The triangulation problem and its role in analysis
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- by Stewart S. Cairns PDF
- Bull. Amer. Math. Soc. 52 (1946), 545-571
References
-
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.
- 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, DOI 10.1090/S0002-9947-1932-1501636-9
- Stewart S. Cairns, On the cellular subdivision of $n$-dimensional regions, Ann. of Math. (2) 33 (1932), no. 4, 671–680. MR 1503083, DOI 10.2307/1968212
- S. Lefschetz and J. H. C. Whitehead, On analytical complexes, Trans. Amer. Math. Soc. 35 (1933), no. 2, 510–517. MR 1501698, DOI 10.1090/S0002-9947-1933-1501698-X
- Stewart S. Cairns, On the triangulation of regular loci, Ann. of Math. (2) 35 (1934), no. 3, 579–587. MR 1503181, DOI 10.2307/1968752
- Georg Nöbeling, Zur Topologie der Mannigfaltigkeiten, Monatsh. Math. Phys. 42 (1935), no. 1, 117–152 (German). MR 1550421, DOI 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.
- L. E. J. Brouwer, Zum Triangulationsproblem, Nederl. Akad. Wetensch., Proc. 42 (1939), 701–706 (German). MR 273
- Hans Freudenthal, Die Triangulation der differenzierbaren Mannigfaltigkeiten, Nederl. Akad. Wetensch., Proc. 42 (1939), 880–901 (German). MR 639
- J. H. C. Whitehead, On $C^1$-complexes, Ann. of Math. (2) 41 (1940), 809–824. MR 2545, DOI 10.2307/1968861
- Stewart S. Cairns, Triangulated manifolds and differentiable manifolds, Lectures in Topology, University of Michigan Press, Ann Arbor, Mich., 1941, pp. 143–157. MR 0005301 15. J. W. Alexander, Some problems in topology, Verhandlungen des Internationalen Mathematiker Kongresses, Zurich, 1932, vol. 1, pp. 249-257.
- P. Alexandroff and H. Hopf, Topologie. I, Die Grundlehren der mathematischen Wissenschaften, Band 45, Springer-Verlag, Berlin-New York, 1974. Berichtigter Reprint. MR 0345087, DOI 10.1007/978-3-642-65614-9
- Carl B. Allendoerfer and André Weil, The Gauss-Bonnet theorem for Riemannian polyhedra, Trans. Amer. Math. Soc. 53 (1943), 101–129. MR 7627, DOI 10.1090/S0002-9947-1943-0007627-9
- Stewart S. Cairns, The generalized theorem of Stokes, Trans. Amer. Math. Soc. 40 (1936), no. 1, 167–174. MR 1501869, DOI 10.1090/S0002-9947-1936-1501869-5
- Stewart S. Cairns, Polyhedral approximations to regular loci, Ann. of Math. (2) 37 (1936), no. 2, 409–415. MR 1503287, DOI 10.2307/1968452
- Stewart S. Cairns, Homeomorphisms between topological manifolds and analytic manifolds, Ann. of Math. (2) 41 (1940), 796–808. MR 2538, DOI 10.2307/1968860
- Stewart S. Cairns, Isotopic deformations of geodesic complexes on the 2-sphere and on the plane, Ann. of Math. (2) 45 (1944), 207–217. MR 10271, DOI 10.2307/1969263
- Stewart S. Cairns, Introduction of a Riemannian geometry on a triangulable 4-manifold, Ann. of Math. (2) 45 (1944), 218–219. MR 10272, DOI 10.2307/1969264 23. E. Cartan, La théorie des groupes finis et continus et la géométrie différentielle traité par la méthode du repère mobile, Gauthier-Villars, 1937.
- Shiing-shen Chern, Integral formulas for the characteristic classes of spere bundles, Proc. Nat. Acad. Sci. U.S.A. 30 (1944), 269–273. MR 11028, DOI 10.1073/pnas.30.9.269
- Shiing-shen Chern, A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Ann. of Math. (2) 45 (1944), 747–752. MR 11027, DOI 10.2307/1969302
- Shiing-Shen Chern, Some new viewpoints in differential geometry in the large, Bull. Amer. Math. Soc. 52 (1946), 1–30. MR 21706, DOI 10.1090/S0002-9904-1946-08487-6
- W. V. D. Hodge, The Theory and Applications of Harmonic Integrals, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1941. MR 0003947
- Oliver Dimon Kellogg, Foundations of potential theory, Die Grundlehren der mathematischen Wissenschaften, Band 31, Springer-Verlag, Berlin-New York, 1967. Reprint from the first edition of 1929. MR 0222317, DOI 10.1007/978-3-642-86748-4 29. S. Lefschetz, Algebraic topology, Amer. Math. Soc. Colloquium Publications, vol. 27, 1942, especially chap. 8.
- 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, DOI 10.1090/coll/018 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.
- Hassler Whitney, Differentiable manifolds, Ann. of Math. (2) 37 (1936), no. 3, 645–680. MR 1503303, DOI 10.2307/1968482
- Hassler Whitney, The imbedding of manifolds in families of analytic manifolds, Ann. of Math. (2) 37 (1936), no. 4, 865–878. MR 1503315, DOI 10.2307/1968624 42. H. Whitney, Topological properties of differentiable manifolds, Bull. Amer. Math. Soc. vol. 43 (1937) pp. 785-805.
- Hassler Whitney, On the topology of differentiable manifolds, Lectures in Topology, University of Michigan Press, Ann Arbor, Mich., 1941, pp. 101–141. MR 0005300
Additional Information
- Journal: Bull. Amer. Math. Soc. 52 (1946), 545-571
- DOI: https://doi.org/10.1090/S0002-9904-1946-08610-3
- MathSciNet review: 0017531