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Transactions of the American Mathematical Society

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

 
 

 

Aposyndetic continua as bundle spaces


Author: James T. Rogers
Journal: Trans. Amer. Math. Soc. 283 (1984), 49-55
MSC: Primary 54F20; Secondary 54F50, 55R10
DOI: https://doi.org/10.1090/S0002-9947-1984-0735408-3
MathSciNet review: 735408
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Abstract: Let $ \mathcal{S}$ be the $ P$-adic solenoid bundle, and let $ \eta :X \to {S^1}$ be a map of the continuum $ X$ onto $ {S^1}$. The bundle space $ B$ of the induced bundle $ {\eta ^{ - 1}}\mathcal{S}$ is investigated. Sufficient conditions are obtained for $ B$ to be connected, to be aposyndetic, and to be homogeneous. Uncountably many aposyndetic, homogeneous, one-dimensional, nonlocally connected continua are constructed. Other classes of continua are placed into this framework.


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

  • [1] R. D. Anderson, One-dimensional continuous curves and a homogeneity theorem, Ann. of Math. (2) 68 (1958), 1-16. MR 0096181 (20:2676)
  • [2] R. H. Bing and F. B. Jones, Another homogeneous plane continuum, Trans. Amer. Math. Soc. 90 (1959), 171-192. MR 0100823 (20:7251)
  • [3] K. Borsuk, Concerning homotopy properties of compacta, Fund. Math. 62 (1968), 223-254. MR 0229237 (37:4811)
  • [4] J. H. Case, Another $ 1$-dimensional homogeneous continuum which contains an arc, Pacific J. Math. 11 (1961), 455-469. MR 0131867 (24:A1714)
  • [5] F. B. Jones, Aposyndesis revisited, Proc. Univ. of Oklahoma Topology Conference, Norman, Oklahoma, 1972, pp. 64-78. MR 0365518 (51:1770)
  • [6] J. Krasinkiewicz and P. Minc, Generalized paths and pointed $ 1$-movability, Fund. Math. 104 (1979), 141-153. MR 551664 (81b:55028)
  • [7] M. C. McCord, Inverse limit sequences with covering maps, Trans. Amer. Math. Soc. 114 (1965), 197-209. MR 0173237 (30:3450)
  • [8] J. T. Rogers, Jr., An aposyndetic homogeneous curve that is not locally connected, Houston J. Math. (to appear). MR 719102 (85f:54073)
  • [9] -, A cohomological characterization of preimages of nonplanar circle-like continua, Proc. Amer. Math. Soc. 49 (1975), 232-236. MR 0365521 (51:1773)
  • [10] -, Solenoids of pseudo-arcs, Houston J. Math. 3 (1977), 531-537. MR 0464193 (57:4128)
  • [11] E. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. MR 0210112 (35:1007)
  • [12] M. Steenrod, The topology of fibre bundles, Princeton Univ. Press, Princeton, N. J., 1951. MR 0039258 (12:522b)
  • [13] A. Trybulec, On shapes of movable spaces, Bull. Acad. Polon. Sci. 21 (1973), 727-733. MR 0334147 (48:12466)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0735408-3
Keywords: Continuum, homogeneous, aposyndetic, solenoid, universal curve, induced bundle
Article copyright: © Copyright 1984 American Mathematical Society

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