Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

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

 
 

 

The triangulation problem and its role in analysis


Author: Stewart S. Cairns
Journal: Bull. Amer. Math. Soc. 52 (1946), 545-571
DOI: https://doi.org/10.1090/S0002-9904-1946-08610-3
MathSciNet review: 0017531
Full-text PDF

References | Additional Information

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

  • 1. B. L. van der Waerden, Topologische Begründung des Kalküls der abzählenden Geometrie, Anhang I, Math. Ann. vol. 102 (1929) pp. 360, 361.
  • 2. S. S. Cairns, The cellular subdivision and approximation of regular spreads, Proc. Nat. Acad. Sci. U.S.A. vol. 16 (1930) pp. 488-491.
  • 3. S. Lefschetz, Topology, Amer. Math. Soc. Colloquium Publications, vol. 12, 1930.
  • 4. B. O. Koopman and A. B. Brown, On the covering of analytic loci by complexes, Trans. Amer. Math. Soc. 34 (1932), no. 2, 231–251. MR 1501636, https://doi.org/10.1090/S0002-9947-1932-1501636-9
  • 5. Stewart S. Cairns, On the cellular subdivision of 𝑛-dimensional regions, Ann. of Math. (2) 33 (1932), no. 4, 671–680. MR 1503083, https://doi.org/10.2307/1968212
  • 6. S. Lefschetz and J. H. C. Whitehead, On analytical complexes, Trans. Amer. Math. Soc. 35 (1933), no. 2, 510–517. MR 1501698, https://doi.org/10.1090/S0002-9947-1933-1501698-X
  • 7. Stewart S. Cairns, On the triangulation of regular loci, Ann. of Math. (2) 35 (1934), no. 3, 579–587. MR 1503181, https://doi.org/10.2307/1968752
  • 8. Georg Nöbeling, Zur Topologie der Mannigfaltigkeiten, Monatsh. Math. Phys. 42 (1935), no. 1, 117–152 (German). MR 1550421, https://doi.org/10.1007/BF01733286
  • 9. H. Seifert, Review of Nöbeling's paper (preceding item), Zentralblatt fũr Mathematik und ihre Grenzgebiete vol. 11 (1935) p. 36.
  • 10. S. S. Cairns, Triangulation of the manifold of class one, Bull. Amer. Math. Soc. vol. 41 (1935) pp. 549-552.
  • 11. L. E. J. Brouwer, Zum Triangulationsproblem, Neder. Akad. Wetensch. vol. 42 (1939) pp. 701-706. MR 273
  • 12. H. Freudenthal, Die Triangulation der differenzierbaren Mannigfaltigkeiten, Neder. Akad. Wetensch. vol. 42 (1939) pp. 880-901. See also Die Triangulation der differenzierbaren Mannigfaltigkeiten. Nachtrag., ibid. vol. 43 (1940) p. 619. MR 639
  • 13. J. H. C. Whitehead, On C1-complexes, Ann. of Math. vol. 41 (1940) pp. 809-824. MR 2545
  • 14. S. S. Cairns, Triangulated manifolds and differentiable manifolds, Lectures in topology, The University of Michigan Press, 1941, pp. 143-157. MR 5301
  • 15. J. W. Alexander, Some problems in topology, Verhandlungen des Internationalen Mathematiker Kongresses, Zurich, 1932, vol. 1, pp. 249-257.
  • 16. P. Alexandroff and H. Hopf, Topologie, I, Springer, 1935, chap. 3. MR 345087
  • 17. C. B. Allendoerfer, and A. Weil, The Gauss-Bonnet theorem for Riemannian polyhedra, Trans. Amer. Math. Soc. vol. 53 (1943) pp. 101-129. MR 7627
  • 18. Stewart S. Cairns, The generalized theorem of Stokes, Trans. Amer. Math. Soc. 40 (1936), no. 1, 167–174. MR 1501869, https://doi.org/10.1090/S0002-9947-1936-1501869-5
