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On the triangulation of stratified sets and singular varieties


Author: F. E. A. Johnson
Journal: Trans. Amer. Math. Soc. 275 (1983), 333-343
MSC: Primary 58A35; Secondary 32B25, 54E60, 57R05
DOI: https://doi.org/10.1090/S0002-9947-1983-0678354-5
MathSciNet review: 678354
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Abstract: We show that every compact stratified set in the sense of Thom can be triangulated as a simplicial complex. The proof uses that author's description of a stratified set as the geometric realisation of a certain type of diagram of smooth fibre bundles and smooth imbeddings, and the triangulability of smooth fibre bundles.

As a consequence, we obtain proofs of the classical triangulation theorems for analytic and subanalytic sets, and a correct proof of Yang's theorem that the orbit space of a smooth compact transformation group is triangulable.


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  • [1] A. Borel and J. P. Serre, Corners and arithmetic groups, Comment. Math. Helv. 48 (1973), 436-491. MR 0387495 (52:8337)
  • [2] -, Cohomologie d'immeubles et de groupes $ S$-arithmétiques, Topology 15 (1976), 211-232. MR 0447474 (56:5786)
  • [3] J. Cerf, Topologie de certains espaces de plongements, Bull. Soc. Math. France 89 (1961), 227-380. MR 0140120 (25:3543)
  • [4] A. Douady and L. Herault, Appendix to [1].
  • [5] H. Hironaka, Triangulation of algebraic sets, Proc. Amer. Math. Soc. Inst. Algebra Geom. Arcata (1974). MR 0374131 (51:10331)
  • [6] S. Illman, Smooth equivariant triangulations of $ G$-manifolds for $ G$ a finite group, Math. Ann. 233 (1978), 199-220. MR 0500993 (58:18474)
  • [7] F. E. A. Johnson, Triangulation of stratified sets, Thesis, University of Liverpool, 1972.
  • [8] -, A triangulation criterion, Mathematika 25 (1978), 110-114. MR 0500988 (58:18470)
  • [9] -, On the triangulation of smooth fibre bundles, Fund. Math. (to appear). MR 736287 (86a:57020)
  • [10] -, On the presentation of stratified sets and singular varieties, Mathematika (to appear). MR 673514 (84h:58009)
  • [11] B. O. Koopman and A. B. Brown, On the covering of analytic loci by complexes, Trans. Amer. Math. Soc. 34 (1932), 231-251. MR 1501636
  • [12] S. Lefschetz and J. H. C. Whitehead, On analytical complexes, Trans. Amer. Math. Soc. 35 (1933), 510-517. MR 1501698
  • [13] S. Lojasiewicz, Triangulation of semi-analytic sets, Ann. Sculoa Norm. Sup. Pisa Cl. Sci. 18 (1964), 449-473. MR 0173265 (30:3478)
  • [14] B. Teissier, Théorèmes de finitude en géométrie analytique (d'après Heisuke Hironaka), Sém. Bourbaki 1973/74, no. 451, Lecture Notes in Math., vol. 431, Springer-Verlag, Berlin and New York, 1975. MR 0477119 (57:16663)
  • [15] R. Thom, Ensembles et morphismes stratifié, Bull. Amer. Math. Soc. 75 (1969), 240-284. MR 0239613 (39:970)
  • [16] B. L. Van der Waerden, Topolgische Begründung des Kalküls der abzählen Geometrie, Math. Ann. 102 (1929), 360.
  • [17] J. H. C. Whitehead, On $ {C^1}$ complexes, Ann. of Math. (2) 41 (1940), 809-824. MR 0002545 (2:73d)
  • [18] H. Whitney, Tangents to an analytic variety, Ann. of Math. (2) 81 (1965), 496-549. MR 0192520 (33:745)
  • [19] C. T. Yang, The triangulability of the orbit space of a differentiable transformation group, Bull. Amer. Math. Soc. 69 (1963), 405-408. MR 0146291 (26:3813)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0678354-5
Article copyright: © Copyright 1983 American Mathematical Society

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