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Recently I found the following definition:

Let $M^{n}$ be an $n-$dimensional topological manifold. Then $N^{k}\subseteq M^{n}$ is a locally flat submanifold if for every $x\in N$ there exists an open set $U$ in $M$ such that the pair $(U,U\cap N)$ is homeomorphic to the pair $(R^{n},R^{k})$.

The first thing I noted is that if $N$ is a locally flat submanifold of $M$, then $N$ in fact is a sumbanifold of $M$. The second thing I noted is that in the category of smooth manifolds the notion of locally smooth submanifold and submanifold are equivalent.

I would like to know if there is an example of a topological submanifold of a manifold that is not a locally flat submanifold or if the notions of locally flat submanifold and submanifold are equivalent in the category of topological manifolds?

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Consider a knot $K\subset S^3$. Then think of the $4$-ball $B^4$ as the cone on $S^3$: $B^4=C(S^3)$. Consider the cone on $K$ inside the $4$-ball, $C(K)\subset B^4$. This is homeomorphic to a disk and is a submanifold of $B^4$. However it is not locally flat since any small ball around the cone point will have a knot on the boundary, so won't be locally standard, as in the definition of locally flat.

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  • $\begingroup$ I'm not sure about the example, does it works with a trivial knot? Or the knot must be "knotted"? $\endgroup$ – Antonio Oct 29 '11 at 18:43
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    $\begingroup$ @Antonio: Yes the knot must be knotted. It doesn't work with the trivial knot. $\endgroup$ – Cheerful Parsnip Oct 29 '11 at 18:52
  • $\begingroup$ Ok, let me study this example more carefully to be completely sure that I understand it. Thank you so much. $\endgroup$ – Antonio Oct 29 '11 at 18:55
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    $\begingroup$ I was not expecting that :D $\endgroup$ – Mariano Suárez-Álvarez Oct 29 '11 at 23:04
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    $\begingroup$ @MattN.: the tameness of $K$ is a separate issue. I was actually thinking that $K$ is tame, but the cone on $K$ is still not locally flat. A knot $K$ in $\mathbb R^3$ is tame if and only of it is locally flat. $\endgroup$ – Cheerful Parsnip Oct 22 '12 at 23:32

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