Topology through the centuries: Low dimensional manifolds
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Abstract:
This note will provide a lightning tour through the centuries, concentrating on the study of manifolds of dimension 2, 3, and 4. Further comments and more technical details about many of the sections may be found in the Appendix.References
- S. I. Adyan, Algorithmic unsolvability of problems of recognition of certain properties of groups, Dokl. Akad. Nauk SSSR (N.S.) 103 (1955), 533–535 (Russian). MR 0081851
- 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
- J. W. Alexander, A proof and extension of the Jordan-Brouwer separation theorem, Trans. Amer. Math. Soc. 23 (1922), no. 4, 333–349. MR 1501206, DOI 10.1090/S0002-9947-1922-1501206-6
- \author{J. W. Alexander [1924],} On the subdivision of $3$-space by a polyhedron, Proc. Nat. Acad. Sci. 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
- \author{P. Alexandroff and H. Hopf [1935],} Topologie, Springer, Berlin.
- R. J. Eden, J. C. Polkinghorne, G. Källén, and J. J. Sakurai, Lectures in theoretical physics. Vol. 1, W. A. Benjamin, Inc., New York, 1962. 1961 Brandeis Summer Institute. MR 0151123
- Ludwig Bieberbach, Über die Bewegungsgruppen der Euklidischen Räume (Zweite Abhandlung.) Die Gruppen mit einem endlichen Fundamentalbereich, Math. Ann. 72 (1912), no. 3, 400–412 (German). MR 1511704, DOI 10.1007/BF01456724
- A. Borel, Commensurability classes and volumes of hyperbolic $3$-manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981), no. 1, 1–33. MR 616899
- J. W. Cannon, The recognition problem: what is a topological manifold?, Bull. Amer. Math. Soc. 84 (1978), no. 5, 832–866. MR 494113, DOI 10.1090/S0002-9904-1978-14527-3
- Andrew J. Casson, Three lectures on new-infinite constructions in $4$-dimensional manifolds, À la recherche de la topologie perdue, Progr. Math., vol. 62, Birkhäuser Boston, Boston, MA, 1986, pp. 201–244. With an appendix by L. Siebenmann. MR 900253
- \author{A. Clebsch [1865],} Ueber diejenigen ebenen Curven, deren Coordinaten rationale Functionen eines Parameters sind, J. reine ang. Math. 64, 43–65. http://www.maths.ed.ac.uk/$\sim$aar/papers/clebschgenus.pdf
- Richard H. Crowell and Ralph H. Fox, Introduction to knot theory, Graduate Texts in Mathematics, No. 57, Springer-Verlag, New York-Heidelberg, 1977. Reprint of the 1963 original. MR 0445489, DOI 10.1007/978-1-4612-9935-6
- Stefano De Michelis and Michael H. Freedman, Uncountably many exotic $\textbf {R}^4$’s in standard $4$-space, J. Differential Geom. 35 (1992), no. 1, 219–254. MR 1152230
- Jean Dieudonné, A history of algebraic and differential topology. 1900–1960, Birkhäuser Boston, Inc., Boston, MA, 1989. MR 995842
- S. K. Donaldson, Self-dual connections and the topology of smooth $4$-manifolds, Bull. Amer. Math. Soc. (N.S.) 8 (1983), no. 1, 81–83. MR 682827, DOI 10.1090/S0273-0979-1983-15090-5
- S. K. Donaldson, An application of gauge theory to four-dimensional topology, J. Differential Geom. 18 (1983), no. 2, 279–315. MR 710056, DOI 10.4310/jdg/1214437665
- S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1990. Oxford Science Publications. MR 1079726
- Walther Dyck, Beiträge zur Analysis situs, Math. Ann. 32 (1888), no. 4, 457–512 (German). MR 1510522, DOI 10.1007/BF01443580
- \author{F. Enriques [1905],} Sulla proprietà caratteristica delle superficie algebriche irregolari, Rend. Accad. Sci. Bologna, 9, 5–13.
- Michael Hartley Freedman, A fake $S^{3}\times \textbf {R}$, Ann. of Math. (2) 110 (1979), no. 1, 177–201. MR 541336, DOI 10.2307/1971257
- Michael Hartley Freedman, The topology of four-dimensional manifolds, J. Differential Geometry 17 (1982), no. 3, 357–453. MR 679066
- Michael H. Freedman, There is no room to spare in four-dimensional space, Notices Amer. Math. Soc. 31 (1984), no. 1, 3–6. MR 728340
- Michael H. Freedman and Frank Quinn, Topology of 4-manifolds, Princeton Mathematical Series, vol. 39, Princeton University Press, Princeton, NJ, 1990. MR 1201584
- Michael Freedman, Robert Gompf, Scott Morrison, and Kevin Walker, Man and machine thinking about the smooth 4-dimensional Poincaré conjecture, Quantum Topol. 1 (2010), no. 2, 171–208. MR 2657647, DOI 10.4171/QT/5
- \author{A. Friedman [1922],} Über die Krümmung des Raumes, Z. Physik 10, 377–386.
