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

  • L. C. SiebenmannEmail author
Part of the The Abel Prize book series (AP)


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

…I was trying to study 3-connected 8-manifolds. The case H4=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.


Smooth Manifold Homotopy Class Morse Function Topological Manifold Pontrjagin Class 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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)


  1. [Ad58]
    Adams J.F., On the nonexistence of elements of Hopf invariant one, Bull. Am. Math. Soc. 64 (1958) 279–282. MathSciNetzbMATHGoogle Scholar
  2. [Ad60]
    Adams J.F., On the non-existence of elements of Hopf invariant one, Ann. of Math. (2) 72(1960) 20–104. MathSciNetzbMATHGoogle Scholar
  3. [AdA66]
    Adams J.F. and Atiyah M.F., K-theory and the Hopf invariant, Q. J. Math. Oxford, series II, 17(1966) 31–38. MathSciNetzbMATHGoogle Scholar
  4. [Alr30]
    Alexander J.W., The combinatorial theory of complexes, Ann. Math. 31(1930) 292–320. MathSciNetzbMATHGoogle Scholar
  5. [Alr32]
    Alexander J.W., Some problems in topology, Verhandlungen Kongress Zürich 1932, 1(1932) 249–257. Google Scholar
  6. [Arf41]
    Arf C., Untersuchungen über quadratische Formen in Körpern der Charakteristik 2, J. Reine Angew. Math., 183(1941) 148–167. See [LorR11]. MathSciNetzbMATHGoogle Scholar
  7. [AH59]
    Atiyah M.F. and Hirzebruch F., Riemann-Roch theorems for differentiable manifolds, Bull. Amer. Math. Soc. 65 (1959), 276–281. MathSciNetzbMATHGoogle Scholar
  8. [At61]
    Atiyah M.F., Thom complexes, Proc. Lond. Math. Soc., Ser. III., 11(1961) 291–310. MathSciNetzbMATHGoogle Scholar
  9. [BaJM83]
    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. [BaMT70]
    Barratt M.G., Mahowald M.E., and Tangora M.C., Some differentials in the Adams spectral sequence, II, Topology 9(1970) 309–316. MathSciNetzbMATHGoogle Scholar
  11. [Brie66]
    Brieskorn E., Beispiele zur Differentialtopologie von Singularitäten, Invent. Math. 2(1966) 1–14 MathSciNetzbMATHGoogle Scholar
  12. [Brow62]
    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. [Brow68]
    Browder W., Embedding smooth manifolds, Proc. Internat. Congr. Math. (Moscow, 1966), Mir Moscow 1968, 712–719. Google Scholar
  14. [Brow69]
    Browder W., The Kervaire invariant of framed manifolds and its generalization, Ann. Math. 90(1969) 157–186. MathSciNetzbMATHGoogle Scholar
  15. [Brow72]
    Browder W., Surgery on Simply-Connected Manifolds, Ergebnisse Band 65, Springer Berlin, 1972. zbMATHGoogle Scholar
  16. [BrowH66]
    Browder W. and Hirsch, M.W, Surgery on piecewise linear manifolds and applications, Bull. Am. Math. Soc. 72(1966) 959–964. MathSciNetzbMATHGoogle Scholar
  17. [BP66]
    Brown E.H. and Peterson F.P., The Kervaire invariant of (8k+2)-manifolds, Am. J. Math. 88(1966) 815–826. MathSciNetzbMATHGoogle Scholar
  18. [Bru70]
    Brumfiel G., The homotopy groups of BPL and PL/O III, Mich. Math. J. 17(1970) 217–224. MathSciNetzbMATHGoogle Scholar
  19. [Cai44]
    Cairns S., Introduction of a Riemannian geometry on a triangulable 4-manifold, Ann. of Math. (2) 45(1944) 218–219. MathSciNetzbMATHGoogle Scholar
  20. [Cap00]
    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. zbMATHGoogle Scholar
  21. [Cap01]
    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. zbMATHGoogle Scholar
  22. [CrE03]
    Crowley D. and Escher C., A classification of S 3-bundles over S 4, Differ. Geom. Appl. 18(2003), No. 3, 363–380. MathSciNetzbMATHGoogle Scholar
  23. [DaM89]
