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
DOI:
https://doi.org/10.1090/S0002-9947-1984-0742426-8
MathSciNet review:
742426
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Abstract | References | Similar Articles | Additional Information
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.
- D. G. Bourgin, Cones and Vietoris-Begle type theorems, Trans. Amer. Math. Soc. 174 (1972), 155–183 (1973). MR 322854, DOI https://doi.org/10.1090/S0002-9947-1972-0322854-7
- E. G. Skljarenko, Some applications of the theory of sheaves in general topology, Uspehi Mat. Nauk 19 (1964), no. 6 (120), 47–70 (Russian). MR 0171259
- D. G. Bourgin, Modern algebraic topology, The Macmillan Co., New York; Collier-Macmillan Ltd., London, 1963. MR 0160201
- A. D. Wallace, A theorem on acyclicity, Bull. Amer. Math. Soc. 67 (1961), 123–124. MR 124886, DOI https://doi.org/10.1090/S0002-9904-1961-10534-X
- Jimmie D. Lawson, A generalized version of the Vietoris-Begle theorem, Fund. Math. 65 (1969), 65–72. MR 248805, DOI https://doi.org/10.4064/fm-65-1-65-72
- Jimmie D. Lawson, Comparison of taut cohomologies, Aequationes Math. 9 (1973), 201–209. MR 331372, DOI https://doi.org/10.1007/BF01832627
- Glen E. Bredon, Sheaf theory, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1967. MR 0221500
- P. Alexandroff, On the dimension of normal spaces, Proc. Roy. Soc. London Ser. A 189 (1947), 11–39. MR 21312, DOI https://doi.org/10.1098/rspa.1947.0027
- James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606
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Additional Information
Keywords:
Vietoris-Begle theorem,
sheaf,
spectral sequence,
paracompact,
graph,
identification topology
Article copyright:
© Copyright 1984
American Mathematical Society
