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Cones and Vietoris-Begle type theorems


Author: D. G. Bourgin
Journal: Trans. Amer. Math. Soc. 174 (1972), 155-183
MSC: Primary 55B30
DOI: https://doi.org/10.1090/S0002-9947-1972-0322854-7
MathSciNet review: 0322854
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Abstract: Infinite cone constructions are exploited to yield diverse generalizations of the Vietoris-Begle theorem for paracompact spaces and Abelian group sheaves. The constructions suggest natural space, map classifications designated as almost p-solid. The methods are extended to upper semicontinuous closed multivalued maps and homotopies and culminate in a disk fixed point theorem for possibly nonacyclic point images.


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  • [1] K. Borsuk, Concerning some upper-continuous decompositions of compacta, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 11, (1963), 499-503. MR 30 #3470. MR 0173257 (30:3470)
  • [2] D. G. Bourgin, On the Vietoris-Begle theorem, Bull. Amer. Math. Soc. 76 (1970), 747-751. MR 42 #8492a. MR 0273614 (42:8492a)
  • [3] -, Vietoris-Begle type theorems, Rend. Mat. (6) 3 (1970), 19-31. MR 42 #8492a. MR 0273615 (42:8492b)
  • [4] -, Modern algebraic topology, Macmillan, New York, 1963. MR 28 #3415. MR 0160201 (28:3415)
  • [5] E. G. Skljarenko, Some applications of the theory of sheaves in general topology, Uspehi Mat. Nauk 19 (1964), no. 6 (120), 47-70 = Russian Math. Surveys 19 (1964), no. 6, 41-62. MR 30 #1490. MR 0171259 (30:1490)
  • [6] A. Białynicki-Birula, On Vietoris mapping theorem and its inverse, Fund. Math. 53 (1963/64), 133-145. MR 29 #606. MR 0163303 (29:606)
  • [7] G. E. Bredon, Sheaf theory, McGraw-Hill, New York, 1967. MR 36 #4552. MR 0221500 (36:4552)
  • [8] J. Dugundji, Topology, Allyn and Bacon, Boston, Mass., 1966. MR 33 #1824. MR 0193606 (33:1824)
  • [9] S. Kakutani, A generalization of Brouwer's fixed point theorem, Duke Math. J. 8 (1941), 457-459. MR 3, 60. MR 0004776 (3:60c)
  • [10] A. D. Wallace, A theorem on acyclicity, Bull. Amer. Math. Soc. 67 (1961), 123-124. MR 23 #A2196. MR 0124886 (23:A2196)
  • [11] J. D. Lawson, A generalized version of the Vietoris-Begle theorem, Fund. Math. 65 (1969), 65-72. MR 40 #2055. MR 0248805 (40:2055)
  • [12] W. Strothers, Multi-homotopy, Duke Math. J. 22 (1955), 281-285. MR 16, 948. MR 0068834 (16:948b)
  • [13] Edwin H. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. MR 35 #1007. MR 0210112 (35:1007)
  • [14] S. Eilenberg and D. Montgomery, Fixed point theorems for multi-valued transformations, Amer. J. Math. 68 (1946), 214-222. MR 8, 51. MR 0016676 (8:51a)
  • [15] R. L. Wilder, Some mapping theorems with applications to nonlocally connected spaces, Algebraic Geometry and Topology (A Sympos. in Honor of S. Lefschetz), Princeton Univ. Press, Princeton, N. J., 1957, pp. 378-388. MR 19, 303. MR 0087107 (19:303a)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0322854-7
Keywords: Vietoris-Begle theorem, sheaf, paracompact, quotient space, Tychonoff parallelotope, exact sequence, support family, upper semicontinuity, graph
Article copyright: © Copyright 1972 American Mathematical Society

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