    Davis D. and Mahowald M., The image of the stable J-homomorphism, Topology 28(1989) 39–58. MathSciNetzbMATHGoogle Scholar
  24. [Don87]
    Donaldson S.K., Irrationality and the h-cobordism conjecture, J. Diff. Geom. 26 (1987) 141–168. MathSciNetzbMATHGoogle Scholar
  25. [EK61]
    Eells J., and Kuiper N., Closed manifolds which admit nondegenerate functions with three critical points, Indag. Math. 23(1961) 411–417. MathSciNetzbMATHGoogle Scholar
  26. [EK62]
    Eells J., and Kuiper N., An invariant for certain smooth manifolds, Ann. Mat. Pura Appl., ser. IV, v. 60, (1962) 93–110. MathSciNetzbMATHGoogle Scholar
  27. [FRR95]
    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. zbMATHGoogle Scholar
  28. [GrZ00]
    Grove K. and Ziller W., Curvature and symmetry of Milnor spheres, Ann. Math. 152(2000) 331–367. MathSciNetzbMATHGoogle Scholar
  29. [Haef61b]
    Haefliger A., Plongements différentiables de variétés dans variétés, Comment. Math. Helv. 36(1961) 47–82; see Theorem 5.1 on page 76. MathSciNetzbMATHGoogle Scholar
  30. [Haef62]
    Haefliger A., Knotted (4k−1)-spheres in 6k-space, Ann. of Math.(2) (1962) 452–466. Google Scholar
  31. [Haef68]
    Haefliger A., Knotted spheres and related geometric problems, Proc. Internat. Congr. Math. (Moscow, 1966), Mir Moscow, 1968, 437–445. Google Scholar
  32. [Hau14]
    Hausdorff F., Grundzüge der Mengenlehre, Veit Leipzig 1914, 476 S; reprinted by Chelsea, USA, 1965. zbMATHGoogle Scholar
  33. [HillHR09]
    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. [Hir63]
    Hirsch M., Obstruction theories for smoothing manifolds and maps, Bull. Amer. Math. Soc. 69(1963) 352–356. MathSciNetzbMATHGoogle Scholar
  35. [HirM74]
    Hirsch, M. W. and Mazur B., Smoothings of Piecewise Linear Manifolds, Ann. of Math. Study 80, Princeton Univ. Press, Princeton, 1974, ix+134 pages. zbMATHGoogle Scholar
  36. [Hirz53]
    Hirzebruch F., Über die quaternionalen projektiven Räume (On the quaternion projective spaces), Sitzungsber. Math.-Naturw. Kl., Bayer. Akad. Wiss. München, 1953, S. 301–312. zbMATHGoogle Scholar
  37. [Hirz54]
    Hirzebruch F., Some problems on differentiable and complex manifolds, Ann. of Math. (2) 60(1954) 213–236. MathSciNetzbMATHGoogle Scholar
  38. [Hirz56]
    Hirzebruch F., Neue topologische Methoden in der algebraischen Geometrie, Ergebnisse 9. Springer, Berlin, 1956, 165 S. zbMATHGoogle Scholar
  39. [Hirz66a]
    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. zbMATHGoogle Scholar
  40. [Hus66]
    Husemoller D., Fibre Bundles, McGraw-Hill New York, 1966; Springer-Verlag, GTM 20, 1975 and 1994. zbMATHGoogle Scholar
  41. [JW55]
    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. MathSciNetzbMATHGoogle Scholar
  42. [JoW08]
    Joachim M. and Wraith D., Exotic spheres and curvature, Bull. Am. Math. Soc., 45(2008), No. 4, 595–616. MathSciNetzbMATHGoogle Scholar
  43. [John12]
    Johnson N., Visualizing seven-manifolds, an animation presented at the second Abel conference: A mathematical celebration of John Milnor, January 2012, see
  44. [Kahn72]
    Kahn P.J., A note on topological Pontrjagin classes and the Hirzebruch index formula, Ill. J. Math. 16(1972) 243–256. MathSciNetzbMATHGoogle Scholar
  45. [Kerv60]
    Kervaire M.A., A manifold which does not admit any differentiable structure, Comment. Math. Helv. 34(1960) 257–270. MathSciNetzbMATHGoogle Scholar
  46. [KM60]
    (=[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. [KM63]
    (=[M63a]) Kervaire M.A., and Milnor J.W., Groups of homotopy spheres I, Ann. of Math. (2) 77(1963) 504–537. MathSciNetzbMATHGoogle Scholar
