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Transactions of the American Mathematical Society

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Isotoping mappings to open mappings


Author: John J. Walsh
Journal: Trans. Amer. Math. Soc. 250 (1979), 121-145
MSC: Primary 57N37; Secondary 54C10, 57N25, 57N60
DOI: https://doi.org/10.1090/S0002-9947-1979-0530046-8
MathSciNet review: 530046
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Abstract: Let f be a quasi-monotone mapping from a compact, connected manifold $ {M^m}\,(m\, \geqslant \,3)$ onto a space Y; then there is an open mapping g from M onto Y such that, for each $ y\, \in \,Y,\,{g^{ - 1}}(y)$ is not a point and $ {g^{ - 1}}(y)$ and $ {f^{ - 1}}(y)$ are equivalently embedded in M (in particular, $ {g^{ - 1}}(y)$ and $ {f^{ - 1}}(y)$ have the same shape). Applying the result with f equal to the identity mapping on M yields a continuous decomposition of M into cellular sets each of which is not a point.


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

  • [A-1] . R. D. Anderson, Open mappings of compact continua, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 347-349. MR 17, 1230. MR 0078682 (17:1230h)
  • [A-2] . -, Monotone interior dimension raising mappings, Duke Math. J. 19 (1952), 359-366. MR 14, 71. MR 0048798 (14:71c)
  • [A-3] . -, Continuous collections of continuous curves, Duke Math. J. 21 (1954), 363-367. MR 15, 977. MR 0062429 (15:977h)
  • [A-4] . -, On monotone interior mappings in the plane, Trans. Amer. Math. Soc. 73 (1952), 211-222. MR 0050269 (14:305a)
  • [Bi] . R. H. Bing, Complementary domains of continuous curves, Fund. Math. 36 (1949), 303-318. MR 12, 348. MR 0038063 (12:348h)
  • [Ke] . L. V. Keldys, Transformation of a monotone irreducible mapping into a monotone open one and monotone mappings of a cube which raise dimension, Dokl. Akad. Nauk SSSR 114 (1957), 472-475. (Russian) MR 19, 972. MR 0091455 (19:972a)
  • [K-L] . K. Kuratowski and C. Lacher, A theorem on the space of monotone mappings, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 17 (1969), 797-800. MR 43 #1143. MR 0275386 (43:1143)
  • [McA] . L. F. McAuley, Open mappings and open problems, Topology Conf., Arizona State Univ., Tempe, Ariz., 1968, pp. 184-202. MR 39 #2134. MR 0240789 (39:2134)
  • [So] . A. B. Sosinskii, Monotonically-open mappings of a sphere,, Amer. Math. Soc. Transl. (2) 78 (1968), 67-101.
  • [Wa-1] . J. J. Walsh, Monotone and open mappings on manifolds. I, Trans. Amer. Math. Soc. 209 (1975), 419-432. MR 0375326 (51:11521)
  • [Wa-2] . -, Light open and open mappings on manifolds. II, Trans. Amer. Math. Soc. 217 (1976), 271-284. MR 0394674 (52:15475)
  • [Wa-3] . -, Monotone and open mappings onto ANR's, Proc. Amer. Math. Soc. 60 (1976), 286-288. MR 0425888 (54:13838)
  • [Wi-1] . D. C. Wilson, Open mappings on manifolds and a counterexample to the Whyburn conjecture, Duke Math. J. 40 (1973), 705-716. MR 47 #9522. MR 0320989 (47:9522)
  • [Wi-2] . -, Open mappings of the universal curve onto continuous curves, Trans. Amer. Math. Soc. 168 (1972), 487-515. MR 45 #7682. MR 0298630 (45:7682)
  • [Why-1] . G. T. Whyburn, Analytic topology, Amer. Math. Soc. Colloq. Publ., vol. 28, Amer. Math. Soc., Providence, R. I., 1942. MR 0007095 (4:86b)
  • [Why-2] . -, Topological analysis, Princeton Univ. Press, Princeton, N. J., 1958.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1979-0530046-8
Keywords: Open mapping, monotone mapping, manifold, cellular
Article copyright: © Copyright 1979 American Mathematical Society

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