Skip to main content
SpringerLink
Log in

John W. Milnor’s Work on the Classification of Differentiable Manifolds

  • Chapter
  • First Online:
The Abel Prize 2008-2012

Part of the book series: The Abel Prize ((AP))

  • 2345 Accesses

Abstract

Jack Milnor has recently given this account of his unexpected encounter with exoticity:

…I was trying to study 3-connected 8-manifolds. The case H 4=0 seemed too hard. For \(H_{4} = \mathbb{Z}\), one can assume that the 4-skeleton is a 4-sphere. To build an 8-manifold, one can try to fatten it up by taking a tubular normal bundle neighborhood, and then adjoin an 8-cell. This worked so beautifully that I came up with many manifolds which couldn’t possibly exist…

This article begins with a detailed elementary exposition of his revolutionary 1956 article that resolved the mentioned paradoxes, by unveiling exotic smooth structures on the 7-sphere. It continues by extensively describing Milnor’s introduction of ‘surgery’ to classify smooth structures on spheres. It concludes by sketching (all too briefly) the subsequent evolution of surgery theory—always confined to dimensions ≥5. In the hands of several following generations of topologists, it has become and remains the dominant tool for classification of compact, smooth (or piecewise linear or topological) manifolds.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 139.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 179.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 179.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Poincaré used triangulations as an essential tool in defining homology, and in establishing Poincaré duality for closed oriented manifolds. Anticipating [WhdJ40] (as Poincaré seemingly did) one can quickly deduce invariance of his homology under diffeomorphisms and also under homeomorphisms that are pl with respect to smooth triangulations; indeed, exercising hindsight, and using the simple bisection operation of J.W. Alexander [Alr30] and K. Reidemeister [Reid38], this deduction is easy, see [Sieb80].

  2. 2.

    Ever since F. Hausdorff’s 1914 monograph [Hau14], “homeomorphism” has consistently meant a one-to-one continuous map between topological spaces such that the inverse map is also continuous. However, for manifolds, Poincaré used the term “homéomorphe” (in French) to mean sometimes diffeomorphic and sometimes pl homeomorphic. In dynamics and in discussing dimension he did indeed sometimes use “homéomorphe” in its modern meaning!

  3. 3.

    The mimeographed notes for it are still extant at http://www.maths.ed.ac.uk/%7Eaar/surgery/.

  4. 4.

    Why was this additivity left unmentioned in 1956? Perhaps because connected sum of two oriented connected n-manifolds without boundary had not yet been proved to be well-defined up to degree +1 diffeomorphism. However this does not invalidate the additivity as asserted and used here. The tubular neighborhood uniqueness theorem, see [Lang62], for the special case of point submanifolds easily implies well-definition of connected sum.

  5. 5.

    The following two quotes are taken with permission from Milnor’s correspondence with this author in October 2013. Compare the historical comments by Milnor in the introduction to Part I of [M07a].

  6. 6.

    On the other hand, Milnor was able to explicitly construct such Morse functions for further examples of exotic spheres in [M59b]. And, M. Kervaire did as much for his 9-dimensional exotic sphere in [Kerv60].

  7. 7.

    http://en.wikipedia.org/wiki/Homotopy_groups_of_spheres.

  8. 8.

    Beware that number theorists denote this rational number |B 2k|.

  9. 9.

    See http://en.wikipedia.org/wiki/Borel_conjecture, http://www.maths.ed.ac.uk/~aar/surgery/borel.pdf, http://lcs98.free.fr/biblio/exposit/, http://egg.epfl.ch/~nmonod/geneve08/Slides_Lueck.pdf.

  10. 10.

    See http://en.wikipedia.org/wiki/Farrell%E2%80%93Jones_conjecture.

