Recognizing manifolds
Author:
James P. Henderson
Journal:
Proc. Amer. Math. Soc. 94 (1985), 721727
MSC:
Primary 57N20; Secondary 54B15, 54C99, 58B05
MathSciNet review:
792291
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Abstract 
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Abstract: Denote by the subspace of the Hilbert cube consisting of for all but finitely many }. Then following characterization of manifolds modeled on is proven and applied to celllike, upper semicontinuous decompositions of manifolds. An ANR is a manifold if and only if (a) is the countable union of finitedimensional compacta, (b) each compact subset of is a strong set, and (c) for each positive integer , every mapping can be arbitrarily closely approximated by an injection.
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 F. D. Ancel, The role of countable dimensionality in the theory of celllike relations, Trans. Amer. Math. Soc. 287 (1985), 140. MR 766204 (86b:54012)
 [2]
 C. Bessaga and A. Pelczyński, The estimated extension theorem, homogeneous collections and skeletons, and their applications to the topological classification of linear metric spaces and convex sets, Fund. Math. 69 (1970), 153190. MR 0273347 (42:8227)
 [3]
 J. W. Cannon, The recognition problem: what is a topological manifold? Bull. Amer. Math. Soc. 84 (1978), 832866. MR 0494113 (58:13043)
 [4]
 , Shrinking celllike decompositions of manifolds. Codimension three, Ann. of Math. (2) 110 (1979), 83112. MR 541330 (80j:57013)
 [5]
 T. A. Chapman, Dense sigmacompact subsets of infinitedimensional manifolds, Trans. Amer. Math. Soc. 154 (1971), 399426. MR 0283828 (44:1058)
 [6]
 R. J. Daverman, Products of celllike decompositions, Topology Appl. 11 (1980), 121139. MR 572368 (81f:54004)
 [7]
 R. D. Edwards, Approximating certain celllike maps by homeomorphisms, Preprint. See Notices Amer. Math. Soc. 24 (1977), A649, #751G5.
 [8]
 J. P. Henderson and J. J. Walsh, Examples of celllike decompositions of the infinitedimensional manifolds and , Topology Appl. 16 (1983), 143154. MR 712860 (85d:57013)
 [9]
 R. C. Lacher, Celllike mappings and their generalizations, Bull. Amer. Math. Soc. 83 (1977), 495552. MR 0645403 (58:31095)
 [10]
 J. Mogilski, Characterizing the topology of infinitedimensional compact manifolds, Proc. Amer. Math. Soc. 92 (1984), 111118. MR 749902 (85m:57012)
 [11]
 F. Quinn, Ends of maps and applications, Ann. of Math. (2) 110 (1979), 275331. MR 549490 (82k:57009)
 [12]
 H. Torunczyk, Skeletonized sets in complete metric spaces and homeomorphisms of the Hilbert cube, Bull. Acad. Polon. Sci. Sér. Sci. Math. 18 (1970), 119126. MR 0264602 (41:9194)
 [13]
 , On images of the Hilbert cube and characterization of manfolds, Fund. Math. 106 (1980), 3140. MR 585543 (83g:57006)
 [14]
 J. J. Walsh, Detecting finite and infinite dimensional manifolds, Address to 811th meeting of Amer. Math. Soc., April 13, 1984, Richmond, Va. See Abstracts Amer. Math. Soc. 5 (1984), 182, #8115701.
 [15]
 J. West, The ambient homeomorphy of incomplete subspaces of infinitedimensional Hilbert spaces, Pacific J. Math. 34 (1970), 257267. MR 0277011 (43:2748)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198507922914
PII:
S 00029939(1985)07922914
Keywords:
Countable dimensional manifold,
celllike decompositions,
Euclidean injection property
Article copyright:
© Copyright 1985 American Mathematical Society
