trudy Moskov. Mat. Obšč. 6 (1957), 231–298 (Russian). MR 0095872
- J. W. Alexander II, A proof of the invariance of certain constants of analysis situs, Trans. Amer. Math. Soc. 16 (1915), no. 2, 148–154. MR 1501007, DOI 10.1090/S0002-9947-1915-1501007-5
- James W. Alexander, Note on Riemann spaces, Bull. Amer. Math. Soc. 26 (1920), no. 8, 370–372. MR 1560318, DOI 10.1090/S0002-9904-1920-03319-7
- J. W. Alexander, Note on two three-dimensional manifolds with the same group, Trans. Amer. Math. Soc. 20 (1919), no. 4, 339–342. MR 1501131, DOI 10.1090/S0002-9947-1919-1501131-0
- Alexander, J. W. (1924a). An example of a simply connected surface bounding a region which is not simply connected. Proceedings of the National Academy of Sciences 10, 8–10.
- Alexander, J. W. (1924b). On the subdivision of 3-space by a polyhedron. Proceedings of the National Academy of Sciences 10, 6–8.
- J. W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc. 30 (1928), no. 2, 275–306. MR 1501429, DOI 10.1090/S0002-9947-1928-1501429-1
- Alexander, J. W. and G. B. Briggs (1927). On types of knotted curves. Ann. Math. 28, 562–586.
- P. Appell, Quelques remarques sur la théorie des potentiels multiformes, Math. Ann. 30 (1887), no. 1, 155–156 (French). MR 1510440, DOI 10.1007/BF01564536
- Artin, E. (1926). Theorie der Zöpfe. Abh. math. Sem. Univ. Hamburg 4, 47–72.
- Betti, E. (1871). Sopra gli spazi di un numero qualunque di dimensioni. Annali di Matematica pura ed applicata 4, 140–158.
- Bianchi, L. (1898). Sugli spazi a tre dimensioni che ammettono un gruppo continuo di movimenti. Memorie di Matematica e di Fisica della Societa Italiana delle Scienze 11, 267–352. English translation by Robert Jantzen in General Relativity and Gravitation 33, (2001), pp. 2171–2253.
- Brauner, K. (1928). Zur Geometrie der Funktionen zweier komplexer Veränderliche. Abh. Math. Sem. Univ. Hamburg 6, 1–55.
- L. E. J. Brouwer, Beweis der Invarianz der Dimensionenzahl, Math. Ann. 70 (1911), no. 2, 161–165 (German). MR 1511615, DOI 10.1007/BF01461154
- Stewart S. Cairns, On the triangulation of regular loci, Ann. of Math. (2) 35 (1934), no. 3, 579–587. MR 1503181, DOI 10.2307/1968752
- Professor Cayley, Desiderata and Suggestions: No. 2. The Theory of Groups: Graphical Representation, Amer. J. Math. 1 (1878), no. 2, 174–176. MR 1505159, DOI 10.2307/2369306
- Alonzo Church, An Unsolvable Problem of Elementary Number Theory, Amer. J. Math. 58 (1936), no. 2, 345–363. MR 1507159, DOI 10.2307/2371045
- Dehn, M. (1900). Über raumgleiche Polyeder. Gött. Nachr. 1900, 345–354..
- Dehn, M. (1907). Berichtigender Zusatz zu III AB3 Analysis situs. Jber. Deutsch. Math. Verein. 16, 573.
- M. Dehn, Über die Topologie des dreidimensionalen Raumes, Math. Ann. 69 (1910), no. 1, 137–168 (German). MR 1511580, DOI 10.1007/BF01455155
- M. Dehn, Über unendliche diskontinuierliche Gruppen, Math. Ann. 71 (1911), no. 1, 116–144 (German). MR 1511645, DOI 10.1007/BF01456932
- M. Dehn, Transformation der Kurven auf zweiseitigen Flächen, Math. Ann. 72 (1912), no. 3, 413–421 (German). MR 1511705, DOI 10.1007/BF01456725
- M. Dehn, Die beiden Kleeblattschlingen, Math. Ann. 75 (1914), no. 3, 402–413 (German). MR 1511799, DOI 10.1007/BF01563732
- Max Dehn, Papers on group theory and topology, Springer-Verlag, New York, 1987. Translated from the German and with introductions and an appendix by John Stillwell; With an appendix by Otto Schreier. MR 881797, DOI 10.1007/978-1-4612-4668-8
- Dehn, M. and P. Heegaard (1907). Analysis situs. Enzyklopädie der Mathematischen Wissenschaften, vol. III AB3, 153–220, Teubner, Leipzig.