References

  1. Adams J.F., On the nonexistence of elements of Hopf invariant one, Bull. Am. Math. Soc. 64 (1958) 279–282.

    MathSciNet  MATH  Google Scholar 

  2. Adams J.F., On the non-existence of elements of Hopf invariant one, Ann. of Math. (2) 72(1960) 20–104.

    MathSciNet  MATH  Google Scholar 

  3. Adams J.F. and Atiyah M.F., K-theory and the Hopf invariant, Q. J. Math. Oxford, series II, 17(1966) 31–38.

    MathSciNet  MATH  Google Scholar 

  4. Alexander J.W., The combinatorial theory of complexes, Ann. Math. 31(1930) 292–320.

    MathSciNet  MATH  Google Scholar 

  5. Alexander J.W., Some problems in topology, Verhandlungen Kongress Zürich 1932, 1(1932) 249–257.

    Google Scholar 

  6. Arf C., Untersuchungen über quadratische Formen in Körpern der Charakteristik 2, J. Reine Angew. Math., 183(1941) 148–167. See [LorR11].

    MathSciNet  MATH  Google Scholar 

  7. Atiyah M.F. and Hirzebruch F., Riemann-Roch theorems for differentiable manifolds, Bull. Amer. Math. Soc. 65 (1959), 276–281.

    MathSciNet  MATH  Google Scholar 

  8. Atiyah M.F., Thom complexes, Proc. Lond. Math. Soc., Ser. III., 11(1961) 291–310.

    MathSciNet  MATH  Google Scholar 

  9. Barratt M.G., Jones J.D.S., Mahowald M.E., The Kervaire invariant and the Hopf invariant, algebraic topology, Proc. Workshop, Seattle 1985, Lect. Notes Math. 1286(1987) 135–173.

    Google Scholar 

  10. Barratt M.G., Mahowald M.E., and Tangora M.C., Some differentials in the Adams spectral sequence, II, Topology 9(1970) 309–316.

    MathSciNet  MATH  Google Scholar 

  11. Brieskorn E., Beispiele zur Differentialtopologie von Singularitäten, Invent. Math. 2(1966) 1–14

    MathSciNet  MATH  Google Scholar 

  12. W. Browder, Homotopy type of differential manifolds, Proc. Aarhus Topology Conference (1962); Vol 1, London Math. Soc. Lecture Notes 226(1995) 97–100.

    Google Scholar 

  13. Browder W., Embedding smooth manifolds, Proc. Internat. Congr. Math. (Moscow, 1966), Mir Moscow 1968, 712–719.

    Google Scholar 

  14. Browder W., The Kervaire invariant of framed manifolds and its generalization, Ann. Math. 90(1969) 157–186.

    MathSciNet  MATH  Google Scholar 

  15. Browder W., Surgery on Simply-Connected Manifolds, Ergebnisse Band 65, Springer Berlin, 1972.

    MATH  Google Scholar 

  16. Browder W. and Hirsch, M.W, Surgery on piecewise linear manifolds and applications, Bull. Am. Math. Soc. 72(1966) 959–964.

    MathSciNet  MATH  Google Scholar 

  17. Brown E.H. and Peterson F.P., The Kervaire invariant of (8k+2)-manifolds, Am. J. Math. 88(1966) 815–826.

    MathSciNet  MATH  Google Scholar 

  18. Brumfiel G., The homotopy groups of BPL and PL/O III, Mich. Math. J. 17(1970) 217–224.

    MathSciNet  MATH  Google Scholar 

  19. Cairns S., Introduction of a Riemannian geometry on a triangulable 4-manifold, Ann. of Math. (2) 45(1944) 218–219.

    MathSciNet  MATH  Google Scholar 

  20. Cappell S. et al. (editors), Surveys on Surgery Theory. Vol. 1, Papers dedicated to C.T.C. Wall on the occasion to his 60th birthday, Princeton Univ. Press, Princeton, Ann. Math. Stud. 145, 2000.

    MATH  Google Scholar 

  21. Cappell S. et al.(editors), Surveys on Surgery Theory, Vol. 2: Papers dedicated to C.T.C. Wall on the occasion of his 60th birthday, Princeton Univ. Press, Princeton, Ann. Math. Stud. 149, 2001.