  48. [KirS71]
    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. [KirS77]
    Kirby R.C. and Siebenmann L.C., Foundational Essays on Topological Manifolds, Smoothing and Triangulations, Annals of Mathematics Study 88(1977). zbMATHGoogle Scholar
  50. [Kn25]
    Kneser H., Die Topologie der Mannigfaltigkeiten, Jahresber. Dtsch. Math.-Ver. 34(1925) 1–14. zbMATHGoogle Scholar
  51. [Lang62]
    Lang S., Introduction to Differentiable Manifolds, Interscience, John Wiley Sons, New York, 1962. zbMATHGoogle Scholar
  52. [Lev65]
    Levine J., A classification of differentiable knots, Ann. of Math. (2) 82(1965) 15–50. MathSciNetzbMATHGoogle Scholar
  53. [LorR11]
    Lorenz F. and Roquette P., Cahit arf and his invariant. Preprint, June 17, 2011. Current URL is
  54. [MadM79]
    Madsen I. and Milgram R.J., The Classifying Spaces for Surgery and Cobordism of Manifolds, Annals of Mathematics Study 92(1979), xii+279 pages. zbMATHGoogle Scholar
  55. [Mar77]
    Marin A., La transversalité topologique, Ann. Math. (2) 106(1977) 269–293. MathSciNetzbMATHGoogle Scholar
  56. [Maz59]
    Mazur B., On embeddings of spheres, Bull. Amer. Math. Soc. 65(1959) 59–65. MathSciNetzbMATHGoogle Scholar
  57. [Maz61]
    Mazur B., Stable equivalence of differentiable manifolds, Bull. Am. Math. Soc. 67(1961) 377–384. MathSciNetzbMATHGoogle Scholar
  58. [Miller10]
    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. [Milnor]
    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:
  60. [M56d]
    Milnor J.W., On manifolds homeomorphic to the 7-sphere, Ann. of Math. (2) 64(1956) 399–405. MathSciNetzbMATHGoogle Scholar
  61. [M56e]
    Milnor J.W., On the relationship between differentiable manifolds and combinatorial manifolds. First published in [M07a, pp. 19–28]. Google Scholar
  62. [M58b]
    Milnor J.W., The Steenrod algebra and its dual, Ann. of Math. (2) 67(1958) 150–171, see also [M09a, pp. 61–82]. MathSciNetzbMATHGoogle Scholar
  63. [M59b]
    Milnor J.W., Differentiable structures on spheres, Am. J. Math., 81(1959) 962–972. MathSciNetzbMATHGoogle Scholar
  64. [M59c]
    Milnor J.W., Sommes de variétes différentiables et structures différentiables des sphères. Bull. Soc. Math. Fr., 87(1959) 439–444. zbMATHGoogle Scholar
  65. [M59d]
    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. [M61a]
    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. [M61b]
    Milnor J.W., Two complexes which are homeomorphic but combinatorially distinct. Ann. of Math. (2) 74(1961) 575–590. MathSciNetzbMATHGoogle Scholar
  68. [M61f]
    Milnor J.W., Microbundles and differentiable structures, polycopied at Princeton U., 1961; first published as pages 173–190 in [M09a]. Google Scholar
  69. [M63c]
    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. [M63d]
    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). zbMATHGoogle Scholar
  71. [M64b]
    Milnor J.W., Microbundles I, Topology 3 (suppl. 1), 53–80. Google Scholar
  72. [M64f]
    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. zbMATHGoogle Scholar
  73. [M65c]
    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. [M65d]
    Milnor J.W., Lectures on the h-Cobordism Theorem. Notes by L. Siebenmann and J. Sondow. Princeton Univ. Press, Princeton, 1965. (Translated into Russian). zbMATHGoogle Scholar
  75. [M65e]
    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. [M00b]
    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. [M03a]
    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). zbMATHGoogle Scholar
  78. [M06a]
