question about cover maps

Question

Here's a problem I've had a hard time with

If $f: M\rightarrow N$ is a cover map and $M$ is a m-manifold, will $N$ also be a m-manifold? A manifold is a space locally Euclidean space that is Hausdorff and Second Countable.

What if $N$ were a n-manifold, will $M$ also be a $n$-manifold?

Since I'm very new to topology, a heuristic explanation would be enough. I have some idea as to what the answers would be but I am not completely sure.

Answer 1

Suppose $f\colon M\to N$ is a covering map. If $N$ is a (topological) $n$-manifold, then $M$ is too. The converse, however, is true only with the additional assumption that $N$ is Hausdorff.

Here's a counterexample in which $N$ is not Hausdorff. Let $M = \mathbb R^2\smallsetminus \{(0,0)\}$, and let $N$ be the quotient space of $M$ by the equivalence relation generated by $$ (x,y) \sim (2^n x, 2^{-n}y), \qquad n\in\mathbb Z. $$ Then the quotient map $f\colon M\to N$ is a covering map, but $N$ is not Hausdorff, so it's not a manifold.

To see that $N$ is not Hausdorff, it's useful to look at the equivalence classes of $(1,0)$ and $(0,1)$.

Answer 2

Being a manifold is at least partly a local property. This is of course given, since a covering map is locally a homeomorphism and hence looks locally as the Euclidean space! Write this down explicitely to understand it, please. But of course this observation yields the dimension of $M$ which is defined locally!

If $M$ would not be necessary connected, it might fail to be a manifold since you could take an uncountable amount of copies of $M$ to cover $N$. This would fail to be second countable. But reasonable covering spaces of manifolds are manifolds again, since they correspond to a countable indexed subgroup of $\pi_1(N)$ (since $N$ is a manifold) and hence you can pull a countable basis back via the covering map.

Note that the Hausdorff is preserved by coverings. (as well the locally euclidean property as mentioned above, which is trivial being a local property). Indeed, take two points on $M$ and observe whether a local trivial neighborhood of both seperates them, otherwise they will be in different sheets. I leave the details to you.

Make sure you prove the first and last part for yourself in detail, it will be a great help.