http://pi.math.cornell.edu/~hatcher/AT/ATpage.html
On page 208 example 3.8 in Hatcher, we are calculating the cohomology groups of a genus 4 non-orientable surface.
Hatcher makes the arguement that using the universal coefficient theorem, it is easy to find a basis for $H^1(N)$, we just dualize the basis for $H_1(N)$. He then goes on to talk about how we need to find a cocycle to represent each element of our basis for $H^1(N)$. He gives an example of a cocycle by using a dashed line and saying let this be the homomorphism that sends each edge this line intersects to 1, and it is easy to check that it is a coycle.
My question is, how can I tell that this cocycle represents the given homology class? What are the coboundry that I am modding out by?
In general, my question is quite broad I suppose. Given a cocycle, can someone give me general insight into understanding the cohomology class it belongs to? I'm having a hard time understanding the equivalence relation imposed by the coboundaries.