    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. [M07a]
    Milnor J.W., Collected Papers of John Milnor III, Differential Topology, Amer. Math. Soc., Providence, 2007. zbMATHGoogle Scholar
  80. [M09a]
    Milnor J.W., Collected Papers of John Milnor IV, Homotopy, Homology and Manifolds, Edited by J. McCleary, Amer. Math. Soc., Providence, 2009. zbMATHGoogle Scholar
  81. [M09c]
    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. [M11a]
    Milnor J.W., Differential topology forty-six years later, Notices Am. Math. Soc., 58(2011) 804–809. MathSciNetzbMATHGoogle Scholar
  83. [MSp60]
    (=[M60a]) Milnor J.W. and Spanier E., Two remarks on fiber homotopy type, Pac. J. Math. 10(1960) 585–590. MathSciNetzbMATHGoogle Scholar
  84. [MSt74]
    (=[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. [Moi52]
    Moise E.E., Affine structures in 3-manifolds. V., The triangulation theorem and hauptvermutung, Ann. of Math. (2) 56(1952) 96–114. MathSciNetzbMATHGoogle Scholar
  86. [Mors25]
    Morse M., Relation between the critical points of a real function of n independent variables, Trans. Am. Math. Soc. 27(1925) 345–396. MathSciNetzbMATHGoogle Scholar
  87. [Mu63]
    Munkres J.R., Elementary differential topology, Annals of Math Studies 54, Princeton U. Press, Princeton, 1963. zbMATHGoogle Scholar
  88. [Mu66]
    Munkres J.R., Concordance is equivalent to smoothability, Topology 5(1966) 371–389. MathSciNetzbMATHGoogle Scholar
  89. [Mu67]
    Munkres J.R., Concordance of differentiable structures—two approaches. Michigan Math. J. 14(1967) 183–191. MathSciNetzbMATHGoogle Scholar
  90. [Mu68]
    Munkres J.R., Compatibility of imposed differentiable structures. Illinois J. Math. 12(1968) 610–615. MathSciNetzbMATHGoogle Scholar
  91. [Nov62]
    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. zbMATHGoogle Scholar
  92. [Nov65]
    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). zbMATHGoogle Scholar
  93. [Nov00]
    Novikov S.P., Surgery in the 1960’s, pages 31–39 in [Cap00]. Google Scholar
  94. [Pal09]
    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. [PW27]
    Peter F., and Weyl H., Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe, Math. Ann. 97(1927) 737–755. MathSciNetzbMATHGoogle Scholar
  96. [Ran92]
    Ranicki A., Algebraic L-Theory and Topological Manifolds, Cambridge Tracts in Mathematics, 102(1992), Cambridge University Press, Cambridge, viii+358 pages. zbMATHGoogle Scholar
  97. [Ran01]
    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. zbMATHGoogle Scholar
  98. [Ran02]
    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 Google Scholar
  99. [RS11]
    Raussen M. and Skau C., Interview with John Milnor, Newsl. - Eur. Math. Soc., September 2011, (81), pages 31–40. MathSciNetzbMATHGoogle Scholar
  100. [Ravenel]
  101. [Rav12]
    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 and Google Scholar
  102. [Reeb49]
    Reeb G., Stabilité des feuilles compactes à groupe de Poincaré fini, C.R. Acad. Sci., Paris 228(1949) 47–48. MathSciNetzbMATHGoogle Scholar
  103. [Reid38]
    Reidemeister K., Topologie der Polyeder und kombinatorische Topologie der Komplexe, Akademische Verlagsgesellschaft Geest und Portig, Leipzig, 1938; zweite (unveränderte) Auflage 1953. zbMATHGoogle Scholar
  104. [RoS72]
    Rourke C.P. and Sanderson B.J., Introduction to piecewise-linear topology, Ergebnisse, Bd. 69. Springer, Berlin, 1972. zbMATHGoogle Scholar
  105. [Ser51]
    Serre J.-P., Homologie singulière des espaces fibrés, Applications, Ann. of Math. (2) 54(1951) 425–505. MathSciNetzbMATHGoogle Scholar