- Dyck, W. (1884). On the “Analysis Situs” of 3-dimensional spaces. Report of the Brit. Assoc. Adv. Sci., 648.
- Epple, M. (1999a). Die Entstehung der Knotentheorie. Braunschweig: Friedr. Vieweg & Sohn.
- Moritz Epple, Geometric aspects in the development of knot theory, History of topology, North-Holland, Amsterdam, 1999, pp. 301–357. MR 1674917, DOI 10.1016/B978-044482375-5/50012-2
- Michael Hartley Freedman, The topology of four-dimensional manifolds, J. Differential Geometry 17 (1982), no. 3, 357–453. MR 679066
- C. McA. Gordon and J. Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989), no. 2, 371–415. MR 965210, DOI 10.1090/S0894-0347-1989-0965210-7
- C. McA. Gordon, $3$-dimensional topology up to 1960, History of topology, North-Holland, Amsterdam, 1999, pp. 449–489. MR 1674921, DOI 10.1016/B978-044482375-5/50016-X
- H. Guggenheimer, The Jordan curve theorem and an unpublished manuscript by Max Dehn, Arch. Hist. Exact Sci. 17 (1977), no. 2, 193–200. MR 0532231, DOI 10.1007/BF02464980
- Wolfgang Haken, Theorie der Normalflächen, Acta Math. 105 (1961), 245–375 (German). MR 141106, DOI 10.1007/BF02559591
- Heegaard, P. (1898). Forstudier til en topologisk Teori for de algebraiske Fladers sammenhœng. Dissertation, Copenhagen, 1898. Available at http://www.maths.ed.ac.uk/˜aar/papers/heegaardthesis.pdf. [35] is a French translation of this work.
- P. Heegaard, Sur l’"Analysis situs", Bull. Soc. Math. France 44 (1916), 161–242 (French). MR 1504754
- Kneser, H. (1929). Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten. Jber. Deutsch. Math. Verein. 38, 248–260.
- W. B. R. Lickorish, A representation of orientable combinatorial $3$-manifolds, Ann. of Math. (2) 76 (1962), 531–540. MR 151948, DOI 10.2307/1970373
- A. Markov, The insolubility of the problem of homeomorphy, Dokl. Akad. Nauk SSSR 121 (1958), 218–220 (Russian). MR 0097793
- Curtis T. McMullen, The evolution of geometric structures on 3-manifolds, Bull. Amer. Math. Soc. (N.S.) 48 (2011), no. 2, 259–274. MR 2774092, DOI 10.1090/S0273-0979-2011-01329-5
- Möbius, A. F. (1863). Theorie der Elementaren Verwandtschaft. Werke 2: 433–471.
- Edwin E. Moise, Affine structures in $3$-manifolds. V. The triangulation theorem and Hauptvermutung, Ann. of Math. (2) 56 (1952), 96–114. MR 48805, DOI 10.2307/1969769
- Neumann, C. (1865). Vorlesungen über Riemann’s Theorie der Abelschen Integralen. Leipzig: Teubner.
- Noether, E. (1925). Ableitung der Elementarteilertheorie aus der Gruppentheorie. Jber. Deutsch. Math. Verein. 34, 104.
- P. S. Novikov, On the algorithmic insolvability of the word problem in group theory, American Mathematical Society Translations, Ser. 2, Vol. 9, American Mathematical Society, Providence, R.I., 1958, pp. 1–122. MR 0092784
- C. D. Papakyriakopoulos, On Dehn’s lemma and the asphericity of knots, Ann. of Math. (2) 66 (1957), 1–26. MR 90053, DOI 10.2307/1970113
- H. Poincaré, Théorie des groupes fuchsiens, Acta Math. 1 (1882), no. 1, 1–76 (French). MR 1554574, DOI 10.1007/BF02391835
- Poincaré, H. (1892). Sur l’analysis situs. Comptes rendus de l’Academie des Sciences 115, 633–636.
- Poincaré, H. (1895). Analysis situs. J. Éc. Polytech., ser. 2 1, 1–123.
