Decomposition spaces having arbitrarily small neighborhoods with sphere boundaries
Author:
Edythe P. Woodruff
Journal:
Trans. Amer. Math. Soc. 232 (1977), 195204
MSC:
Primary 57A10; Secondary 54B15
MathSciNet review:
0442944
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Abstract: Let G be an u.s.c. decomposition of . Let H denote the set of nondegenerate elements and P be the natural projection of onto . Suppose that each point in the decomposition space has arbitrarily small neighborhoods with 2sphere boundaries which miss . We prove in this paper that this condition implies that is homeomorphic to . This answers a question asked by Armentrout [1, p. 15]. Actually, the hypothesis concerning neighborhoods with 2sphere boundaries is necessary only for the points of .
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 [1]
 Steve Armentrout, Monotone decompositions of , Topology Seminar (Wisconsin, 1965), Ann. of Math. Studies, no. 60, Princeton Univ. Press, Princeton, N.J., 1966, pp. 125. MR 36 #5915. MR 0222865 (36:5915)
 [2]
 , A threedimensional spheroidal space which is not a sphere, Fund. Math. 68 (1970), 183186. MR 42 #5239. MR 0270350 (42:5239)
 [3]
 W. T. Eaton, The sum of solid spheres, Michigan Math. J. 19 (1972), 193207. 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), 4556. MR 45 #4423. MR 0295357 (45:4423)
 [5]
 O. G. Harrold, Jr., A sufficient condition that a monotone image of the threesphere be a topological threesphere, Proc. Amer. Math. Soc. 9 (1958), 846850. MR 21 #2223. MR 0103454 (21:2223)
 [6]
 N. Hosay, Erratum to "The sum of a cube and a crumpled cube is ", 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), 534549. 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), 501507. MR 0388319 (52:9156)
 [10]
 Louis F. McAuley, Some upper semicontinuous decompositions of into , Ann. of Math. (2) 73 (1961), 437457. MR 23 #A3554. MR 0126258 (23:A3554)
 [11]
 , Upper semicontinuous decompositions of into and generalizations to metric spaces, Topology of 3Manifolds and Related Topics (Proc. Univ. of Georgia Inst., 1961), PrenticeHall, Englewood Cliffs, N. J., 1962, pp. 2126. MR 25 #4502.
 [12]
 T. M. Price, A necessary condition that a cellular upper semicontinuous decomposition of yield , Trans. Amer. Math Soc. 122 (1966), 427435. 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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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197704429442
PII:
S 00029947(1977)04429442
Article copyright:
© Copyright 1977
American Mathematical Society