  106. [Seif36]
    Seifert H., La théorie des noeuds, Enseign. Math. 35(1936) 201–212. zbMATHGoogle Scholar
  107. [Sieb71]
    Siebenmann L.C., Topological manifolds, Actes Congr. Internat. Math. 1970, 2, 133–163 (1971). MathSciNetzbMATHGoogle Scholar
  108. [Sieb80]
    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
  109. [Sm60]
    Smale S., The generalized Poincaré conjecture in higher dimensions, Bull. Am. Math. Soc. 66(1960) 373–375. zbMATHGoogle Scholar
  110. [Sm61]
    Smale S., Generalized Poincaré’s conjecture in dimensions greater than four, Ann. of Math. (2) 74(1961) 391–406. MathSciNetzbMATHGoogle Scholar
  111. [Sm62]
    Smale S., On the structure of manifolds, Am. J. Math. 84(1962) 387–399. zbMATHGoogle Scholar
  112. [Spi67]
    Spivak M., Spaces satisfying Poincaré duality, Topology 6(1967) 77–101. MathSciNetzbMATHGoogle Scholar
  113. [Strd51]
    Steenrod N., The Topology of Fibre Bundles, Princeton Univ. Press, Princeton 1951. zbMATHGoogle Scholar
  114. [Stz08]
    Steinitz E., Beiträge zur Analysis Situs, Sitzungsber. Berl. Math. Ges. 7(1908) 29–49. zbMATHGoogle Scholar
  115. [Sull66]
    Sullivan D.P., Triangulating Homotopy Equivalences, Princeton Univ. Press, Princeton, 1966. Google Scholar
  116. [Sull67]
    Sullivan D.P., Triangulating and Smoothing Homotopy Equivalences, Geometric Topology Seminar Notes, Princeton Univ. Press, Princeton, 1967. Google Scholar
  117. [Thom54]
    Thom R., Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28(1954) 17–86. MathSciNetzbMATHGoogle Scholar
  118. [Thom58]
    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. [Thom60]
    Thom R., Des variétés triangulées aux variétés différentiables, Proc. Int. Congr. Math. 1958, 248–255 (1960). MathSciNetzbMATHGoogle Scholar
  120. [Ttz08]
    Tietze H., Über die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten, Monatshefte für Math. u. Physica 19(1908) 1–118. zbMATHGoogle Scholar
  121. [vonN27]
    von Neumann J., Zur Theorie der Darstellungen kontinuierlicher Gruppen, Sitzungsberichte Akad. Berlin 1927, 76–90. Google Scholar
  122. [Wall66]
    Wall C.T.C., An extension of results of Novikov and Browder, Am. J. Math. 88(1966) 20–32. MathSciNetzbMATHGoogle Scholar
  123. [Wall70]
    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). zbMATHGoogle Scholar
  124. [Waa60]
    Wallace A.H., Modifications and cobounding manifolds, Can. J. Math. 12(1960) 503–528. MathSciNetzbMATHGoogle Scholar
  125. [Waa61]
    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. MathSciNetzbMATHGoogle Scholar
  126. [Weier85]
    Weierstrass K., Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen einer reellen Veränderlichen, Berliner Berichte 1885, 633–640, 789–806. Google Scholar
  127. [WhdG42]
    Whitehead G.W., On the homotopy groups of spheres and rotation groups, Ann. of Math. (2) 43(1942) 634–640. MathSciNetzbMATHGoogle Scholar
  128. [WhdJ40]
    Whitehead J.H.C., On C 1-complexes, Ann. of Math. (2) 41(1940) 809–824. MathSciNetzbMATHGoogle Scholar
  129. [WhdJ61]
    Whitehead J.H.C., Manifolds with transverse fields in Euclidean space, Ann. of Math. (2) 73(1961) 154–212. MathSciNetzbMATHGoogle Scholar
  130. [Why36]
    Whitney, H., Differentiable manifolds, Ann. of Math. (2) 37(1936) 645–680. MathSciNetzbMATHGoogle Scholar
  131. [Why44]
    Whitney H., The singularities of a smooth n-manifold in (2n−1)-space, Ann. of Math. (2) 45(1944) 247–293. MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Laboratoire de MathématiqueUniversité de Paris-SudOrsayFrance

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