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Transactions of the American Mathematical Society

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Products of complexes and Fréchet spaces which are manifolds


Author: James E. West
Journal: Trans. Amer. Math. Soc. 166 (1972), 317-337
MSC: Primary 58B05
DOI: https://doi.org/10.1090/S0002-9947-1972-0293679-6
MathSciNet review: 0293679
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Abstract: It is shown that if a locally finite-dimensional simplicial complex is given the ``barycentric'' metric, then its product with any Fréchet space X of suitably high weight is a manifold modelled on X, provided that X is homeomorphic to its countably infinite Cartesian power. It is then shown that if X is Banach, all paracompact X-manifolds may be represented (topologically) by such products.


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  • [1] R. D. Anderson, Topological properties of the Hilbert cube and the infinite product of open intervals, Trans. Amer. Math. Soc. 126 (1967), 200-216. MR 34 #5045. MR 0205212 (34:5045)
  • [2] R. D. Anderson and R. H. Bing, A complete elementary proof that Hilbert space is homeomorphic to the countable infinite product of lines, Bull. Amer. Math. Soc. 74 (1968), 771-792. MR 37 #5847. MR 0230284 (37:5847)
  • [3] W. Barit, Small extensions of small homeomorphisms, Notices Amer. Math. Soc. 16 (1969), 295. Abstract #663-715.
  • [4] R. G. Bartle and L. M. Graves, Mappings between function spaces, Trans. Amer. Math. Soc. 72 (1952), 400-413. MR 13, 951. MR 0047910 (13:951i)
  • [5] C. Bessaga, Topological equivalence of unseparable reflexive Banach spaces. Ordinal resolutions of identity and monotone bases, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 15 (1967), 397-399. MR 36 #4321. MR 0221269 (36:4321)
  • [6] C. Bessaga and V. L. Klee, Every non-normable Fréchet space is homeomorphic with all of its closed convex bodies, Math. Ann. 163 (1966), 161-166. MR 34 #1826. MR 0201949 (34:1826)
  • [7] C. Bessaga and A. Pełczyński, Some remarks on homeomorphisms of F-spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 10 (1962), 265-270. MR 25 #3344. MR 0139917 (25:3344)
  • [8] -, Estimated extension theorem, Fund. Math. 69 (1970), 153-190. MR 0273347 (42:8227)
  • [9] C. H. Dowker, Topology of metric complexes, Amer. J. Math. 74 (1952), 555-577. MR 13, 965. MR 0048020 (13:965h)
  • [10] D. W. Henderson, Infinite-dimensional manifolds are open subsets of Hilbert space, Topology 9 (1969), 25-33. MR 40 #3581. MR 0250342 (40:3581)
  • [11] -, Open subsets of Hilbert space, Compositio Math. 21 (1969), 312-318. MR 40 #4975. MR 0251748 (40:4975)
  • [12] -, Stable classification of infinite-dimensional manifolds by homotopy type, Invent. Math. 12 (1971), 48-56. MR 0290413 (44:7594)
  • [13] V. L. Klee, Some topological properties of convex sets, Trans. Amer. Math. Soc. 78 (1955), 30-45. MR 16, 1030. MR 0069388 (16:1030c)
  • [14] A. H. Kruse, Badly incomplete normed linear spaces, Math. Z. 83 (1964), 314-320. MR 29 #3859. MR 0166586 (29:3859)
  • [15] E. Michael, Continuous selections. I, Ann. of Math. (2) 63 (1956), 361-382. MR 17, 990. MR 0077107 (17:990e)
  • [16] R. S. Palais, Homotopy theory of infinite dimensional manifolds, Topology 5 (1966), 1-16. MR 32 #6455. MR 0189028 (32:6455)
  • [17] H. H. Schaefer, Topological vector spaces, Macmillan, New York and London, 1966. MR 33 #1689. MR 0193469 (33:1689)
  • [18] H. Toruńczyk, Skeletonized sets in complete metric spaces and homeomorphisms of the Hilbert cube, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 68 (1970), 119-126. MR 0264602 (41:9194)
  • [19] S. Weingram, On the triangulation of the realization of a semisimplicial complex, Illinois J. Math. 12 (1968), 403-413. MR 38 #6579. MR 0238303 (38:6579)
  • [20] J. E. West, Infinite products which are Hilbert cubes, Trans. Amer. Math. Soc. 150 (1970), 1-25. MR 0266147 (42:1055)
  • [21] J. H. C. Whitehead, Combinatorial homotopy. I, Bull. Amer. Math. Soc. 55 (1949), 213-245. MR 11, 48. MR 0030759 (11:48b)
  • [22] -, A certain exact sequence, Ann. of Math. (2) 52 (1950), 51-110. MR 12, 43. MR 0035997 (12:43c)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0293679-6
Keywords: Fréchet manifold, Banach manifold, metric simplicial complex, homotopy type
Article copyright: © Copyright 1972 American Mathematical Society

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