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

Homotopy equivalences on $ 3$-manifolds


Author: Gerard A. Venema
Journal: Proc. Amer. Math. Soc. 91 (1984), 637-642
MSC: Primary 57N10; Secondary 55P10, 57N13, 57N60
MathSciNet review: 746105
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Abstract: Suppose $ {M^3}$ is a $ 3$-manifold and $ f:{M^3} \to X$ is a homotopy equivalence onto an ANR $ X$. In this paper the cellularity properties of point preimages under $ f$ are studied. It is shown that for every open cover $ \alpha $ of $ X$ there exists an open cover $ \beta $ of $ X$ such that if $ f$ is a $ \beta $-equivalence then each $ {f^{ - 1}}\left( x \right)$ is $ \alpha $-cellular in $ M_ + ^3 \times {{\mathbf{R}}^1}$. In fact, the (open) cellularity occurs in a continuous fashion and so the map $ f$ can be approximated by a Euclidean bundle map.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1984-0746105-8
PII: S 0002-9939(1984)0746105-8
Keywords: $ 3$-manifold, absolute neighborhood retract, $ \alpha $-equivalence, cell-like set, cellular set
Article copyright: © Copyright 1984 American Mathematical Society