Singular regular neighborhoods and local flatness in codimension one

Daverman, Robert J.

doi:10.1090/S0002-9939-1976-0420630-7

Singular regular neighborhoods and local flatness in codimension one
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Proc. Amer. Math. Soc. 57 (1976), 357-362 Request permission

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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