    MATH  Google Scholar 

  22. Crowley D. and Escher C., A classification of S 3-bundles over S 4, Differ. Geom. Appl. 18(2003), No. 3, 363–380.

    MathSciNet  MATH  Google Scholar 

  23. Davis D. and Mahowald M., The image of the stable J-homomorphism, Topology 28(1989) 39–58.

    MathSciNet  MATH  Google Scholar 

  24. Donaldson S.K., Irrationality and the h-cobordism conjecture, J. Diff. Geom. 26 (1987) 141–168.

    MathSciNet  MATH  Google Scholar 

  25. Eells J., and Kuiper N., Closed manifolds which admit nondegenerate functions with three critical points, Indag. Math. 23(1961) 411–417.

    MathSciNet  MATH  Google Scholar 

  26. Eells J., and Kuiper N., An invariant for certain smooth manifolds, Ann. Mat. Pura Appl., ser. IV, v. 60, (1962) 93–110.

    MathSciNet  MATH  Google Scholar 

  27. Ferry S., Ranicki A., and Rosenberg J. (editors), Novikov Conjectures, Index Theorems and Rigidity, Vol. 1 and Vol. 2, LMS Lecture Note Series 226 and 227, Cambridge Univ. Press, Cambridge, 1995.

    MATH  Google Scholar 

  28. Grove K. and Ziller W., Curvature and symmetry of Milnor spheres, Ann. Math. 152(2000) 331–367.

    MathSciNet  MATH  Google Scholar 

  29. Haefliger A., Plongements différentiables de variétés dans variétés, Comment. Math. Helv. 36(1961) 47–82; see Theorem 5.1 on page 76.

    MathSciNet  MATH  Google Scholar 

  30. Haefliger A., Knotted (4k−1)-spheres in 6k-space, Ann. of Math.(2) (1962) 452–466.

    Google Scholar 

  31. Haefliger A., Knotted spheres and related geometric problems, Proc. Internat. Congr. Math. (Moscow, 1966), Mir Moscow, 1968, 437–445.

    Google Scholar 

  32. Hausdorff F., Grundzüge der Mengenlehre, Veit Leipzig 1914, 476 S; reprinted by Chelsea, USA, 1965.

    MATH  Google Scholar 

  33. Hill M., Hopkins M. and Ravenel D., On the non-existence of elements of Kervaire invariant one, article arXiv:0908.3724, revision of November 2010.

  34. Hirsch M., Obstruction theories for smoothing manifolds and maps, Bull. Amer. Math. Soc. 69(1963) 352–356.

    MathSciNet  MATH  Google Scholar 

  35. Hirsch, M. W. and Mazur B., Smoothings of Piecewise Linear Manifolds, Ann. of Math. Study 80, Princeton Univ. Press, Princeton, 1974, ix+134 pages.

    MATH  Google Scholar 

  36. Hirzebruch F., Über die quaternionalen projektiven Räume (On the quaternion projective spaces), Sitzungsber. Math.-Naturw. Kl., Bayer. Akad. Wiss. München, 1953, S. 301–312.

    MATH  Google Scholar 

  37. Hirzebruch F., Some problems on differentiable and complex manifolds, Ann. of Math. (2) 60(1954) 213–236.

    MathSciNet  MATH  Google Scholar 

  38. Hirzebruch F., Neue topologische Methoden in der algebraischen Geometrie, Ergebnisse 9. Springer, Berlin, 1956, 165 S.

    MATH  Google Scholar 

  39. Hirzebruch F., Topological Methods in Algebraic Geometry, Springer Berlin, 1966, 232 pages. This is a translation and expansion of [Hirz56] with an appendix by R.L.E. Schwarzenberger.

    MATH  Google Scholar 

  40. Husemoller D., Fibre Bundles, McGraw-Hill New York, 1966; Springer-Verlag, GTM 20, 1975 and 1994.

    MATH  Google Scholar 

  41. James I.M., and Whitehead J.H.C., The homotopy theory of sphere bundles over spheres. II, Proc. London Math. Soc. (3) 5, (1955) 148–166.

