Let $(m,n)\in\mathbb{Z}^2$ and let define the following map: $$f:\left\{\begin{array}{ccc} \mathbb{S}^1&\rightarrow&\mathbb{S}^1\times\mathbb{S}^1\\ \zeta&\mapsto&(\zeta^m,\zeta^n) \end{array}\right..$$ I have to prove the following:
Proposition. The following statements are equivalent:
There exists a continuous map $g:\mathbb{S}^1\times\mathbb{S}^1\rightarrow\mathbb{S}^1$ such that $g\circ f=\textrm{id}_{\mathbb{S}^1}$.
$m$ and $n$ are relatively prime.
Proof.
If $m$ and $n$ are relatively prime, using Bézout's there exists $(u,v)\in\mathbb{Z}^2$ such that $um+vn=1$ and the following map: $$g:\left\{\begin{array}{ccc} \mathbb{S}^1\times\mathbb{S}^1&\rightarrow&\mathbb{S}^1\\ (\zeta_1,\zeta_2)&\mapsto&{\zeta_1}^u{\zeta_2}^v \end{array}\right.$$ is continuous and satisfies $g\circ f=\textrm{id}_{\mathbb{S}_1}$.
If there exists $g:\mathbb{S}^1\times\mathbb{S}^1\rightarrow\mathbb{S}^1$ such that $g\circ f=\textrm{id}_{\mathbb{S}^1}$, then $g_*\circ f_*=\textrm{id}_{\pi_1\left(\mathbb{S}^1,1\right)}$ and $f_*$ is injective. $\Box$ I understand that one has: $$f_*:\left\{\begin{array}{ccc} \pi_1\left(\mathbb{S}^1,1\right)&\rightarrow&\pi_1\left(\mathbb{S}^1\times\mathbb{S}^1,(1,1)\right)\\ [\gamma]&\mapsto&[(\gamma^n,\gamma^m)]. \end{array}\right.$$ Despite I tried to use that $\left[e^{2i\pi\cdot}\right]$ is a generator of $\pi_1\left(\mathbb{S}^1,1\right)$, I am not able to determine further $f_*$. Any help will be greatly appreciated.