- Robert Friedman and John W. Morgan, Complex versus differentiable classification of algebraic surfaces, Proceedings of the 1987 Georgia Topology Conference (Athens, GA, 1987), 1989, pp. 135–139. MR 1007985, DOI 10.1016/0166-8641(89)90050-3
- \author{C. F. Gauss [1833],} Zur mathematischen Theorie der electrodynamischen Wirkung, Königliche Gesellschaft der Wissenschaften zu Göttingen 5, 602–629.
- \author{H. Gieseking [1912],} Analytische Untersuchungen über topologische Gruppen, Thesis, Muenster.
- Robert E. Gompf, Three exotic $\textbf {R}^{4}$’s and other anomalies, J. Differential Geom. 18 (1983), no. 2, 317–328. MR 710057
- Robert E. Gompf, An exotic menagerie, J. Differential Geom. 37 (1993), no. 1, 199–223. MR 1198606
- 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
- Jeremy Gray, Henri Poincaré, Princeton University Press, Princeton, NJ, 2013. A scientific biography. MR 2986502
- Michael Gromov, Hyperbolic manifolds (according to Thurston and Jørgensen), Bourbaki Seminar, Vol. 1979/80, Lecture Notes in Math., vol. 842, Springer, Berlin-New York, 1981, pp. 40–53. MR 636516
- Wolfgang Haken, Über das Homöomorphieproblem der 3-Mannigfaltigkeiten. I, Math. Z. 80 (1962), 89–120 (German). MR 160196, DOI 10.1007/BF01162369
- Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982), no. 2, 255–306. MR 664497
- \author{P. Heegaard [1898],} Forstudier til en topologisk teori for algebraiske Sammenhäng, Copenhagen, det Nordiske Forlag Ernst Bojesen. http://www.maths.ed.ac.uk/$\sim$aar/papers/heegaardthesis.pdf http://www.maths.ed.ac.uk/$\sim$aar/papers/heegaardenglish.pdf
- John Hempel, $3$-Manifolds, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR 0415619
- \author{F. Hirzebruch [1966],} Topological methods in algebraic geometry, Classics in Mathematics, Springer-Verlag, Berlin.
- Friedrich Hirzebruch, Emmy Noether and topology, The heritage of Emmy Noether (Ramat-Gan, 1996) Israel Math. Conf. Proc., vol. 12, Bar-Ilan Univ., Ramat Gan, 1999, pp. 57–65. MR 1665435
- Friedrich E. P. Hirzebruch and Matthias Kreck, On the concept of genus in topology and complex analysis, Notices Amer. Math. Soc. 56 (2009), no. 6, 713–719. MR 2536790
- \author{H. Hopf [1925],} Zum Clifford-Kleinschen Raumproblem, Math. Ann. 95, no. 1, 313–319 .
