function lifting on $S^1 \times S^1$

Question

Let $f:S^1 \times S^1 \to S^1 \times S^1$ a continuous function and $p:\mathbb{R}^2 \to S^1 \times S^1: (t,s) \mapsto (e^{2\pi i t},e^{2\pi i s})$ a covering map. if $F: \mathbb R ^2 \to \mathbb R ^2 $ is a lifting of $f \circ p$ prove that there exists $(d_1,d_2),(e_1,e_2) \in \mathbb{Z}^2$ such that for all $n,m \in \mathbb Z$ $$ \forall (t,s) \in \mathbb R^2 \quad F(t+m,s+n) = F(t,s) + n(d_1,d_2) + m(e_1,e_2) $$

Answer 1

What is the question here? Given $ F $, notice that $ F'(t,s) = F(t + 1, s) $ is also a lift. So then $ p \circ F = p \circ F' $, and since the kernel of $ p $ is discrete, and $ F, F' $ are continuous, they $ F - F'$ is a constant element of the kernel. Then you can show the same for the second argument.