    MathSciNet  MATH  Google Scholar 

  42. Joachim M. and Wraith D., Exotic spheres and curvature, Bull. Am. Math. Soc., 45(2008), No. 4, 595–616.

    MathSciNet  MATH  Google Scholar 

  43. Johnson N., Visualizing seven-manifolds, an animation presented at the second Abel conference: A mathematical celebration of John Milnor, January 2012, see http://www.nilesjohnson.net/seven-manifolds.html.

  44. Kahn P.J., A note on topological Pontrjagin classes and the Hirzebruch index formula, Ill. J. Math. 16(1972) 243–256.

    MathSciNet  MATH  Google Scholar 

  45. Kervaire M.A., A manifold which does not admit any differentiable structure, Comment. Math. Helv. 34(1960) 257–270.

    MathSciNet  MATH  Google Scholar 

  46. (=[M60c]) Kervaire M.A., and Milnor, J.W., Bernoulli numbers, homotopy groups, and a theorem of Rohlin, Proc. Internat. Congress Math. Edinburgh 1958, pages 454–458. Cambridge Univ. Press, New York, 1960.

    Google Scholar 

  47. (=[M63a]) Kervaire M.A., and Milnor J.W., Groups of homotopy spheres I, Ann. of Math. (2) 77(1963) 504–537.

    MathSciNet  MATH  Google Scholar 

  48. Kirby R.C. and Siebenmann L.C., Some theorems on topological manifolds, in manifolds—Amsterdam 1970, Proc. NUFFIC Summer School Manifolds 1970, Lect. Notes Math. 197, 1–7, 1971.

    Google Scholar 

  49. Kirby R.C. and Siebenmann L.C., Foundational Essays on Topological Manifolds, Smoothing and Triangulations, Annals of Mathematics Study 88(1977).

    MATH  Google Scholar 

  50. Kneser H., Die Topologie der Mannigfaltigkeiten, Jahresber. Dtsch. Math.-Ver. 34(1925) 1–14.

    MATH  Google Scholar 

  51. Lang S., Introduction to Differentiable Manifolds, Interscience, John Wiley Sons, New York, 1962.

    MATH  Google Scholar 

  52. Levine J., A classification of differentiable knots, Ann. of Math. (2) 82(1965) 15–50.

    MathSciNet  MATH  Google Scholar 

  53. Lorenz F. and Roquette P., Cahit arf and his invariant. Preprint, June 17, 2011. Current URL is http://www.rzuser.uni-heidelberg.de/%7Eci3/arf3-withpicture.pdf.

  54. Madsen I. and Milgram R.J., The Classifying Spaces for Surgery and Cobordism of Manifolds, Annals of Mathematics Study 92(1979), xii+279 pages.

    MATH  Google Scholar 

  55. Marin A., La transversalité topologique, Ann. Math. (2) 106(1977) 269–293.

    MathSciNet  MATH  Google Scholar 

  56. Mazur B., On embeddings of spheres, Bull. Amer. Math. Soc. 65(1959) 59–65.

    MathSciNet  MATH  Google Scholar 

  57. Mazur B., Stable equivalence of differentiable manifolds, Bull. Am. Math. Soc. 67(1961) 377–384.

    MathSciNet  MATH  Google Scholar 

  58. Miller H., Kervaire invariant one [after M.A. Hill, M.J. Hopkins, and D.C. Ravenel], Séminaire Bourbaki no 1029, Novembre 2010, 63ème année.

    Google Scholar 

  59. List of Publications for John Willard Milnor. There are currently three sources: (1) This volume, Chapter 22; (2) Any future volume of in the series: Collected Papers of John Milnor, published by the American Mathematical Society; (3) Internet: http://www.math.sunysb.edu/%7Ejack/milnor-pub.pdf.