- William Jaco and Peter B. Shalen, A new decomposition theorem for irreducible sufficiently-large $3$-manifolds, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 71–84. MR 520524
- I. M. James (ed.), History of topology, North-Holland, Amsterdam, 1999. MR 1674906
- Klaus Johannson, Homotopy equivalences of $3$-manifolds with boundaries, Lecture Notes in Mathematics, vol. 761, Springer, Berlin, 1979. MR 551744, DOI 10.1007/BFb0085406
- Troels Jørgensen, Compact $3$-manifolds of constant negative curvature fibering over the circle, Ann. of Math. (2) 106 (1977), no. 1, 61–72. MR 450546, DOI 10.2307/1971158
- John W. Milnor and Michel A. Kervaire, Bernoulli numbers, homotopy groups, and a theorem of Rohlin, Proc. Internat. Congress Math. 1958., Cambridge Univ. Press, New York, 1960, pp. 454–458. MR 0121801
- Michel A. Kervaire and John W. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504–537. MR 148075, DOI 10.1090/S0273-0979-2015-01504-1
- Robion C. Kirby, The topology of $4$-manifolds, Lecture Notes in Mathematics, vol. 1374, Springer-Verlag, Berlin, 1989. MR 1001966, DOI 10.1007/BFb0089031
- R. C. Kirby and L. C. Siebenmann, On the triangulation of manifolds and the Hauptvermutung, Bull. Amer. Math. Soc. 75 (1969), 742–749. MR 242166, DOI 10.1090/S0002-9904-1969-12271-8
- Robion C. Kirby and Laurence C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Annals of Mathematics Studies, No. 88, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1977. With notes by John Milnor and Michael Atiyah. MR 0645390, DOI 10.1515/9781400881505
- Olav Arnfinn Laudal and Ragni Piene (eds.), The legacy of Niels Henrik Abel, Springer-Verlag, Berlin, 2004. Papers from the Abel Bicentennial Conference held at the University of Oslo, Oslo, June 3–8, 2002; With 1 CD-ROM. MR 2074480, DOI 10.1007/978-3-642-18908-1
- S. Lefschetz, L’analysis situs et la géométrie algébrique, Gauthier-Villars, Paris, 1950 (French). MR 0033557
- \author{S. Lhuilier and J. Gergonne [1812-1813],} Mémoire sur la polyédrométrie $\cdots$, Ann. Math. Pures et Appl. (= Ann. de Gergonne) 3, 169–189. http://www.numdam.org/item?id=AMPA_1812-1813__3__169_0
- W. B. Raymond Lickorish, An introduction to knot theory, Graduate Texts in Mathematics, vol. 175, Springer-Verlag, New York, 1997. MR 1472978, DOI 10.1007/978-1-4612-0691-0
- \author{J. B. Listing [1862],} Der Census räumlicher Complexe oder Verallgemeinerung des Euler’schen Satzes von den Polyedern, Abhandlungen königlichen Gesellschaft der Wissenschaften zu Göttingen 10, 97–180. http://www.maths.ed.ac.uk/$\sim$aar/papers/listing2.pdf
- \author{C. Manolescu [2014],} An introduction to knot Floer homology, ArXiv:1401.7107.
- A. Markov, The insolubility of the problem of homeomorphy, Dokl. Akad. Nauk SSSR 121 (1958), 218–220 (Russian). MR 0097793
- John Milnor, On simply connected $4$-manifolds, Symposium internacional de topología algebraica International symposium on algebraic topology, Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958, pp. 122–128. MR 0103472
- J. Milnor, A unique decomposition theorem for $3$-manifolds, Amer. J. Math. 84 (1962), 1–7. MR 142125, DOI 10.2307/2372800
- John Milnor, The work of M. H. Freedman, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 13–15. MR 934211
- John Milnor, Differential topology forty-six years later, Notices Amer. Math. Soc. 58 (2011), no. 6, 804–809. MR 2839925
- John Milnor and Dale Husemoller, Symmetric bilinear forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73, Springer-Verlag, New York-Heidelberg, 1973. MR 0506372, DOI 10.1007/978-3-642-88330-9
- \author{A. F. Möbius [1863],} Theorie der elementaren Verwandtschaft, Berichte Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig 15, 18–57. http://www.maths.ed.ac.uk/$\sim$aar/papers/mobiussurf.pdf
- 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
- John W. Morgan, Definition of the Seiberg-Witten (SW) invariants of 4-manifolds, Low dimensional topology, New Stud. Adv. Math., vol. 3, Int. Press, Somerville, MA, 2003, pp. 1–11. MR 2052242
- John Morgan and Gang Tian, Ricci flow and the Poincaré conjecture, Clay Mathematics Monographs, vol. 3, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2007. MR 2334563, DOI 10.1305/ndjfl/1193667709
- John Morgan and Gang Tian, The geometrization conjecture, Clay Mathematics Monographs, vol. 5, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2014. MR 3186136
- G. D. Mostow, Quasi-conformal mappings in $n$-space and the rigidity of hyperbolic space forms, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 53–104. MR 236383, DOI 10.1007/BF02684590
- Walter D. Neumann and Don Zagier, Volumes of hyperbolic three-manifolds, Topology 24 (1985), no. 3, 307–332. MR 815482, DOI 10.1016/0040-9383(85)90004-7
- 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
- \author{E. Picard and G. Simart [1897, 1906],} Théorie des fonctions algébriques de deux variables indépendantes I and II, Gauthier-Villars, Paris.
- \author{H. Poincaré [1900],} Second complément à l’analysis situs, Proc. London Math. Soc. 32, no. 1, 277–308 (Oeuvres 6, 338–370).
- \author{H. Poincaré [1902],} Quatriéme complément à l’analysis situs, Journ. de Math. 8, 169–214 (Oeuvres 6, 397–434).
