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Transactions of the American Mathematical Society

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Finiteness theorems for approximate fibrations


Authors: D. S. Coram and P. F. Duvall
Journal: Trans. Amer. Math. Soc. 269 (1982), 383-394
MSC: Primary 55R65; Secondary 55P55, 57N55
DOI: https://doi.org/10.1090/S0002-9947-1982-0637696-9
MathSciNet review: 637696
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Abstract: This paper concerns conditions on the point inverses of a mapping between manifolds which insure that it is an approximate fibration almost everywhere. The primary condition is $ {\pi _i}$-movability, which says roughly that nearby point inverses include isomorphically on the $ i$th shape group into a mutual neighborhood. Suppose $ f:{M^m} \to {N^n}$ is a $ U{V^1}$ mapping which is $ {\pi _i}$-movable for $ i \leqslant k - 1$, and $ n \geqslant k + 1$. An earlier paper proved that $ f$ is an approximate fibration when $ m \leqslant 2k - 1$. If instead $ m = 2k$, this paper proves that there is a locally finite set $ S \subset N$ such that $ f\vert{f^{ - 1}}(N - S)$ is an approximate fibration. Also if $ m = 2k + 1$ and all of the point inverses are FANR's with the same shape, then there is a locally finite set $ E \subset N$ such that $ f\vert{f^{ - 1}}(N - E)$ is an approximate fibration.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1982-0637696-9
Keywords: Approximate fibration, movability, mapping between manifolds
Article copyright: © Copyright 1982 American Mathematical Society

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