Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Neighborhoods of algebraic sets

Author: Alan H. Durfee
Journal: Trans. Amer. Math. Soc. 276 (1983), 517-530
MSC: Primary 32B20; Secondary 14G30, 32B25
MathSciNet review: 688959
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In differential topology, a smooth submanifold in a manifold has a tubular neighborhood, and in piecewise-linear topology, a subcomplex of a simplicial complex has a regular neighborhood. The purpose of this paper is to develop a similar theory for algebraic and semialgebraic sets. The neighborhoods will be defined as level sets of polynomial or semialgebraic functions.

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

  • [W] Barth, Lokale Cohomologie bei isolierten Singularitäten analytischer Mengen, Schr. Math. Inst. Univ. Munster 2 (5) (1971). MR 0302936 (46:2079)
  • [P] Church, Differentiable open maps on $ n$-manifolds, Trans. Amer. Math. Soc. 109 (1963), 87-100. MR 0154296 (27:4245)
  • [H] Hamm, On the vanishing of local homotopy groups for isolated singularites of complex spaces, J. Reine Angew. Math. 323 (1981), 172-176. MR 611450 (82i:32023)
  • [H] Hironaka, Triangulations of algebraic sets, Proc. Sympos. Pure Math., vol. 29, Amer. Math. Soc. Providence, R.I., 1975, pp. 165-185. MR 0374131 (51:10331)
  • [J] Hudson, Piecewise linear topology, Benjamin, New York, 1969. MR 0248844 (40:2094)
  • [K] Kähler, Über die Verzweigung einer algebraischen Funktion zweier Veränderlichen in der Umgebung einen singularen Stelle, Math. Z. 30 (1929), 188-204. MR 1545053
  • [J] Mather, Notes on topological stability, Harvard Univ. 1970.
  • [J] Milnor, 1. Morse theory, Ann. of Math. Studies, no. 51, Princeton Univ. Press, Princeton, N.J., 1963. MR 0163331 (29:634)
  • [2] Singular points of complex hypersurfaces, Ann. of Math. Studies, no. 61, Princeton Univ. Press, Princeton, N.J., 1968. MR 0239612 (39:969)
  • [D] Mumford, The topology of normal singularities of an algebraic surface and a criterion for simplicity, Publ. Math. Inst. Hautes Etudes Scientifiques, no. 9 (1961). MR 0153682 (27:3643)
  • [R] Thom, Les structures différentiables des boules et des spheres, Colloque de Géométrie Différentielle Globale, Bruxelles 1958, Gauthier-Villars, Paris, 1959. MR 0121805 (22:12535)
  • [C] T. C. Wall, Regular stratifications, Dynamical Systems, Warwick 1974, Lecture Notes in Math., vol. 468, Springer-Verlag, Berlin and New York, 1975, pp. 332-344. MR 0649271 (58:31184)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 32B20, 14G30, 32B25

Retrieve articles in all journals with MSC: 32B20, 14G30, 32B25

Additional Information

Keywords: Tubular neighborhood, regular neighborhood, rug function, link of singularity, stratification, Milnor fibration theorem, semialgebraic sets
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society