Take the 2-minute tour ×  For any positive integer $n$ the map
$p:S^1\to S^1$, defined by $p(z)=z^n$ is a covering map.  I
think it's easy, but as I'm a really beginner I'm struggling to
prove it. By the way, what kind of operation is that $z^n$?
share|improve this question Consider $S^1$ as $\{z\in \Bbb
C:|z|=1\}$ –  Dominik Dec 10 '12 at 20:50 add comment  1 Answer
up vote 4 down vote accepted  I'm in a rush, but here's a hint:
EDIT: Now that I have more time, let me see if I can explain
this better.  Let's begin by thinking of the circle as a
paramaterized curve. In particular, let's consider
parameterizing $S^1$ as $\gamma(t)=e^{2\pi it}$ for $t=[0,1]$.
The curve $\gamma$ can be thought of as traversing the unit
circle once, counterclockwise. While this parameterization will
not really make any of our work any easier, it will at least
make it clear that the statement is true.  Ok, so note then that
if $f:z\mapsto z^n$ is the $n^{\text{th}}$ power map, then we
can get a handle of what $f$ does to $S^1$