lifting a product of commutators of standard generators on 2-manifolds I have a problem with understand the proof http://www.ams.org/journals/proc/1972-032-01/S0002-9939-1972-0295352-2/S0002-9939-1972-0295352-2.pdf
I don't understand this part: "(...) we can easily construct $p: \tilde{F} \rightarrow F$ to be a six sheeted covering corresponding to the kernel of an appropriate map $\pi_1(F) \rightarrow \Sigma_3$ (such covering that: if $f$ represents a loop without sef-intersections and which is a product of commutators of "standard generators" then $f$ does not lift to a loop in $\tilde{F}$)."
$\hspace{1.5cm}$ 1. why we know that such covering exist?
$\hspace{1.5cm}$ 2. why this covering is  six sheeted?
$\hspace{1.5cm}$ 3. why kernel doesn't contain $f$?

I think that talking about "products of commutators" is not useful (probably the fashion of the time was more algebraic). Cut along your null-homologous curve, so you get two pieces each with one boundary component, and restrict your attention to one of these (say, $S_1$). You want to find a nontrivial covering of $S_1$ with a single boundary component (which will be of length a multiple of the original boundary). You can then verify Hempel's claim by some educational cutting and pasting.
