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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

A Hurewicz-type theorem for approximate fibrations


Authors: D. S. Coram and P. F. Duvall
Journal: Proc. Amer. Math. Soc. 78 (1980), 443-448
MSC: Primary 55R65; Secondary 55P05
MathSciNet review: 553392
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Abstract: This paper concerns conditions on point inverses which insure that a mapping between locally compact, separable, metric ANR's is an approximate fibration. Roughly a mapping is said to be $ {\pi _i}$-movable [respectively, $ {H_i}$-movable] provided that nearby fibers include isomorphically into mutual neighborhoods on $ {\pi _i}$ [resp. $ {H_i}$]. An earlier result along this line is that $ {\pi _i}$-movability for all i implies that a mapping is an approximate fibration. The main result here is that for a $ U{V^1}$ mapping, $ {\pi _i}$-movability for $ i \leqslant k - 1$ plus $ {H_k}$- and $ {H_{k + 1}}$-movability imply $ {\pi _k}$-movability of the mapping. Hence a $ U{V^1}$ mapping which is $ {H_i}$-movable for all i is an approximate fibration. Also, if a $ U{V^1}$ mapping is $ {\pi _i}$-movable for $ i \leqslant k$ and k is at least as large as the fundamental dimension of any point inverse, then it is an approximate fibration. Finally, a $ U{V^1}$ mapping $ f:{M^m} \to {N^n}$ between manifolds is an approximate fibration provided that f is $ {\pi _i}$-movable for all $ i \leqslant \max \{ m - n,\tfrac{1}{2}(m - 1)\} $.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1980-0553392-4
PII: S 0002-9939(1980)0553392-4
Keywords: Approximate fibration, UV property
Article copyright: © Copyright 1980 American Mathematical Society