The following is an exercise taken from "Manifolds and Differenial Geometry" by Jeffrey M. Lee.
Let $\widetilde M$ and M be (connected) $C^r$ manifolds. Let $f: \widetilde M \to M$ be a proper map which is a local diffeomorphism
from $\widetilde M$ onto $M$.Prove: $f$ is a covering map.
Attempt:
Let $x \in M$. $\{x\}$ is compact hence $f^{-1}(x)$ is compact. $f$ is a local diffeomorphism so every $y \in f^{-1}(x)$ is contained in some open neighborhood $U_y$ that is mapped $C^r$-diffeomorphically to $f(U_y)$ that is: $\left.f\right|_{U_y} : U_y \to f(U_y)$ is a $C^r$- diffeomorphism. It follows that $f^{-1}(x) \cup U_y$ = $\{y\}$.
$\{U_y| y \in f^{-1}(x)\}$ is an open cover of $f^{-1}(x)$ which has a finite subcover.
Let $\{U_i\}_{ 1\le i \le k}$ be a finite subcover. ($U_i$ denoting the set that contains $y_i)$.
By the preceding arguments $f^{-1}(x)=\{y_1,y_2, \dots,y_k \}$.
This is where i got stuck…
I can't seem to find the appropriate neighborhood of $x$ that will be evenly covered by
it's preimage. I know it can't be just $\bigcap_{i=1}^k f(U_i)$…
Any help would be appreciated.
Side note: Only recently did I start to read about manifolds and to find a simple exercise that's so out of my reach is really off putting. Would you say I'm better off putting the book aside and picking up an easier one? (It would be a shame since so far i really like it).
Best Answer
In the case of manifolds, you can use the characterization of covering map via the path-lifting property:
$f$ is a covering if for every $x\in M$, for every continuous path $\sigma:[0,1]\to M$ with $\sigma(0)=x$, and for every $\tilde x\in\tilde M$ that prjects to $x$, there is a unique lift $\tilde\sigma:[0,1]\to \tilde M$ that projects to $\sigma$ (i.e. $f(\tilde\sigma(t))=\sigma(t)\ \forall t$).
Now, the properness of $f$ plus the local diffeo porperty ensure the path-lifting property:
Let $x,\tilde,\sigma$ be as above. Let $D\subset[0,1]$ the set of times for which $\tilde\sigma$ exists: $D=\{t\in[0,1]: \tilde\sigma$ is defined on $[0,s]$ for every $s\leq t\}$.
We show that $D=[0,1]$ by showing that it is open and closed and non-empty. Clearly $D\neq\emptyset$ because it contains $0$ by definition of $\tilde x$. The local diffeo property of $f$ implies that $D$ is open.
To see that $D$ is closed we use properness. Let $T=\sup D$ so that $\bar D=[0,T]$ is compact. By properness $f^{-1}([0,1])$ is compact. Thus its connected components are compacts. Let $S$ be the connected component containing $\tilde x$. $S$ is compact hence is closed. Since $T=\sup D$ there is a sequence $t_n\to T$ with $t_n\in D$. Let $s_n=\tilde\sigma(t_n)$ be the corresponding sequence in $S$. As $S$ is compact $s_n$ has an adherence point $s$ in $S$. By continuity $f(s)=\sigma(T)$. The local diffeo property of $f$ now concludes that $T\in D$, whence $\bar D=D$ and $D$ is closed.