Hereditary ball-covers for some Banach manifolds

West, James E.

doi:10.1090/S0002-9939-1972-0336747-8

Hereditary ball-covers for some Banach manifolds
HTML articles powered by AMS MathViewer

by James E. West

Proc. Amer. Math. Soc. 33 (1972), 132-136

DOI: https://doi.org/10.1090/S0002-9939-1972-0336747-8

PDF | Request permission

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.

References

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 (English, with Russian summary). MR 221269
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 139917
C. H. Dowker, Topology of metric complexes, Amer. J. Math. 74 (1952), 555–577. MR 48020, DOI 10.2307/2372262
K. H. Dauker, Affine and euclidean complexes, Dokl. Akad. Nauk SSSR 128 (1959), 655–656 (Russian). MR 0117708
David W. Henderson, Infinite-dimensional manifolds, Proc. Internat. Sympos. on Topology and its Applications (Herceg-Novi, 1968) Savez Društava Mat. Fiz. i Astronom., Belgrade, 1969, pp. 183–185. MR 0285036
David W. Henderson, Infinite-dimensional manifolds are open subsets of Hilbert space, Topology 9 (1970), 25–33. MR 250342, DOI 10.1016/0040-9383(70)90046-7
David W. Henderson and R. Schori, Topological classification of infinite dimensional manifolds by homotopy type, Bull. Amer. Math. Soc. 76 (1970), 121–124. MR 251749, DOI 10.1090/S0002-9904-1970-12392-8
James E. West, Products of complexes and Fréchet spaces which are manifolds, Trans. Amer. Math. Soc. 166 (1972), 317–337. MR 293679, DOI 10.1090/S0002-9947-1972-0293679-6