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Decomposition spaces having arbitrarily small neighborhoods with $ 2$-sphere boundaries


Author: Edythe P. Woodruff
Journal: Trans. Amer. Math. Soc. 232 (1977), 195-204
MSC: Primary 57A10; Secondary 54B15
DOI: https://doi.org/10.1090/S0002-9947-1977-0442944-2
MathSciNet review: 0442944
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Abstract: Let G be an u.s.c. decomposition of $ {S^3}$. Let H denote the set of nondegenerate elements and P be the natural projection of $ {S^3}$ onto $ {S^3}/G$. Suppose that each point in the decomposition space has arbitrarily small neighborhoods with 2-sphere boundaries which miss $ P(H)$. We prove in this paper that this condition implies that $ {S^3}/G$ is homeomorphic to $ {S^3}$. This answers a question asked by Armentrout [1, p. 15]. Actually, the hypothesis concerning neighborhoods with 2-sphere boundaries is necessary only for the points of $ P(H)$.


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

  • [1] Steve Armentrout, Monotone decompositions of $ {E^3}$, Topology Seminar (Wisconsin, 1965), Ann. of Math. Studies, no. 60, Princeton Univ. Press, Princeton, N.J., 1966, pp. 1-25. MR 36 #5915. MR 0222865 (36:5915)
  • [2] -, A three-dimensional spheroidal space which is not a sphere, Fund. Math. 68 (1970), 183-186. MR 42 #5239. MR 0270350 (42:5239)
  • [3] W. T. Eaton, The sum of solid spheres, Michigan Math. J. 19 (1972), 193-207. MR 46 #8227. MR 0309116 (46:8227)
  • [4] Robert D. Edwards and Leslie C. Glaser, A method for shrinking decompositions of certain manifolds, Trans. Amer. Math. Soc. 165 (1972), 45-56. MR 45 #4423. MR 0295357 (45:4423)
  • [5] O. G. Harrold, Jr., A sufficient condition that a monotone image of the three-sphere be a topological three-sphere, Proc. Amer. Math. Soc. 9 (1958), 846-850. MR 21 #2223. MR 0103454 (21:2223)
  • [6] N. Hosay, Erratum to "The sum of a cube and a crumpled cube is $ {S^3}$", Notices Amer. Math. Soc. 11 (1964), p. 152.
  • [7] L. V. Keldyš, Topological imbeddings in Euclidean space, Proc. Steklov Inst. Math. 81 (1966); English transl., Amer. Math. Soc., Providence, R. I., 1968. MR 34 #6745; 38 #696. MR 0232371 (38:696)
  • [8] L. L. Lininger, Some results on crumpled cubes, Trans. Amer. Math. Soc. 118 (1965), 534-549. MR 31 #2717. MR 0178460 (31:2717)
  • [9] A. Marin and Y. M. Visetti, A general proof of Bing's shrinkability criterion, Proc. Amer. Math. Soc. 53 (1975), 501-507. MR 0388319 (52:9156)
  • [10] Louis F. McAuley, Some upper semi-continuous decompositions of $ {E^3}$ into $ {E^3}$, Ann. of Math. (2) 73 (1961), 437-457. MR 23 #A3554. MR 0126258 (23:A3554)
  • [11] -, Upper semicontinuous decompositions of $ {E^3}$ into $ {E^3}$ and generalizations to metric spaces, Topology of 3-Manifolds and Related Topics (Proc. Univ. of Georgia Inst., 1961), Prentice-Hall, Englewood Cliffs, N. J., 1962, pp. 21-26. MR 25 #4502.
  • [12] T. M. Price, A necessary condition that a cellular upper semicontinuous decomposition of $ {E^n}$ yield $ {E^n}$, Trans. Amer. Math Soc. 122 (1966), 427-435. MR 33 #1843. MR 0193627 (33:1843)
  • [13] Myra J. Reed, Decomposition spaces and separation properties, Doctoral Dissertation, SUNY, Binghamton, 1971.
  • [14] Gordon Thomas Whyburn, Analytic topology, Amer. Math. Soc. Colloq. Publ., vol. 28, Amer. Math. Soc., Providence, R.I., 1942. MR 4, 86. MR 0007095 (4:86b)

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DOI: https://doi.org/10.1090/S0002-9947-1977-0442944-2
Article copyright: © Copyright 1977 American Mathematical Society

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