  • 19. Stewart S. Cairns, Polyhedral approximations to regular loci, Ann. of Math. (2) 37 (1936), no. 2, 409–415. MR 1503287, https://doi.org/10.2307/1968452
  • 20. S. S. Cairns, Homeomorphisms between topological manifolds and analytic manifolds, Ann. of Math. vol. 41 (1940) pp. 796-808. MR 2538
  • 21. S. S. Cairns, Isotopic deformations of geodesic complexes on the 2-sphere and on the plane, Ann. of Math. vol. 45 (1944) pp. 207-217. MR 10271
  • 22. S. S. Cairns, Introduction of a Riemannian geometry on a triangulable 4-manifold, Ann. of Math. vol. 45 (1944) pp. 218-219. MR 10272
  • 23. E. Cartan, La théorie des groupes finis et continus et la géométrie différentielle traité par la méthode du repère mobile, Gauthier-Villars, 1937.
  • 24. S. S. Chern, Integral formulas for the characteristic classes of sphere bundles, Proc. Nat. Acad. Sci. U.S.A. vol. 30 (1944) pp. 269-273. MR 11028
  • 25. S. S. Chern, A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Ann. of Math. vol. 45 (1944) pp. 747-752. MR 11027
  • 26. S. S. Chern, Some new viewpoints in differentiable geometry in the large, Bull. Amer. Math. Soc. vol. 52 (1946) pp. 1-30. MR 21706
  • 27. W. V. D. Hodge, The theory and applications of harmonic integrals, Cambridge University Press, 1941, chap. 1. MR 3947
  • 28. O. D. Kellogg, Foundations of potential theory, Springer, 1929, chap. 4. MR 222317
  • 29. S. Lefschetz, Algebraic topology, Amer. Math. Soc. Colloquium Publications, vol. 27, 1942, especially chap. 8.
  • 30. Marston Morse, The calculus of variations in the large, American Mathematical Society Colloquium Publications, vol. 18, American Mathematical Society, Providence, RI, 1996. Reprint of the 1932 original. MR 1451874
  • 31. W. F. Osgood, Lehrbuch der Funktionentheorie I, Teubner, 1928, especially chap. 5.
  • 32. G. de Rham, Sur l'analysis situs des variétés à n dimensions, Journal de Mathématique (9) vol. 10 (1931) pp. 115-200.
  • 33. H. Seifert and W. Threlfall, Variationsrechnung im Grossen (Theorie von Marston Morse), Teubner, 1938.
  • 34. H. A. Schwarz, Gesammelte mathematische Abhandlungen, Springer, 1890, vol. 2, pp. 309-311.
  • 35. E. Stiefel, Richtungsfelder und Fernparallelismus in n-dimensionalen Mannigfaltigkeiten, Comment. Math. Helv. vol. 8 (1936) pp. 305-343.
  • 36. W. Threlfall. See H. Seifert.
  • 37. O. Veblen and J. H. C. Whitehead, The foundations of differential geometry, Cambridge Tracts, No. 29, Cambridge University Press, 1932.
  • 38. A. Weil. See C. B. Allendoerfer.
  • 39. J. H. C. Whitehead. See O. Veblen.
  • 40. Hassler Whitney, Differentiable manifolds, Ann. of Math. (2) 37 (1936), no. 3, 645–680. MR 1503303, https://doi.org/10.2307/1968482
  • 41. Hassler Whitney, The imbedding of manifolds in families of analytic manifolds, Ann. of Math. (2) 37 (1936), no. 4, 865–878. MR 1503315, https://doi.org/10.2307/1968624
  • 42. H. Whitney, Topological properties of differentiable manifolds, Bull. Amer. Math. Soc. vol. 43 (1937) pp. 785-805.
  • 43. H. Whitney, On the topology of differentiable manifolds, Lectures in topology, Michigan, 1941, pp. 101-141. MR 5300


Additional Information

DOI: https://doi.org/10.1090/S0002-9904-1946-08610-3

American Mathematical Society