  60. Milnor J.W., On manifolds homeomorphic to the 7-sphere, Ann. of Math. (2) 64(1956) 399–405.

    MathSciNet  MATH  Google Scholar 

  61. Milnor J.W., On the relationship between differentiable manifolds and combinatorial manifolds. First published in [M07a, pp. 19–28].

    Google Scholar 

  62. Milnor J.W., The Steenrod algebra and its dual, Ann. of Math. (2) 67(1958) 150–171, see also [M09a, pp. 61–82].

    MathSciNet  MATH  Google Scholar 

  63. Milnor J.W., Differentiable structures on spheres, Am. J. Math., 81(1959) 962–972.

    MathSciNet  MATH  Google Scholar 

  64. Milnor J.W., Sommes de variétes différentiables et structures différentiables des sphères. Bull. Soc. Math. Fr., 87(1959) 439–444.

    MATH  Google Scholar 

  65. Milnor J.W., Differentiable manifolds which are homotopy spheres, mimeographed at Princeton, dated January 23 1959, first published as pp. 65–88 of [M07a].

    Google Scholar 

  66. Milnor J.W., A procedure for killing homotopy groups of differentiable manifolds. In Proc. Sympos. Pure Math., V. III, pages 39–55, Amer. Math. Soc., Providence, 1961.

    Google Scholar 

  67. Milnor J.W., Two complexes which are homeomorphic but combinatorially distinct. Ann. of Math. (2) 74(1961) 575–590.

    MathSciNet  MATH  Google Scholar 

  68. Milnor J.W., Microbundles and differentiable structures, polycopied at Princeton U., 1961; first published as pages 173–190 in [M09a].

    Google Scholar 

  69. Milnor J.W., Topological manifolds and smooth manifolds, in Proc. Internat. Congr. Math., Stockholm 1962, pages 132–138, Inst. Mittag-Leffler, Djursholm, Sweden 1963; also published as pages 191–197 in [M09a].

    Google Scholar 

  70. Milnor J.W., Morse Theory, Based on Lecture Notes by M. Spivak and R. Wells, Annals of Math. Studies, no. 51. Princeton Univ. Press, Princeton, 1963. (Translated into Russian, Japanese, Korean).

    MATH  Google Scholar 

  71. Milnor J.W., Microbundles I, Topology 3 (suppl. 1), 53–80.

    Google Scholar 

  72. Milnor J.W., Differential Topology, pages 165–183 in Lectures on Modern Mathematics, vol. II, ed. T. Saaty, Wiley 1964 New York; also in Russian, Uspehi Mat. Nauk, 20(1965) no. 6, 41–54.

    MATH  Google Scholar 

  73. Milnor J.W., Remarks concerning spin manifolds, In Differential and Combinatorial Topology, a Symposium in Honor of Marston Morse, S.S. Cairns, editor, pages 55–62. Princeton Univ. Press, Princeton, 1965 and [CP-3, 299–306].

    Google Scholar 

  74. Milnor J.W., Lectures on the h-Cobordism Theorem. Notes by L. Siebenmann and J. Sondow. Princeton Univ. Press, Princeton, 1965. (Translated into Russian).

    MATH  Google Scholar 

  75. Milnor J.W., Topology from the differentiable viewpoint, based on notes by David W. Weaver, University Press of Virginia, Charlottesville, Revised reprint, Princeton Landmarks in Mathematics, Princeton Univ. Press, Princeton, NJ, 1997. (Translated into Russian, Japanese).

    Google Scholar 

  76. Milnor J.W., Classification of (n−1)-connected 2n-dimensional manifolds and the discovery of exotic spheres. In Surveys on Surgery Theory, vol. 1, Annals of Math. Studies, no. 145, pages 25–30, Princeton Univ. Press, Princeton, 2000.

    Google Scholar 

  77. Milnor J.W., Towards the Poincaré conjecture and the classification of 3-manifolds, Not. Am. Math. Soc., 50(10) (2003) 1226–1233. Also available in Gaz. Math. S.M.F. 99 (2004) 13–25 (in French).

