Let $V_g$ be a compact connected orientable genus $g$ surface without boundary. Further, let $X$ be $V_2$ with two disjoint open disks removed, and let $X'$ be $X$ with three distinct points identified. Calculate $H^*(X,\mathbb Z)$ and $H^∗(X',\mathbb Z)$.
I was thinking of drawing a $4g$-hedron and after gluing it we will get $1$ vertex and $2g$ edges with identified vertices and diagonal edges, which will be a wedge product of $2g + 1$ circles. I don't know whether my process is right. Please help.