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Triangulations of subanalytic sets and locally subanalytic manifolds


Authors: M. Shiota and M. Yokoi
Journal: Trans. Amer. Math. Soc. 286 (1984), 727-750
MSC: Primary 32B20; Secondary 57Q15
MathSciNet review: 760983
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Abstract: If two polyhedrons are locally subanalytically homeomorphic (that is, the graph is locally subanalytic), they are $ {\text{PL}}$ homeomorphic. A locally subanalytic manifold is one whose coordinate transformations are locally subanalytic. It is proved that a locally subanalytic manifold has a unique $ {\text{PL}}$ manifold structure. A semialgebraic manifold also is considered.


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  • [1] R. D. Edwards, The double suspension of a certain homology $ 3$-sphere is $ {S^5}$ (unpublished).
  • [2] Christopher G. Gibson, Klaus Wirthmüller, Andrew A. du Plessis, and Eduard J. N. Looijenga, Topological stability of smooth mappings, Lecture Notes in Mathematics, Vol. 552, Springer-Verlag, Berlin-New York, 1976. MR 0436203
  • [3] Robert M. Hardt, Triangulation of subanalytic sets and proper light subanalytic maps, Invent. Math. 38 (1976/77), no. 3, 207–217. MR 0454051
  • [4] Heisuke Hironaka, Subanalytic sets, Number theory, algebraic geometry and commutative algebra, in honor of Yasuo Akizuki, Kinokuniya, Tokyo, 1973, pp. 453–493. MR 0377101
  • [5] Heisuke Hironaka, Triangulations of algebraic sets, Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974) Amer. Math. Soc., Providence, R.I., 1975, pp. 165–185. MR 0374131
  • [6] S. Lojasiewicz, Triangulation of semi-analytic sets, Ann. Scuola Norm. Sup. Pisa (3) 18 (1964), 449–474. MR 0173265
  • [7] -, Ensembles semi-analytique, Inst. Hautes Etudes Sci., Paris, 1965.
  • [8] John N. Mather, Stratifications and mappings, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971) Academic Press, New York, 1973, pp. 195–232. MR 0368064
  • [9] Barry Mazur, A note on some contractible 4-manifolds, Ann. of Math. (2) 73 (1961), 221–228. MR 0125574
  • [10] John Milnor, Two complexes which are homeomorphic but combinatorially distinct, Ann. of Math. (2) 74 (1961), 575–590. MR 0133127
  • [11 J] James R. Munkres, Elementary differential topology, Lectures given at Massachusetts Institute of Technology, Fall, vol. 1961, Princeton University Press, Princeton, N.J., 1963. MR 0163320
  • [12] Colin Patrick Rourke and Brian Joseph Sanderson, Introduction to piecewise-linear topology, Springer Study Edition, Springer-Verlag, Berlin-New York, 1982. Reprint. MR 665919
  • [13] M. G. Scharlemann and L. C. Siebenmann, The Hauptvermutung for smooth singular homeomorphisms, Manifolds—Tokyo 1973 (Proc. Internat. Conf., Tokyo, 1973) Univ. Tokyo Press, Tokyo, 1975, pp. 85–91. MR 0372871
  • [14] Masahiro Shiota, Classification of Nash manifolds, Ann. Inst. Fourier (Grenoble) 33 (1983), no. 3, 209–232 (English, with French summary). MR 723954
  • [15] -, Piecewise linearization of real analytic functions, Publ. RIMS, Kyoto Univ. (to appear).
  • [16] L. C. Siebenmann, Topological manifolds, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 133–163. MR 0423356
  • [17] Jean-Louis Verdier, Stratifications de Whitney et théorème de Bertini-Sard, Invent. Math. 36 (1976), 295–312 (French). MR 0481096

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1984-0760983-2
Keywords: Triangulation, subanalytic, polyhedron, $ {\text{PL}}$ homeomorphism
Article copyright: © Copyright 1984 American Mathematical Society