Definition of Submanifold of Topological Manifold

Question

Analogy to smooth manifold, I want to define the submanifold of topological manifold.

There are two ways. Let $M$ be a topological manifold, and $N\subset M$.

• If $N$ is a topological manifold, then we call $N$ is a submanifold of $M$.

• For any $p\in N$, there exists $(U,\phi)$, we have $\phi(U\cap N)=\phi(U)\cap\mathbb R^n\times\{0\}$. Then we call $N$ is a submanifold of $M$.

Actually the second definition is same to the smooth case. Also, second $\Longrightarrow$ first. But can first definition deduce the second? Any advice is helpful. Thank you.

Answer 1

The Alexander Horned Sphere is a great example. Here's another one. Take any nontrivial knot $K\subset S^3$ regarded as the boundary of the $4$-ball. Then consider taking the cone of that knot to the center of $B^4$. This is homeomorphic to a disk, but does not have a "flat" neighborhood as in in the second condition.

By the way, the second condition is called local flatness.

Answer 2

1st does not imply the 2nd, see Alexander horned sphere.

Addendum: Every (proper) topological embedding of a 1-dimensional manifold into a 2-dimensional manifold is tame (i.e., is a submanifold in the 2nd sense). A proof of this and more (tameness of embeddings of graphs) can be found e.g. in Chapter 10 of

E.E. Moise, "Geometric topology in dimensions 2 and 3", GTM, vol. 47, 1977.