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Recently I found the next 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 a submanifold 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?

Ben McKay
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Antonio
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    Alexander's Horned Sphere is the standard example, see any textbook on knot theory. Rolfsen's book has a detailed example. – Ryan Budney Oct 28 '11 at 21:30
  • Alexander's Horned Sphere is technically an example of a topological subspace of a manifold which is also a manifold. So if that's your definition of "submanifold" then it qualifies. – Ryan Budney Oct 28 '11 at 21:32
  • Alexander's Horned Sphere is not a submanifold of R^{3} is a manifold on its own but not a submanifold. Perhaps I dont understand the notion of locally flat submanifold. – Antonio Oct 28 '11 at 21:35
  • What definition of "submanifold" are you using? You've given a definition of "locally flat submanifold" but you haven't given a definition of "submanifold". – Ryan Budney Oct 28 '11 at 21:46
  • You should tell us your definition of topological submanifold, probably. – Mariano Suárez-Álvarez Oct 28 '11 at 21:46
  • For me N is a d-dimensional submanifold of an n-dimensional manifold M if for every point x in N there exists an open subset of M such that x belongs to M and $U\cap N$ is homeomorphic to an open set in $R^{d}$ – Antonio Oct 28 '11 at 21:54
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    Then Alexander's Horned Sphere is a submanifold of $\mathbb R^3$. – Ryan Budney Oct 28 '11 at 21:58
  • I think you're wrong I read that the Horned Sphere is not a submanifold of $R^{3}$ – Antonio Oct 28 '11 at 22:10
  • Either you misread the statement, or the author of the statement was wrong. IMO your question would maybe be more appropriate on math.stackexchange.com, as it seems like your question has more to do with interpreting standard point-set topology definitions. – Ryan Budney Oct 29 '11 at 00:20
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    Another example: Take a knot $K$ in $S^3$ and consider the cone on the knot in $B^4$. This is a submanifold of $B^4$ homeomorphic to a disk, but it is not flat at the cone point. – Jim Conant Oct 29 '11 at 01:18
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    Some authors use "submanifold" to mean what some other authors mean by "locally flat submanifold". Unfortunate, maybe, but undoubtedly true. – Tom Goodwillie Oct 29 '11 at 01:26
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    Please do not ask questions here and on math.stackexchange.com at the same time. http://math.stackexchange.com/questions/76977/locally-flat-submanifold – Mariano Suárez-Álvarez Oct 29 '11 at 23:03
  • See http://mathoverflow.net/questions/58061/how-can-there-be-topological-4-manifolds-with-no-differentiable-structure – Benoît Kloeckner Apr 05 '14 at 12:02

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