It's well-known that every path $s(t): I \to S^1$ has a lifting, i.e. mapping $\widetilde{s(t)}: I \to \mathbb{R}$, so that $e^{i\widetilde{s(t)}} = s(t)$. The main idea of constructing $\widetilde{s(t)}$ is the following:
- Find by uniform continuity for $[0, t]$ such an $n$ that $|x-y| < 1/n \implies |s(x) - s(y)| < 1$.
- Devide $[0, t]$ on $n$ parts.
- For each part find $\phi_{n}$ is the angle between values in the left and right bounds of this interval.
- Set $\widetilde{s(t)} = \alpha + \sum \phi_n$, where $\alpha$ is the initial value of the lifting.
There are two questions concerning this reasoning:
- I dont's see any problems when trying to apply it to $s: S^1 \to S^1$, although in this case the statement is wrong.
- How to modify this proof to be used for $s: I \times I -> S^1$