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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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    [Local flatness definition on Wikipedia](http://en.wikipedia.org/wiki/Local_flatness).2012-10-22

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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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    I'm not sure about the example, does it works with a trivial knot? Or the knot must be "knotted"?2011-10-29
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    @Antonio: Yes the knot must be knotted. It doesn't work with the trivial knot.2011-10-29
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    Ok, let me study this example more carefully to be completely sure that I understand it. Thank you so much.2011-10-29
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    I don't know if you can give some comments about the definition of locally flat submanifold and also if you can give me some bibliography where I can found about the topic?2011-10-29
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    @Antonio: The way I think of it, a locally flat submanifold mimics all of the nice properties of a smooth submanifold. I really don't know any good references. I learned about this stuff talking to my advisor in grad school.2011-10-29
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    I was not expecting that :D2011-10-29
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    @Antonio: The manifold atlas project http://www.map.him.uni-bonn.de/Main_Page might be a good place to get started.2011-10-29
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    Rolfsen's "Knots and links" textbook is a good source for this material. Roseman's "Elementary Topology" would also be another good source.2011-10-31
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    @JimConant Do you make any assumption on tameness of $K$? I am wondering whether a knot is tame if and only if it is locally flat or whether we can have tame knots that aren't locally flat.2012-10-22
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    @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.2012-10-22
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    Dear @JimConant, thank you very much for your comment!2012-10-24
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    Just to be sure: in $\mathbb R^3$ we have "$K$ is tame" <=> "$K$ is locally flat" but in $S^3$ this doesn't hold?2012-10-24
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    @MattN: No, it's still true in $S^3.$ I just wanted to point out that it's not true in other pairs of dimensions. My cone example is tame but not locally flat.2012-10-24