Lifts of Non-Trivial Elements In The Fundamental Group In The Universal Covering

Question

Say $p: \hat{X}\rightarrow X$ is a universal covering, $X$ is path-connected, locally path-connected, and semi-locally simply connected. Now say $\gamma\in\pi_1(X,x)$, if I can show $\gamma$ can only be lifted to a loop in $\hat{X}$, am I safe to conclude that $\gamma$ is homotopic to the constant loop? I think this is true and can be shown with some simple arguments, but I'm a bit stuck. Thanks.

Answer 1

This is true, and simple.

Suppose $\gamma$ lifts to a loop $\hat\gamma$ in $\hat X$. Using that $\hat X$ is simply connected, there exists a continuous function $\hat H : [0,1] \times [0,1] \to \hat X$ which is a path homotopy from $\hat\gamma$ to a constant path.

It follows that $H = p \circ \hat H : [0,1] \times [0,1] \to X$ is a path homotopy from $\gamma$ to a constant path.