# lifting theorem

Let $p\colon\thinspace E\to B$ be a covering map and $f\colon\thinspace X\to B$ 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 $\tilde{f}\colon\thinspace X\to E$ with $\tilde{f}(x)=e$ 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 $\tilde{f}$ is unique (provided it exists of course).

The following diagrams might be useful: To check