Lifting closed curves to finite covers of free groups

In this article, we show that given any integer $l\ge 2$, every closed curve $\gamma$ on the bouquet of $n$-circles $\mathrm{\Gamma }$, admits a lift to a finite $l$-sheeted normal covering of $\mathrm{\Gamma }$. Equivalently, identifying the free group $F_n$ of $n$ generators with the fundamental group of $\mathrm{\Gamma }$, this statement asserts that $F_n$ is a union of $l$-index normal subgroups for any $l\ge 2.$ The proof proceeds by explicitly constructing families of $l$-sheeted normal coverings of $\mathrm{\Gamma }$, together with a characterization, in terms of necessary and sufficient conditions, of when a closed curve $\gamma$ on $\mathrm{\Gamma }$ lifts to these covers.