2
$\begingroup$

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.

$\endgroup$
5
$\begingroup$

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.

$\endgroup$
3
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ Out of curiosity, do you know a one-dimensional example for $N\subset \mathbb{R}^2$ such that $N$ is a topological manifold but not locally flat? $\endgroup$ – Daniel Fischer Nov 5 '14 at 22:36
  • $\begingroup$ @DanielFischer: Such examples do not exist. This should be in Moise's book. $\endgroup$ – Moishe Kohan Nov 5 '14 at 22:42
  • $\begingroup$ Thanks. Wasn't sure whether they don't, or whether I could only not think weird enough to find one. $\endgroup$ – Daniel Fischer Nov 5 '14 at 22:44

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.