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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Alexander duality and Hurewicz fibrations


Author: Steven C. Ferry
Journal: Trans. Amer. Math. Soc. 327 (1991), 201-219
MSC: Primary 55R65; Secondary 55R10, 57N15
DOI: https://doi.org/10.1090/S0002-9947-1991-1028308-9
MathSciNet review: 1028308
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Abstract: We explore conditions under which the restriction of the projection map $ p:{S^n} \times B \to B$ to an open subset $ U \subset S^n \times B$ is a Hurewicz fibration. As a consequence, we exhibit Hurewicz fibrations $ p:E \to I$ such that: (i) $ p:E \to I$ is not a locally trivial bundle, (ii) $ p^{ - 1}(t)$ is an open $ n$-manifold for each $ t$, and (iii) $ p\; \circ \;{\text{proj:E}} \times {R^1} \to I$ is a locally trivial bundle. The fibers in our examples are distinguished by having nonisomorphic fundamental groups at infinity. We also show that when the fibers of a Hurewicz fibration with open $ n$-manifold fibers have finitely generated $ (n - 1){\text{st}}$ homology, then all fibers have the same finite number of ends. This last shows that the punctured torus and the thrice punctured two-sphere cannot both be fibers of a Hurewicz fibration $ p:E \to I$ with open $ 2$-manifold fibers.


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DOI: https://doi.org/10.1090/S0002-9947-1991-1028308-9
Keywords: Hurewicz fibration, Čech cohomology, acyclic space
Article copyright: © Copyright 1991 American Mathematical Society