Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 
 

 

The golden age of immersion theory in topology: 1959-1973. A mathematical survey from a historical perspective


Author: David Spring
Journal: Bull. Amer. Math. Soc. 42 (2005), 163-180
MSC (2000): Primary 57R42, 01A60
DOI: https://doi.org/10.1090/S0273-0979-05-01048-7
Published electronically: January 25, 2005
MathSciNet review: 2133309
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We review the history of modern immersion-theoretic topology during the period 1959-1973, beginning with the work of S. Smale followed by the important contributions from the Leningrad school of topology, including the work of M. Gromov. We discuss the development of the major geometrical ideas in immersion-theoretic topology during this period. Historical remarks are included and technical concepts are introduced informally.


References [Enhancements On Off] (What's this?)

  • 1. Adachi, M., 1993. Embeddings and immersions. Transl. Math. Monographs, Vol. 124, Amer. Math. Soc., Providence, RI. MR 1225100 (95a:57039)
  • 2. Bierstone, E., 1974. Equivariant Gromov Theory. Topology 13, 327-345. MR 0353334 (50:5818)
  • 3. Cohen, R.L., 1985. The immersion conjecture for differentiable manifolds. Ann. of Math. 122, 237-328. MR 0808220 (86m:57030)
  • 4. Datta, M., Mukherjee, A., 1998. Parametric homotopy principle of some partial differential relations. Math. Slovaca 48, 411-421. MR 1693501 (2000f:58021)
  • 5. Ehresmann, C., 1951/1984. Les prolongements d'une variété différentiable, I, II, III. C. R. Acad. Sci. Paris 233, 598-600; 777-779; 1081-1083. Reprinted in: Charles Ehresmann: \oeuvres complètes et commentées, I-1, I-2. Topologie algébrique et géométrie différentielle. Cahiers Topologie et Géom.Différentielle 24 (1983), suppl.1 (1984), 343-345; 346-348; 349-351. MR 13:386a, MR 13:584b, MR 13:584c; MR 0794193 (86i:01059)
  • 6. Eliashberg, Ja.M., 1970. Singularities of folding type. Izv. Akad. Nauk SSSR Ser. Mat. 34, 1110-1126 (Russian). English translation: Math. USSR Izvestia 4 (1970), 1119-1134. MR 0278321 (43:4051)
  • 7. Eliashberg, Ja.M., 1972. Surgery of singularities of smooth mappings. Izv. Akad. Nauk SSSR Ser. Mat. 36, 1321-1347 (Russian). English translation: Math. USSR Izvestia 6 (1972), 1302-1326. MR 0339261 (49:4021).
  • 8. Eliashberg, Y., Mishachev, N., 1997. Wrinkling of smooth mappings and its applications, I. Invent. Math. 130, 345-369. MR 1474161 (99d:57021)
  • 9. Eliashberg, Y., Mishachev, N., 2000. Wrinkling of smooth mappings.II. Wrinkling of embeddings and K. Igusa's theorem, Topology 39, 711-732. MR 1760426 (2001g:57058)
  • 10. Eliashberg, Y., Mishachev, N., 2001. Holonomic approximation and Gromov's $h$-principle. Essays on geometry and related topics. Monogr. Enseign. Math. 38, Vols. 1, 2, 271-285. MR 1929330 (2003k:58006)
  • 11. Eliashberg, Y., Mishachev, N., 2002. Introduction to the $h$-principle. Grad. Stud. in Math., Vol. 48, Amer. Math. Soc., Providence, RI. MR 1909245 (2003g:53164)
  • 12. Feldman, E.A., 1965. The geometry of immersions, I. Trans. Amer. Math. Soc. 120, 185-224. MR 0185602 (32:3065)
  • 13. Feldman, E.A. 1968. Deformations of closed space curves. J. Diff. Geom. 2, 67-75. MR 0232400 (38:725)
  • 14. Feit, S.D., 1969. $k$-mersions of manifolds. Acta Math. 122, 173-195. MR 0243541 (39:4862)
  • 15. Gromov, M.L., 1968. Transversal mappings of foliations. Dokl. Akad. Nauk SSSR. 182, 255-258 (Russian). English translation: Soviet Math. Dokl. 9 (1968), 1126-1129. MR 0238352 (38:6628)
  • 16. Gromov, M.L., 1969. Stable maps of foliations into manifolds. Izv. Akad. Nauk SSSR Ser. Mat. 33, 707-734 (Russian). English translation: Trans. Math. USSR Izvestia 3 (1969), 671-694. MR 0263103 (41:7708)
  • 17. Gromov, M.L., 1972. Smoothing and inversion of differential operators. Math. Sbornik 88 (130), 382-441 (Russian). English translation: Math. USSR Sbornik 17 (1972), 381-435. MR 0310924 (46:10022)
