I am computing homology group of space $X=S^{2}\cup T^{2}/\sim$ such that $S^{2}$ and $T^{2}$ gluing by there points on 2-sphere $s_{1}, s_{2}, s_{3}$ and three points on 2-torus $t_{1}, t_{2}, t_{3}$ where $s_{i}\sim t_{i}$. How can i found cell structure of this space and homology groups?
It looks like $S^{2}\lor S^{1}\lor S^{1}\lor S^{1}\lor S^{1}$ but I am not sure, maybe I should add one more $S^{2}$, I couldn't imagine it! I totally confused.
Thank you for your help!
You can use the Mayer Vietoris sequence with $X_1 = S^2$, $X_2 = T^2$, $X_1 \cap X_2 = \{a, b, c\}$ and $X = X_1 \cup X_2$.