I want to use the technique from hatcher section 3.2 to compute the cup product structure of a punctured torus (with $\mathbb{Z}$ coefficient), but I found that I still don't know how to do this when I have new spaces. How should I start?
I know that a punctured torus deformation retracts to wedge of circles $S^1 \vee S^1$, this gives us a CW complex structure, one 0-cell $e$ and two 1-cells $a,b$. Then I denote the map acting on $a,b$ as $\alpha, \beta$, and I want to see what $\alpha \cup \alpha$, $\alpha \cup \beta$ and $\beta \cup \beta$ should be. Any advice?