John Hempel proved that fundamental groups of $2$-manifolds are residually finite. I want to understand this proof so I have some questions:
Why if we have $S(f)=\emptyset$ then $f$ represents either a "standard generator" or a product of commutators of "standard generators"? -Let's look at connected sum of two tori. I have 4 generators $a_1,b_1,a_2,b_2$ such that $a_1$, $b_1$ are contain in one torus. Then we have $a_1 a_2^{-1}$ a loop which go through both holes and which singular set is empty. So can we present that loop as product of commutators?
How and why can we construct six sheeted covering in the second case?
Thanks for any advice
Let $S$ be the surface. I will be using the following facts:
1) There are two types of simple closed curves, separating and non-separating. Separating closed curves are zero in $H^1(S)$.
2) $H^1(S)=\frac{\pi_1(S)}{[\pi_1(s),\pi_1(S)]}$, (Theorem 2A.1 of Hatcher).
3) For any two non-separating curves $a$ and $b$, there exists a homeomorphism $\phi$ such that $\phi(a)=b$. This follows from classification of surfaces.
Now if image of $f$ is non-separating (i.e. non-trivial in homology), then post composing it by a homeomorphism as in $3)$ we can assume that it is a standard generator (as the standard generators are non-separating).
If image of $f$ is separating, then it is non-trivial in the fundamental group but trivial in $H^1(S)$. Hence by $2)$, it belongs to ${[\pi_1(s),\pi_1(S)]}$. And therefore is a product of the commutators of the standard generators.