Characterizing the topology of infinitedimensional compact manifolds
Author:
Jerzy Mogilski
Journal:
Proc. Amer. Math. Soc. 92 (1984), 111118
MSC:
Primary 57N20; Secondary 54C55
MathSciNet review:
749902
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Abstract: A metric space , which is a countable union of finitedimensional compacta, is a manifold modelled on the space all but finitely many iff is an ANR and the following condition holds: given , a pair of finitedimensional compacta and a map such that is an embedding, there is an embedding such that and for all . An analogous condition characterizes manifolds modelled on the space .
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 [1]
 R. D. Anderson, On sigmacompact subsets of finitedimensional spaces, preprint.
 [2]
 R. D. Anderson, D. W. Curtis and J. van Mill, A fake topological Hilbert space, Trans. Amer. Math. Soc. 272 (1982), 311321. MR 656491 (83j:57009)
 [3]
 C. Bessaga and A. Pelczynski, The estimated extension theorem, homogeneous collections and skeletons, and their application to the topological classification of linear space and convex sets, Fund. Math. 69 (1970), 153190. MR 0273347 (42:8227)
 [4]
 , Selected topics in infinitedimensional topology, PWN, Warsaw, 1975.
 [5]
 D. W. Curtis and N. T. Nhu, Hyperspaces of finite subsets which are homeomorphic to dimensional linear metric spaces, Topology Appl. (to appear). MR 831185 (87g:54028)
 [6]
 D. W. Curtis, T. Dobrowolski and J. Mogilski, Some applications of the topological characterizations of the sigmacompact spaces , and , Trans. Amer. Math. Soc. (to appear).
 [7]
 T. Dobrowolski and J. Mogilski, Sigmacompact locally convex metric linear spaces universal for compacta are homeomorphic, Proc. Amer. Math. Soc. 85 (1982), 653658. MR 660623 (83i:57006)
 [8]
 R. Geoghegan, Manifolds of piecewise linear maps and a related normed linear space, Bull. Amer. Math. Soc. 77 (1971), 629632. MR 0277010 (43:2747)
 [9]
 R. Geoghegan (Editor), Open problems in infinite dimensional topology, Topology Proc. 4 (1979), 287338. MR 583711 (82a:57015)
 [10]
 R. Geoghegan and W. Haver, On the space of piecewiselinear homeomorphisms of a manifold, Proc. Amer. Math. Soc. 55 (1976), 145151. MR 0402785 (53:6599)
 [11]
 J. P. Henderson and J. J. Walsh, Examples of celllike decompositions of the infinite dimensinal manifolds and , Topology Appl. 16 (1983), 143154. MR 712860 (85d:57013)
 [12]
 J. Mogilski, decornposition ofmanifolds, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 27 (1979), 309314. MR 552055 (81e:57018)
 [13]
 J. Mogilski and H. Roslaniec, Celllike decompositions of ANR's, Topology Appl. (submitted).
 [14]
 H. Torunczyk, Skeletons and absorbing sets in complete meric spaces, Doctoral thesis, Institute of Mathematics of the Polish Academic of Sciences, 1970.
 [15]
 H. Torunczyk, Concerning locally homotopy negligible sets and characterization of manifolds, Fund. Math. 101 (1978), 93110. MR 518344 (80g:57019)
 [16]
 , An introduction to infinitedimensional topology, in preparation.
 [17]
 , Absolute retracte as factors of normed linear spaces, Fund. Math. 86 (1974), 5367. MR 0365471 (51:1723)
 [18]
 , On images of the Hilbert cube and characterization of manifolds, Fund. Math. 106 (1980), 3140. MR 585543 (83g:57006)
 [19]
 , Characterizing Hilbert space topology, Fund. Math. 111 (1981), 247262. MR 611763 (82i:57016)
 [20]
 J. West, The ambient homeomorphy of incomplete subspaces of infinitedimensional Hilbert spaces, Pacific J. Math. 34 (1970), 257267. MR 0277011 (43:2748)
 [21]
 Ph. Bowers, M. Bestvina, J. Mogilski and J. Walsh, Characterizing Hilbert space manifolds revisited, in preparation.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198407499028
PII:
S 00029939(1984)07499028
Keywords:
Sigmacompact metric ANR's,
the strong universality property for compacta,
infinitedimensional sigmacompact manifolds,
sets,
nearhomeomorphisms
Article copyright:
© Copyright 1984
American Mathematical Society
