I have a question about the relation between non-separating surfaces and the nontrivial 2nd homology of a 3-manifold. The question is:
Let $\Sigma$ be a closed nonseparating surface embedded in a closed $3-$manifold $M,$ is it true that $\Sigma$ represents a nontrivial element in $H_{2}(M; \mathbb{Z}_{2})$? And on the other hand, is it true that every nontrivial element in $H_{2}(M;\mathbb{Z}_{2})$ can be represented by a nonseparating surface in $M$?
The answer is Yes.
In fact this there is a construction due to Stallings that lets you construct non-separating surfaces in this way.
Take a $3$--manifold $M$, which is compact and look at $f:M\rightarrow S^{1}$. From compactness, we know that $H_{1}(M)$ has at least a $\mathbb{Z}$ component. In particular the map $f_{*}$ can be defined so that, $f_{*}:H_{1}(M)\rightarrow H_{1}(S^1)=\mathbb{Z}$. Now we can look at an inverse image of a point $a\in\mathbb{Z}$; that is $f^{-1}(a)$ is a surface and in particular we have defined an element in $H^{1}(M)$. By duality this gives an element in $H_{2}(M)$ which is non-zero and hence non-separating. I have left out some details, let me know if you need them and I'll fill them in.