- 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
- Jean-Claude Pont, Petite enfance de la topologie algébrique, Enseign. Math. (2) 20 (1974), 111–126 (French). MR 373806
- Gopal Prasad, Strong rigidity of $\textbf {Q}$-rank $1$ lattices, Invent. Math. 21 (1973), 255–286. MR 385005, DOI 10.1007/BF01418789
- \author{T. Radó [1924],} Über den Begriff der Riemannschen Fläche, Acta Sc. Math. Szeged 2, 101–121.
- \author{B. Riemann [1857],} Theorie der Abel’schen Functionen, J. reine ang. Math 54. (See pages 89 and 97 of the following.) https://ia700400.us.archive.org/16/items/bernardrgesamm00riemric/bernardrgesamm00riemrich.pdf
- Robert Riley, A personal account of the discovery of hyperbolic structures on some knot complements, Expo. Math. 31 (2013), no. 2, 104–115. MR 3057120, DOI 10.1016/j.exmath.2013.01.003
- Saburo Ide, On the theory of curves in an $n$-dimensional space with the metrics $s=\int (A_ix^{\prime \prime i}+B)^{1/p}dt$. II, Tensor (N.S.) 2 (1952), 89–98. MR 52191
- V. A. Rohlin, Relations between characteristic classes of four-dimensional manifolds. , Kolomen. Ped. Inst. Uč. Zap. Ser. Fiz.-Mat. 2 (1958), no. 1, 3–17 (Russian). MR 0133138
- Dale Rolfsen, Knots and links, Mathematics Lecture Series, No. 7, Publish or Perish, Inc., Berkeley, Calif., 1976. MR 0515288
- \author{D. Salamon [1999],} Spin geometry and Seberg-Witten invariants, at http://www.math.ethz.ch/$\sim$salamon/PREPRINTS/witsei.pdf.
- Alexandru Scorpan, The wild world of 4-manifolds, American Mathematical Society, Providence, RI, 2005. MR 2136212
- \author{H. Seifert and W. Threlfall [1934],} Lehrbuch der Topologie, Teubner, Leipzig. (English translation, Academic Press 1980).
- Jean-Pierre Serre, Cours d’arithmétique, Collection SUP: “Le Mathématicien”, vol. 2, Presses Universitaires de France, Paris, 1970 (French). MR 0255476
- Stephen Smale, Generalized Poincaré’s conjecture in dimensions greater than four, Ann. of Math. (2) 74 (1961), 391–406. MR 137124, DOI 10.2307/1970239
- John Stallings, The piecewise-linear structure of Euclidean space, Proc. Cambridge Philos. Soc. 58 (1962), 481–488. MR 149457, DOI 10.1017/S0305004100036756
- Clifford Henry Taubes, Gauge theory on asymptotically periodic $4$-manifolds, J. Differential Geom. 25 (1987), no. 3, 363–430. MR 882829
- \author{W. Thurston [1980],} The geometry and topology of three-manifolds, Princeton University Lecture Notes, available at http://www.msri.org/publications/books/gt3m.
- William P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357–381. MR 648524, DOI 10.1090/S0273-0979-1982-15003-0
- William P. Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, vol. 35, Princeton University Press, Princeton, NJ, 1997. Edited by Silvio Levy. MR 1435975, DOI 10.1515/9781400865321
- Friedhelm Waldhausen, On irreducible $3$-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56–88. MR 224099, DOI 10.2307/1970594
- J. H. C. Whitehead, On simply connected, $4$-dimensional polyhedra, Comment. Math. Helv. 22 (1949), 48–92. MR 29171, DOI 10.1007/BF02568048
- J. H. C. Whitehead, Combinatorial homotopy. I, Bull. Amer. Math. Soc. 55 (1949), 213–245. MR 30759, DOI 10.1090/S0002-9904-1949-09175-9
- Don Zagier, Hyperbolic manifolds and special values of Dedekind zeta-functions, Invent. Math. 83 (1986), no. 2, 285–301. MR 818354, DOI 10.1007/BF01388964
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Additional Information
- John Milnor
- Affiliation: Institute for Mathematical Sciences, Stony Brook University, New York
- MR Author ID: 125060
- Email: jack@math.sunysb.edu
- Published electronically: July 1, 2015
- © Copyright 2015 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 52 (2015), 545-584
- MSC (2010): Primary 57N05--57N13; Secondary 01A55, 01A60
- DOI: https://doi.org/10.1090/bull/1507
- MathSciNet review: 3393347
Dedicated: Based on the Abel Lecture at the 2014 International Congress of Mathematicians in Seoul