  • 18. Gromov, M.L., 1973a. Degenerate smooth mappings. Mat. Zametki 14, 509-516 (Russian). English translation: Math. Notes 14 (1973), 849-853. MR 0339209 (49:3972)
  • 19. Gromov, M.L., 1973b. Convex integration of differential relations.I. Izv. Akad. Nauk SSSR Ser. Mat. 37, 329-343 (Russian). English translation: Math. USSR Izvestia 7 (1973), 329-343. MR 0413206 (54:1323)
  • 20. Gromov, M.L., 1986. Partial differential relations. Ergebnisse der Mathematik und ihrer Grenzgebiete 3, Folge $\cdot$ Band 9, Springer-Verlag, Berlin. MR 0864505 (90a:58201)
  • 21. Gromov, M.L., Eliashberg, Ja.M., 1971a. Removal of singularities of smooth mappings. Izv. Akad. Nauk SSSR Ser. Mat. 35, 600-626 (Russian). English translation: Math. USSR Izvestia 5 (1971), 615-639. MR 0301748 (46:903)
  • 22. Gromov, M.L., Eliashberg, Ja.M., 1971b. Nonsingular maps of Stein manifolds. Funkcional. Anal. i Prilozhen 2, 82-83 (Russian). English translation: Functional Anal. Appl. 5 (1971), 156-157. MR 0301236 (46:394)
  • 23. Gromov, M., Eliashberg, Y., 1992. Embeddings of Stein manifolds of dimension $n$ into the affine space of dimension $(3n/2)+1$. Ann. of Math. (2) 136, 123-135. MR 1173927 (93g:32037)
  • 24. Haefliger, A., Poénaru, V., 1964. La classification des immersions combinatoires. Inst. Hautes Études Sci. Publ. Math. 23, 75-91. MR 0172296 (30:2515)
  • 25. Haefliger, A., 1970. Feuilletages sur les variétés ouvertes. Topology 9, 183-194. MR 0263104 (41:7709)
  • 26. Haefliger, A., 1971. Lectures on the theorem of Gromov. Proceedings of Liverpool Singularities Symposium, II (1969/1970), pp. 128-141. Lecture Notes in Math., Vol. 209, Springer. MR 0334241 (48:12560)
  • 27. Hirsch, M.W., 1959. Immersions of manifolds. Trans. Amer. Math. Soc. 93, 242-276. MR 0119214 (22:9980)
  • 28. Hirsch, M.W., 1961. On imbedding differentiable manifolds in euclidean space. Ann. of Math. 73, 566-571. MR 0124915 (23:A2223)
  • 29. Hirsch, M.W., 1993. The work of Stephen Smale in differential topology. From Topology to Computation: Proceedings of the Smalefest (M.W. Hirsch, J.E. Marsden, M. Shub, eds.), Springer-Verlag, New York, 83-106. MR 1246112 (97b:57001)
  • 30. Kirby, R.C., 1969. Stable homeomorphisms and the annulus conjecture. Ann. of Math. 89, 575-582. MR 0242165 (39:3499)
  • 31. Kirby, R.C., Siebenmann, L.C., 1969. On the triangulation of manifolds and the Hauptvermutung. Bull. Amer. Math. Soc. 75, 742-749. MR 0242166 (39:3500)
  • 32. Kuiper, N.H., 1955. On $C^1$-isometric imbeddings, I, II. Indag. Math. 17, 545-556, 683-689. MR 0075640 (17:782c)
  • 33. Lashof, R., Rothenberg, M., 1969. Triangulation of manifolds, I, II. Bull. Amer. Math. Soc. 75, 750-754, 755-757. MR 0247631 (40:895)
  • 34. Lawson, H.B., 1974. Foliations. Bull. Amer. Math. Soc. 80, 369-418. MR 0343289 (49:8031)
  • 35. Lees, J.A., 1969. Immersions and surgeries of topological manifolds. Bull. Amer. Math. Soc. 75, 529-534. MR 0239602 (39:959)
  • 36. Levy, S., Maxwell, D., Munzner, T., 1995. Outside In (Video). Accompanying text, Making waves, University of Minnesota Geometry Center, Minneapolis, MN. Distributed by A K Peters Ltd., Wellesley, MA. MR 1357901 (96m:00012)
  • 37. Levine, H.I., 1965. Elimination of cusps. Topology 3, suppl. 2, 263-296. MR 0176484 (31:756)
  • 38. Lohkamp, J., 1994a. Negative bending of open manifolds. J. Differential Geom. 40, 461-474. MR 1305978 (96k:53061)
  • 39. Lohkamp, J., 1994b. Metrics of negative Ricci curvature. Ann. of Math (2) 140, 655-683. MR 1307899 (95i:53042)
  • 40. Nash, J., 1954. $C^1$-isometric imbeddings. Ann. of Math. 63, 383-396. MR 0065993 (16:515e)
  • 41. Palais, R.S., 1965 Seminar on the Atiyah-Singer index theorem. Annals of Math. Studies, Vol. 57, Princeton University Press. MR 0198494 (33:6649)
