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 Free Access

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] Wolf Barth, Lokale Cohomologie bei isolierten Singularitäten analytischer Mengen, Schr. Math. Inst. Univ. Münster (2) 5 (1971), i+59 (German). MR 0302936
  • [P] P. T. Church, Differentiable open maps on manifolds, Trans. Amer. Math. Soc. 109 (1963), 87–100. MR 0154296, 10.1090/S0002-9947-1963-0154296-6
  • [H] Helmut A. Hamm, On the vanishing of local homotopy groups for isolated singularities of complex spaces, J. Reine Angew. Math. 323 (1981), 172–176. MR 611450, 10.1515/crll.1981.323.172
  • [H] 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
  • [J] J. F. P. Hudson, Piecewise linear topology, University of Chicago Lecture Notes prepared with the assistance of J. L. Shaneson and J. Lees, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0248844
  • [K] Erich Kähler, Über die Verzweigung einer algebraischen Funktion zweier Veränderlichen in der Umgebung einer singulären Stelle, Math. Z. 30 (1929), no. 1, 188–204 (German). MR 1545053, 10.1007/BF01187762
  • [J] Mather, Notes on topological stability, Harvard Univ. 1970.
  • [J] J. Milnor, Morse theory, Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. MR 0163331
  • [2] John Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. MR 0239612
  • [D] David Mumford, The topology of normal singularities of an algebraic surface and a criterion for simplicity, Inst. Hautes Études Sci. Publ. Math. 9 (1961), 5–22. MR 0153682
  • [R] R. Thom, Les structures différentiables des boules et des sphères, Colloque Géom. Diff. Globale (Bruxelles, 1958) Centre Belge Rech. Math., Louvain, 1959, pp. 27–35 (French). MR 0121805
  • [C] C. T. C. Wall, Regular stratifications, Dynamical systems—Warwick 1974 (Proc. Sympos. Appl. Topology and Dynamical Systems, Univ. Warwick, Coventry, 1973/1974; presented to E. C. Zeeman on his fiftieth birthday), Springer, Berlin, 1975, pp. 332–344. Lecture Notes in Math., Vol. 468. MR 0649271

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