    MATH  Google Scholar 

  78. Milnor J.W., The Poincaré conjecture one hundred years later. In The Millennium Prize Problems, pages 71–83. Clay Math. Inst., Cambridge, 2006.

    Google Scholar 

  79. Milnor J.W., Collected Papers of John Milnor III, Differential Topology, Amer. Math. Soc., Providence, 2007.

    MATH  Google Scholar 

  80. Milnor J.W., Collected Papers of John Milnor IV, Homotopy, Homology and Manifolds, Edited by J. McCleary, Amer. Math. Soc., Providence, 2009.

    MATH  Google Scholar 

  81. Milnor J.W., Fifty years ago: topology of manifolds in the 50s and 60s. In Low Dimensional Topology, volume 15 of IAS/Park City Math. Ser., T. Mrowka and P. Osváth, editors, pages 9–20, Amer. Math. Soc. Providence, 2009, see [M09a, p. 345–356].

    Google Scholar 

  82. Milnor J.W., Differential topology forty-six years later, Notices Am. Math. Soc., 58(2011) 804–809.

    MathSciNet  MATH  Google Scholar 

  83. (=[M60a]) Milnor J.W. and Spanier E., Two remarks on fiber homotopy type, Pac. J. Math. 10(1960) 585–590.

    MathSciNet  MATH  Google Scholar 

  84. (=[M57a]) Milnor J.W. and Stasheff J.D., Characteristic Classes. Annals of Math. Studies, no. 76. Princeton University Press, Princeton, 1974. (Translated into Russian, Japanese).

    Google Scholar 

  85. Moise E.E., Affine structures in 3-manifolds. V., The triangulation theorem and hauptvermutung, Ann. of Math. (2) 56(1952) 96–114.

    MathSciNet  MATH  Google Scholar 

  86. Morse M., Relation between the critical points of a real function of n independent variables, Trans. Am. Math. Soc. 27(1925) 345–396.

    MathSciNet  MATH  Google Scholar 

  87. Munkres J.R., Elementary differential topology, Annals of Math Studies 54, Princeton U. Press, Princeton, 1963.

    MATH  Google Scholar 

  88. Munkres J.R., Concordance is equivalent to smoothability, Topology 5(1966) 371–389.

    MathSciNet  MATH  Google Scholar 

  89. Munkres J.R., Concordance of differentiable structures—two approaches. Michigan Math. J. 14(1967) 183–191.

    MathSciNet  MATH  Google Scholar 

  90. Munkres J.R., Compatibility of imposed differentiable structures. Illinois J. Math. 12(1968) 610–615.

    MathSciNet  MATH  Google Scholar 

  91. Novikov S.P., Diffeomorphisms of simply connected manifolds Sov. Math. Dokl. 3(1962) 540–543; translation from Russian Dokl. Akad. Nauk SSSR 143(1962) 1046–1049.

    MATH  Google Scholar 

  92. Novikov S.P., Topological invariance of rational Pontrjagin classes (English), Sov. Math. Dokl. 6 (1965) 921–923; translation from Dokl. Akad. Nauk SSSR 163 (1965) 298–300 (Russian).

    MATH  Google Scholar 

  93. Novikov S.P., Surgery in the 1960’s, pages 31–39 in [Cap00].

    Google Scholar 

  94. Palais R.S., Gleason’s contribution to the solution of Hilbert’s fifth problem, Not. Am. Math. Soc. 56(2009) 1243–1248.

    Google Scholar 

  95. Peter F., and Weyl H., Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe, Math. Ann. 97(1927) 737–755.

    MathSciNet  MATH  Google Scholar 

  96. Ranicki A., Algebraic L-Theory and Topological Manifolds, Cambridge Tracts in Mathematics, 102(1992), Cambridge University Press, Cambridge, viii+358 pages.

