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Hereditary ball-covers for some Banach manifolds

Author: James E. West
Journal: Proc. Amer. Math. Soc. 33 (1972), 132-136
MSC: Primary 57A20
MathSciNet review: 0336747
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Abstract: At a problem seminar in Ithaca, New York, during January 1969, James Eells raised the question (numbered 33 on the circulated list) of whether a paracompact Fréchet manifold admits a locally finite cover by open sets, all of whose intersections are contractible. This had been established in the separable case by David Henderson, who obtained star-finite covers. This note settles the case that the model space is a Banach space homeomorphic to its countably infinite Cartesian power. The cover elements and all nonempty intersections are homeomorphic to the model. A short proof that the nerve of the cover has the homotopy type of the manifold is also included.

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  • [1] 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)
  • [2] C. Bessaga and A. Pelczyński, 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)
  • [3] C. H. Dowker, Topology of metric complexes, Amer. J. Math. 74 (1952), 555-577. MR 13, 965. MR 0048020 (13:965h)
  • [4] -, Affine and euclidean complexes, Dokl. Akad. Nauk SSSR 128 (1959), 655-656. (Russian) MR 22 #8483. MR 0117708 (22:8483)
  • [5] D. W. Henderson, Infinite-dimensional manifolds, Proc. Internat. Sympos. on Topology and its Applications, Herceg Novi, Yugoslavia, 1968. MR 0285036 (44:2260)
  • [6] -, Infinite-dimensional manifolds are open subsets of Hilbert space, Topology 9 (1969), 25-33. MR 40 #3581. MR 0250342 (40:3581)
  • [7] D. W. Henderson and R. M. Schori, Topological classification of infinite dimensional manifolds by homotopy type, Bull. Amer. Math. Soc. 76 (1970), 121-124. MR 40 #4976. MR 0251749 (40:4976)
  • [8] J. E. West, Products of complexes and Fréchet spaces which are manifolds, Trans. Amer. Math. Soc. (to appear). MR 0293679 (45:2756)

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Keywords: Banach manifold, locally-finite cover, nerve, homotopy type
Article copyright: © Copyright 1972 American Mathematical Society

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