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Transactions of the American Mathematical Society

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Singular Vietoris-Begle theorems for relations

Authors: D. G. Bourgin and Robert M. Nehs
Journal: Trans. Amer. Math. Soc. 284 (1984), 281-318
MSC: Primary 55N30; Secondary 54A10, 54C60, 55T25
MathSciNet review: 742426
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Abstract: The Vietoris-Begle theorem with singularities, for three spaces $ X$, $ Y$, $ T$, is extended to the case that a closed relation replaces a continuous map and more generally to set valued maps. The developments are carried out based on modification of the topology of $ T$ so that in general it is no longer even Hausdorff. This entails interpretation of dimension of singulars sets in terms of considertions in $ Y$ rather than $ T$. The techniques are those of sheaf and spectral sequence theory.

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

  • [1] D. G. Bourgin, Cones and Vietoris-Begle type theorems, Trans. Amer. Math. Soc. 174 (1972), 155-183. MR 0322854 (48:1215)
  • [2] E. G. Skljarenko, Some applications of the theory of sheaves in general topology, Uspehi Mat. Nauk 19 (1964), 47-49 = Russian Math. Surveys 19 (1964), 41-62. MR30#1490. MR 0171259 (30:1490)
  • [3] D. G. Bourgin, Modern algebraic topology, Macmillan, New York, 1963. MR28#3415 MR 0160201 (28:3415)
  • [4] A. P. Wallace, A theorem on acylicity, Bull. Amer. Math. Soc. 67 (1961), 123-124. MR 0124886 (23:A2196)
  • [5] J. D. Lawson, A generalized version of the Vietoris-Begle theorem, Fund. Math. 65 (1969), 65-72. MR40#2055 MR 0248805 (40:2055)
  • [6] -, Comparison of taut cohomologies, Aequationes Math. 9 (1973), 201-209. MR 0331372 (48:9705)
  • [7] G. E. Bredon, Sheaf theory, McGraw-Hill, New York, 1967. MR36#4552 MR 0221500 (36:4552)
  • [8] P. Alexandroff, On the dimension of normal spaces, Proc. Roy. Soc. London Ser. A 189 (1947), 11-39. MR 0021312 (9:52a)
  • [9] J. Dugundji, Topology, Allyn and Bacon, Boston, Mass., 1966. MR33#1824 MR 0193606 (33:1824)

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Keywords: Vietoris-Begle theorem, sheaf, spectral sequence, paracompact, graph, identification topology
Article copyright: © Copyright 1984 American Mathematical Society

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