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Proceedings of the American Mathematical Society

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

 

 

Singular regular neighborhoods and local flatness in codimension one


Author: Robert J. Daverman
Journal: Proc. Amer. Math. Soc. 57 (1976), 357-362
MSC: Primary 57A45
DOI: https://doi.org/10.1090/S0002-9939-1976-0420630-7
MathSciNet review: 0420630
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Abstract: For an $ (n - 1)$-manifold $ S$ topologically embedded as a closed subset of an $ n$-manifold $ N$, we define what it means for $ S$ to have a singular regular neighborhood in $ N$. The principal result demonstrates that $ S$ has a singular regular neighborhood in $ N$ if and only if the homotopy theoretic condition holds that $ N - S$ is locally simply connected ($ 1$-LC) at each point of $ S$. Consequently, $ S$ has a singular regular neighborhood in $ N$ if and only if $ S$ is locally flatly embedded $ (n \ne 4)$.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0420630-7
Keywords: Singular regular neighborhood, locally singularly collared, $ 1$-ULC embedding, locally flat embedding, degree one mapping of manifolds
Article copyright: © Copyright 1976 American Mathematical Society