This question is from an old exam. I was completely lost on it and not sure where to start- hoping for even a point in the right direction.
Let $\Sigma_2$ be the genus 2 surface and $C\subset\Sigma_2$ a separating circle. So that $\Sigma_2/C$ is the wedge of two tori. Study the quotient map and your knowledge of the cup product structure on a torus to compute the cohomology ring of $\Sigma_2$.
Now I'm pretty shaky on using the cup product. I do know the cohomology ring of the torus is isomorphic to the exterior algebra on two variables. I don't know where the quotient map comes into play here at all. I have a feeling that degree comes into play somehow, but am not sure.
Any help would be much appreciated. Thank you.