    MATH  Google Scholar 

  97. Ranicki A., An introduction to algebraic surgery, Surveys on Surgery Theory, Vol. 2, 81–163, Ann. of Math. Study 149(2001), Princeton Univ. Press, Princeton.

    MATH  Google Scholar 

  98. Ranicki, A., The structure set of an arbitrary space, the algebraic surgery exact sequence and the total surgery obstruction, pages 515–538, Topology of High-Dimensional Manifolds, Nos. 1, 2 (Trieste, 2001), ICTP Lect. Notes, 9, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2002. (See http://lcs98.free.fr/biblio/exposit/.)

    Google Scholar 

  99. Raussen M. and Skau C., Interview with John Milnor, Newsl. - Eur. Math. Soc., September 2011, (81), pages 31–40.

    MathSciNet  MATH  Google Scholar 

  100. Ravenel, D., See http://www.math.rochester.edu/people/faculty/doug

  101. Ravenel D., A solution to the Arf–Kervaire invariant problem, illustrated lecture, Second Abel conference, A Mathematical Celebration of John Milnor, U. of Minn, Minneapolis, February 2012. (Preprint slides available at http://www.math.rochester.edu/people/faculty/doug/AKtalks.html and http://www.maths.ed.ac.uk/%7Eaar/hhrabel.pdf.)

    Google Scholar 

  102. Reeb G., Stabilité des feuilles compactes à groupe de Poincaré fini, C.R. Acad. Sci., Paris 228(1949) 47–48.

    MathSciNet  MATH  Google Scholar 

  103. Reidemeister K., Topologie der Polyeder und kombinatorische Topologie der Komplexe, Akademische Verlagsgesellschaft Geest und Portig, Leipzig, 1938; zweite (unveränderte) Auflage 1953.

    MATH  Google Scholar 

  104. Rourke C.P. and Sanderson B.J., Introduction to piecewise-linear topology, Ergebnisse, Bd. 69. Springer, Berlin, 1972.

    MATH  Google Scholar 

  105. Serre J.-P., Homologie singulière des espaces fibrés, Applications, Ann. of Math. (2) 54(1951) 425–505.

    MathSciNet  MATH  Google Scholar 

  106. Seifert H., La théorie des noeuds, Enseign. Math. 35(1936) 201–212.

    MATH  Google Scholar 

  107. Siebenmann L.C., Topological manifolds, Actes Congr. Internat. Math. 1970, 2, 133–163 (1971).

    MathSciNet  MATH  Google Scholar 

  108. Siebenmann L.C., Les Bisections expliquent le théorème de Reidemeister–Singer—un retour aux sources, dont la Section 5 (suite) l’Histoire des bisections, prépublication d’Orsay, 1980. (Printed version at http://lcs98.free.fr/biblio/prepub/.)

  109. Smale S., The generalized Poincaré conjecture in higher dimensions, Bull. Am. Math. Soc. 66(1960) 373–375.

    MATH  Google Scholar 

  110. Smale S., Generalized Poincaré’s conjecture in dimensions greater than four, Ann. of Math. (2) 74(1961) 391–406.

    MathSciNet  MATH  Google Scholar 

  111. Smale S., On the structure of manifolds, Am. J. Math. 84(1962) 387–399.

    MATH  Google Scholar 

  112. Spivak M., Spaces satisfying Poincaré duality, Topology 6(1967) 77–101.

    MathSciNet  MATH  Google Scholar 

  113. Steenrod N., The Topology of Fibre Bundles, Princeton Univ. Press, Princeton 1951.

    MATH  Google Scholar 

  114. Steinitz E., Beiträge zur Analysis Situs, Sitzungsber. Berl. Math. Ges. 7(1908) 29–49.

    MATH  Google Scholar 

  115. Sullivan D.P., Triangulating Homotopy Equivalences, Princeton Univ. Press, Princeton, 1966.

    Google Scholar 

  116. Sullivan D.P., Triangulating and Smoothing Homotopy Equivalences, Geometric Topology Seminar Notes, Princeton Univ. Press, Princeton, 1967.

