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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A note on paired fibrations


Author: Patricia Tulley McAuley
Journal: Proc. Amer. Math. Soc. 34 (1972), 534-540
MSC: Primary 55D05
MathSciNet review: 0423336
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Abstract: Consider pairs $ (\mathcal{X},\mathcal{A})$ where $ \mathcal{X} = (X,p,B)$ and $ \mathcal{A} = (A,p\vert A,B)$ are Hurewicz fibrations mapping onto B and $ A \subset X$. It is proved that $ (\mathcal{X},\mathcal{A})$ is a cofibration if and only if $ (\mathcal{X}{ \cup _f}\mathcal{Y},\mathcal{Y})$ is a strongly-paired fibration for each fibration $ \mathcal{Y} = (Y,q,B)$ and fiber map $ f:\mathcal{A} \to \mathcal{Y}$. It follows as a corollary that the notions of fiber homotopy equivalence and strong fiber homotopy equivalence [5] coincide for all Hurewicz fibrations. That $ (\mathcal{X},\mathcal{A})$ be ``strongly-paired'' requires more than that each lifting function for $ \mathcal{A}$ be extendable to $ \mathcal{X}$. This and other notions of pairing are studied.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0423336-X
Keywords: Hurewicz fibration, adjunction space, fiber homotopy equivalence, cofibration
Article copyright: © Copyright 1972 American Mathematical Society