Cantor sets and homotopy connectedness of manifolds
Author:
David G. Wright
Journal:
Proc. Amer. Math. Soc. 50 (1975), 463470
MSC:
Primary 57A15
MathSciNet review:
0370594
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Abstract: We prove that a topological manifold of dimension is connected if each Cantor set in is contained in an open ball of . An immediate consequence is that a compact manifold of dimension is homeomorphic to the sphere if and only if every Cantor set of is contained in an open ball of . This consequence generalizes a dimensional theorem of Doyle and Hocking.
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 [1]
 R. H. Bing, An alternative proof that manifolds can be triangulated, Bull. Amer. Math. Soc. 62 (1956), 179180.
 [2]
 , Necessary and sufficient conditions that a manifold be , Ann. of Math. (2) 68 (1958), 1737. MR 20 # 1973. MR 0095471 (20:1973)
 [3]
 W. A. Blankinship, Generalization of a construction of Antoine, Ann. of Math. (2) 53 (1951), 276297. MR 12, 730. MR 0040659 (12:730c)
 [4]
 M. Brown, A proof of the generalized Schoenflies theorem, Bull. Amer. Math. Soc. 66 (1960), 7476. MR 22 # 8470b. MR 0117695 (22:8470b)
 [5]
 E. H. Connell, A topological cobordism theorem for , Illinois J. Math. 11 (1967), 300309. MR 35 #3673. MR 0212808 (35:3673)
 [6]
 P. H. Doyle and J. G. Hocking, Dimensional invertibility, Pacific J. Math. 12 (1962), 12351240. MR 26 #6977. MR 0149490 (26:6977)
 [7]
 W. T. Eaton, A generalization of the dog bone space to , Proc. Amer. Math. Soc. 39 (1973), 379387. MR 48 #1238. MR 0322877 (48:1238)
 [8]
 E. E. Moise, Affine structures in manifolds. V. The triangulation theorem and Hauptvermutung, Ann. of Math. (2) 56 (1952), 96114. MR 14, 72. MR 0048805 (14:72d)
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 R. P. Osborne, Embedding Cantor sets in a manifold. Part III: Approximating spheres (preprint).
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 T. B. Rushing, Topological embeddings, Academic Press, New York, 1973. MR 0348752 (50:1247)
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 H. Seifert and W. Threlfall, Lehrbuch der Topologie, Teubner, Leipzig, 1934; reprint, Chelsea, New York, 1947.
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 E. H. Spainer, Algebraic topology, McGrawHill, New York, 1966. MR 35 #1007. MR 0210112 (35:1007)
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 J. Stallings, The piecewiselinear structure of Euclidean space, Proc. Cambridge Philos. Soc. 58 (1962), 481488. MR 26 #6945. MR 0149457 (26:6945)
 [14]
 R. L. Wilder, Topology of manifolds, Amer. Math. Soc. Colloq. Publ., vol. 32, Amer. Math. Soc., Providence, R. I., 1963. MR 32 #440. MR 0182958 (32:440)
 [15]
 D. G. Wright, Cantor sets and homotopy connectedness of manifolds, Doctoral Thesis, University of Wisconsin, Madison, Wisconsin, 1973.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197503705949
PII:
S 00029939(1975)03705949
Keywords:
Cantor set,
homotopy connected,
engulfing,
characterization of sphere
Article copyright:
© Copyright 1975 American Mathematical Society
