I stumbled over the following question.
Imagine we have a two homotopic curves on the sphere $\mathbb{S}^1$ namely $\gamma_1,\gamma_2$. Then we can write them as $\gamma_{i}(t) = e^{i \alpha_i (t)}$ for some function $\alpha_i$. Both paths satisfy $\gamma_1(0)= \gamma_2(0) = -1$ and $\gamma_1(1)=\gamma_2(1)=1.$ Can we conclude from this that the difference $\alpha_2(1)-\alpha_2(0) = \alpha_1(1)-\alpha_1(0)$?
If anything is unclear, please let me know.