# lifting theorem

Let $p:E\to B$ be a covering map and $f:X\to B$ be a (continuous^{})
map where $X$, $B$ and $E$ are path connected and locally path connected (http://planetmath.org/LocallyConnected).
Also let $x\in X$ and $e\in E$ be points such that $f(x)=p(e)$.
Then $f$ lifts to a map $\stackrel{~}{f}:X\to E$ with $\stackrel{~}{f}(x)=e$ if and only if
${\pi}_{1}(f)$ maps ${\pi}_{1}(X,x)$ inside the image
${\pi}_{1}(p)\left({\pi}_{1}(E,e)\right)$, where ${\pi}_{1}$ denotes the fundamental
group functor. Furthermore $\stackrel{~}{f}$ is unique (provided it exists of course).

The following diagrams might be useful: To check