- H. Poincaré, Second Complement a l’Analysis Situs, Proc. Lond. Math. Soc. 32 (1900), 277–308. MR 1576227, DOI 10.1112/plms/s1-32.1.277
- Poincaré, Sur certaines surfaces algébriques. Troisième complément à l’Analysis sitûs, Bull. Soc. Math. France 30 (1902), 49–70 (French). MR 1504408
- Poincaré, H. (1904). Cinquième complément à l’analysis situs. Rendiconti del Circolo matematico di Palermo 18, 45–110.
- Henri Poincaré, Papers on Fuchsian functions, Springer-Verlag, New York, 1985. Translated from the French and with an introduction by John Stillwell. MR 809181, DOI 10.1007/978-1-4612-5148-4
- Henri Poincaré, Papers on topology, History of Mathematics, vol. 37, American Mathematical Society, Providence, RI; London Mathematical Society, London, 2010. Analysis situs and its five supplements; Translated and with an introduction by John Stillwell. MR 2723194, DOI 10.1090/hmath/037
- Reidemeister, K. (1926). Knoten und Gruppen. Abh. math. Sem. Univ. Hamburg 5, 7–23.
- Reidemeister, K. (1932). Einführung in die kombinatorische Topologie. Braunschweig: Vieweg.
- Riemann, G. F. B. (1851). Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse. Werke, 2nd ed., 3–48.
- Joachim H. Rubinstein, An algorithm to recognize the $3$-sphere, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 601–611. MR 1403961
- Herbert Seifert and William Threlfall, Seifert and Threlfall: a textbook of topology, Pure and Applied Mathematics, vol. 89, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. Translated from the German edition of 1934 by Michael A. Goldman; With a preface by Joan S. Birman; With “Topology of $3$-dimensional fibered spaces” by Seifert; Translated from the German by Wolfgang Heil. MR 575168
- C. Weber and H. Seifert, Die beiden Dodekaederräume, Math. Z. 37 (1933), no. 1, 237–253 (German). MR 1545392, DOI 10.1007/BF01474572
- Z. Sela, The isomorphism problem for hyperbolic groups. I, Ann. of Math. (2) 141 (1995), no. 2, 217–283. MR 1324134, DOI 10.2307/2118520
- Smith, H. J. S. (1861). On systems of linear indeterminate equations and congruences. Philosophical Transactions 111, 293–326. In his Collected Mathematical Papers, Vol. I, pp. 367–409.
- Sommerfeld, A. (1897). Über verzweigte Potential im Raum. Proc. Lond. Math. Soc. 28, 395–429.
- John Stillwell, Letter to the editors, Math. Intelligencer 1 (1978/79), no. 4, 192. MR 547746, DOI 10.1007/BF03028232
- John Stillwell, Classical topology and combinatorial group theory, 2nd ed., Graduate Texts in Mathematics, vol. 72, Springer-Verlag, New York, 1993. MR 1211642, DOI 10.1007/978-1-4612-4372-4
- Heinrich Tietze, Über die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten, Monatsh. Math. Phys. 19 (1908), no. 1, 1–118 (German). MR 1547755, DOI 10.1007/BF01736688
- A. M. Turing, On Computable Numbers, with an Application to the Entscheidungsproblem, Proc. London Math. Soc. (2) 42 (1936), no. 3, 230–265. MR 1577030, DOI 10.1112/plms/s2-42.1.230
- Klaus Volkert, The early history of Poincaré’s conjecture, Henri Poincaré: science et philosophie (Nancy, 1994) Publ. Henri-Poincaré-Arch., Akademie Verlag, Berlin, 1996, pp. 241–250, 580 (English, with French summary). MR 1384995
- Volkert, K. (2002). Das Homöomorphieproblem, insbesondere der 3-Mannigfaltigkeiten in der Topologie 1892–1935. Philosophia Scientiae. Paris: Editions Kimé.
Additional Information
- John Stillwell
- Affiliation: University of San Francisco, San Francisco, California; and Monash University, Melbourne, Australia
- MR Author ID: 167425
- Received by editor(s): June 9, 2012
- Published electronically: July 23, 2012
- © Copyright 2012 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 49 (2012), 555-576
- MSC (2010): Primary 57-03
- DOI: https://doi.org/10.1090/S0273-0979-2012-01385-X
- MathSciNet review: 2958930