  • 42. Phillips, A., 1967. Submersions of open manifolds. Topology 6, 171-206. MR 0208611 (34:8420)
  • 43. Phillips, A., 1969. Foliations on open manifolds, II. Comment. Math. Helv. 44, 367-370. MR 0253364 (40:6579)
  • 44. Phillips, A., 1970. Smooth maps transverse to a foliation. Bull. Amer. Math. Soc. 76, 792-797. MR 0263106 (41:7711)
  • 45. Phillips, A., 1974. Smooth maps of constant rank. Bull. Amer. Math. Soc. 80, 513-517. MR 0334238 (48:12557)
  • 46. duPlessis, A., 1975. Maps without certain singularities. Comment. Math. Helv. 50, 363-382. MR 0397779 (53:1637)
  • 47. Poénaru, V., 1962. Sur la théorie des immersions. Topology 1, 81-100. MR 0153023 (27:2992)
  • 48. Poénaru, V., 1966. On regular homotopy in codimension 1. Ann. of Math. 83, 257-265. MR 0192507 (33:732)
  • 49. Poénaru, V., 1968 Extensions des immersions en codimension 1 (d'après S. Blank). Sém. Bourbaki, Exposé 342, 1-33. Reprinted in: Séminaire Bourbaki, Vol. 10, Années 1966/67-1967/68, Société Math. de France, Paris (1995), 473-505. MR 1610469, MR 1610884 (99f:00042)
  • 50. Poénaru, V., 1971. Homotopy theory and differentiable singularities. Manifolds-Amsterdam 1970 (Proc.Nuffic Summer School), pp. 106-132, Lecture Notes in Math., Vol. 197, Springer. MR 0285026 (44:2250)
  • 51. Smale, S., 1958. A classification of immersions of the two-sphere. Trans. Amer. Math. Soc. 90, 281-290. MR 0104227 (21:2984)
  • 52. Smale, S., 1959. The classification of immersions of spheres in Euclidean spaces. Ann. Math. 69, 27-344. MR 0105117 (21:3862)
  • 53. Spring, D., 1993. Note on the history of immersion theory. From Topology to Computation: Proceedings of the Smalefest (M.W. Hirsch, J.E. Marsden, M. Shub, eds.), Springer-Verlag, New York, 114-116. MR 1246113 (97b:57002)
  • 54. Spring, D., 1998. Convex integration theory: solutions to the $h$-principle in geometry and topology. Monographs in Mathematics, Vol. 92, Birkhäuser-Verlag, Basel. MR 1488424 (99e:58024)
  • 55. Spring, D., 2002. Directed embeddings and the simplification of singularities. Commun. Contemp. Math. 4, 107-144. MR 1890079 (2003h:58055)
  • 56. Thom, R., 1957. La classification des immersions (d'après Smale). Séminaire Bourbaki, Exposé 157, 11 pp. MR 0105693 (21:4429)
  • 57. Thom, R., 1959. Remarques sur les problèmes comportant des inéquations différentielles globales. Bull. Soc. Math. France 87, 455-461. MR 0121807 (22:12537)
  • 58. Thurston, W.P., 1974. The theory of foliations of codimension greater than one. Comment. Math. Helv. 49, 214-231. MR 0370619 (51:6846)
  • 59. Thurston, W.P., 1976. Existence of codimension-one foliations. Ann. of Math. 104, 249-268. MR 0425985 (54:13934)
  • 60. Vassiliev, V.A., 1992. Complements of discriminants of smooth maps: topology and applications. Translations of Mathematical Monographs, 98, Amer. Math. Soc., Providence, RI. MR 1168473 (94i:57020)
  • 61. Whitney, H., 1937. On regular closed curves in the plane. Compositio Math. 4, 276-284.
  • 62. Whitney, H., 1944. The singularities of a smooth $n$-manifold in $(2n-1)$-space. Ann. of Math. 45, 247-293. MR 0010275 (5:274a)
  • 63. Whitney, H., 1955. On singularities of mappings of euclidean spaces.I. Mappings of the plane into the plane. Ann. of Math. 62, 374-410. MR 0073980 (17:518d)

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 57R42, 01A60

Retrieve articles in all journals with MSC (2000): 57R42, 01A60


Additional Information

David Spring
Affiliation: Department of Mathematics, Glendon College, York University, 2275 Bayview Avenue, Toronto, Ontario, Canada M4N 3M6
Email: dspring@glendon.yorku.ca

DOI: https://doi.org/10.1090/S0273-0979-05-01048-7
Keywords: Topology, immersion theory, $h$-principle, Smale's theory, Leningrad school
Received by editor(s): June 16, 2004
Published electronically: January 25, 2005
Article copyright: © Copyright 2005 American Mathematical Society

American Mathematical Society