Following is an argument given by Hempel where I am unable to understand his comment about choosing a loop close enough to a surface. Can somebody please elucidate this: 


**Lemma:** If $F$ is a compact connected surface properly embedded in a $3$-manifold $M$ and if $image(i\_\*:H\_1(F;Z/2Z)\rightarrow{H\_1(M;Z/2Z)})=0$, then $F$ is 2-sided in $M$. 


**Proof:** By regular neighborhood theory, it suffices to show that $F$ separates some connected neighborhood of $F$. If this is not the case then there is a loop $J\subset{M}$ such that $J\cap{F}$ is a single point, with transverse intersection. **We may choose $J$ close enough to $F$** so that $J$ is homologous to zero (mod 2) in $M$. This contradicts homological invariance (mod 2) of intersection numbers. QED. 


**Doubt:** Everything else is clear to me except the bold part in the proof. I don't think we will be able to bring $J$ close to $F$ unless it already bounds a disc. So I can see the proof only in the simply connected $M$ case where after suitably perturbing the loop and using the loop theorem to bound, I can apply an ambient isotopy to bring the disc itself closer to $F$. But how do we see this for the non-simply connected $M$? A complicated loop in $M$ might go all around it. 


**Reference:** Lemma 2.1, Chapter 2 (Heegard Splittings), 3-manifolds(book) by Hempel. 



Can you explain a step in a proof about 2-sided surfaces in 3-manifolds?
We want to prove that in the case F is not one-sided, we may replace J by a curve J' that is contained in a small neighborhood of F and interesects F in the same way as J. By assumtion F is one sided. Consider the boundary B of a small neighborhood N of $F$. Since F is one-sided, B is connected. Now, conisder the intersection of J with B. There are even number of intersections, since B is the boundary. So you can throw the part of J that does not belong to the neighbohood N an close it to a connected curve J' by segments in B (we assumed that B is connected). This explanes the words written in bold.


