Surface groups are residually finite I am looking at the short paper by Hempel "Residual finiteness of surface groups".  Let $F$ be a compact orientable surface and let $f : S^1 \to F$ be a map that represents a nontrivial element of the fundamental group of $F$.  Further assume that $f$ is an embedding.  We want to construct a covering of $F$ by a compact surface $\tilde{F}$ such that the map $f$ can not be lifted to this covering.  
First suppose that $f$ is a parameterization for one of the "standard generators" - then Hempel claims that we can construct a two-sheeted covering of $F$ that $f$ does not lift to.  What is this construction?? 
Alternatively, suppose that the image of $f$ is homologically trivial (i.e. the image of $f$ is a product of commutators in $\pi_1(F)$.  Then Hempel claims that we can construct a six-sheeted cover of $F$ corresponding to the kernel of a suitable map from $\pi_1(F)$ to the fundamental group of a genus 3 surface.  What is this map and what is this construction?
This question was asked before and some parts of it were answered but I am confused about some of the unanswered parts:
John Hempel's proof of residual finiteness of surface groups

Correction: Hempel used  $\Sigma_3$ to denote symmetric group of three letters not the surface of genus three. Its order is $6$, therefore any surjective map will have a kernel of index $6$.
Now to answer to your question, let us consider the possibilities of the components of a given curve $x$. The complement of the curve $x$ can be either connected (non-separating) or disconnected (separating).  If the complement is connected then by the classification of surfaces, any two such curves are homeomorphic, hence start with a standard generator. 
Case-1: Non-separating.
Recall that the algebraic intersection number $\hat{i}(x,y)$ is defined as the sum of the indices of the intersection points of $x$ and $y$, where an intersection point is of index $+1$ if orientation agrees with the surface, $-1$ otherwise. Also $\hat{i}(x,y)$ only depends on the homology classes of $x$ and $y$.
Now given any non-separating curve $x$ you can always find $y$ such that $\hat{i}(x,y)=1$. 
Define the homomorphism $\hat{i}(\,\,,y)$ from $\pi_1(F)$ to $\mathbb{Z}_2$. The kernel is the required subgrop ad the cover ($x$ does not belong to the kernel).
Case-2: Separating.
Let $x=\prod_{i=1}^n[x_i,y_i]$. Define the homomorphism $\phi:\pi_1{F}\rightarrow\Sigma_3$ as:
1) $\phi(x_1)=(1,2),\,\, \phi(y_1)=(1,3).$
2) $\phi(x_2)=(1,2), \,\, \phi(y_2)=(2,3).$
3) $\phi(x_i)=\phi(y_i)=e$ for all $i\neq 1,2.$ 
Check that the product of the commutators is $e$, therefore this is a homomorphism and $x$ is not in the kernel.