    Google Scholar 

  117. Thom R., Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28(1954) 17–86.

    MathSciNet  MATH  Google Scholar 

  118. Thom R., Les classes caractéristiques de Pontrjagin des variétés triangulées, International Symposium on Algebraic Topology, Universidad Nacional Autonoma de Mexico and UNESCO, Mexico City, Mexico, 1958, 54–67.

    Google Scholar 

  119. Thom R., Des variétés triangulées aux variétés différentiables, Proc. Int. Congr. Math. 1958, 248–255 (1960).

    MathSciNet  MATH  Google Scholar 

  120. Tietze H., Über die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten, Monatshefte für Math. u. Physica 19(1908) 1–118.

    MATH  Google Scholar 

  121. von Neumann J., Zur Theorie der Darstellungen kontinuierlicher Gruppen, Sitzungsberichte Akad. Berlin 1927, 76–90.

    Google Scholar 

  122. Wall C.T.C., An extension of results of Novikov and Browder, Am. J. Math. 88(1966) 20–32.

    MathSciNet  MATH  Google Scholar 

  123. Wall C.T.C., Surgery on Compact Manifolds, London Mathematical Society Monographs, No.1., Academic Press, New York, 1970, 280 pp.; 2nd edition (edited by A. Ranicki), Amer. Math. Soc. 302 pages (1999).

    MATH  Google Scholar 

  124. Wallace A.H., Modifications and cobounding manifolds, Can. J. Math. 12(1960) 503–528.

    MathSciNet  MATH  Google Scholar 

  125. Wallace A.H., Modifications and cobounding manifolds II, Journal of Mathematics and Mechanics 10(1961), 773–809. This journal’s current name is Indiana University Mathematics Journal.

    MathSciNet  MATH  Google Scholar 

  126. Weierstrass K., Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen einer reellen Veränderlichen, Berliner Berichte 1885, 633–640, 789–806.

    Google Scholar 

  127. Whitehead G.W., On the homotopy groups of spheres and rotation groups, Ann. of Math. (2) 43(1942) 634–640.

    MathSciNet  MATH  Google Scholar 

  128. Whitehead J.H.C., On C 1-complexes, Ann. of Math. (2) 41(1940) 809–824.

    MathSciNet  MATH  Google Scholar 

  129. Whitehead J.H.C., Manifolds with transverse fields in Euclidean space, Ann. of Math. (2) 73(1961) 154–212.

    MathSciNet  MATH  Google Scholar 

  130. Whitney, H., Differentiable manifolds, Ann. of Math. (2) 37(1936) 645–680.

    MathSciNet  MATH  Google Scholar 

  131. Whitney H., The singularities of a smooth n-manifold in (2n−1)-space, Ann. of Math. (2) 45(1944) 247–293.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Mathématique, Université de Paris-Sud, 91405, Orsay, France

    L. C. Siebenmann

Authors
  1. L. C. Siebenmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. C. Siebenmann .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Helge Holden

  2. Department of Mathematics, University of Oslo, Oslo, Norway

    Ragni Piene

1 Electronic Supplementary Material

Below are the links to the electronic supplementary material.

A lecture by Prof. Ghys in connection with the Abel Prize 2011 to John Milnor (MP4 399 MB)

A lecture by Prof. Hopkins in connection with the Abel Prize 2011 to John Milnor (MP4 306 MB)

A lecture by Prof. McMullen in connection with the Abel Prize 2011 to John Milnor (MP4 362 MB)

The Abel Lecture by John Milnor, the Abel Laureate 2011 (MP4 379 MB)

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Siebenmann, L.C. (2014). John W. Milnor’s Work on the Classification of Differentiable Manifolds. In: Holden, H., Piene, R. (eds) The Abel Prize 2008-2012. The Abel Prize. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39449-2_21

Download citation

Publish with us

Policies and ethics

Search

Navigation