Is this "the winding number"? > Note that $p:\mathbb{C}\rightarrow \mathbb{C}\setminus\\{0\\}:z\mapsto e^z$ is a covering map. > > Let $\alpha:[0,1]\rightarrow \mathbb{C}\setminus\\{0\\}$ be a closed curve. > > Let $\gamma$ be any lift of $\alpha$ such that $\alpha=p\circ \gamma$. > > Call $\frac{\gamma(1)-\gamma(0)}{2\pi i}$ **the winding number of $\alpha$**. Is it okay to define "winding number" in this way? That is, does this definition exactly mean the winding number?

This is correct, yes. If you have a closed curve $\alpha$, then from beginning to end it only differs by a phase $e^{2\pi i n}$. Using the exponential map as you have, you see that this gives you exactly the number of times that your curve has wound around the origin.