I found a proof of following statement (which is available here ), but I'm not sure if we must assume that fundamental group of surface $M$ is non-abelian:
Lemma: Let $M$ be a compact orientable surface (possibly bounded) with $\pi_1(M)$ non-abelian and let $f: S^1 \to M$ be a non-nullhomotopic closed curve. Then there exists a degree 8 normal cover $M \to \tilde{M}$ such that one of the following holds.
- $f$ does not lift to a closed curve on $\tilde{M}$.
- $f$ lifts to a closed curve $\tilde{f}: S^1 → \tilde{M}$ with $i( \tilde{f}) < i( f )$.
where $i(f)$ is self-intersection number of the loop (minimum over all curves $f'$ which are freely homotopic to $f$ of the quantity $|\{(x,y)\mid x,y \in S^1, x \neq y, f'(x)=f'(y) \} |/2$)
So, the question is: Is this lemma is true for surfaces with abelian fundamental group?