I'm trying to solve the following problem: let $X$ be the space obtained from the disjoint union of two 2-tori $A,B$ be identifying them along 2 pairs points (resp. three pairs of points).
If we identify two pairs of points, we obtain a space which is homotopy equivalent to a wedge sum $A\vee \mathbb{S}^1 \vee B$, so Van Kampen's theorem solves the problem.
I am having a hard time picturing $X$ when we identify 3 pairs of points though and I could use some insight. Is $X$ simply homotopy equivalent to $A\vee \mathbb{S}^1 \vee \mathbb{S}^1 \vee B$?
That looks right.
And the obvious generalization to $n$-pairs should be right also, using a wedge sum of $n-1$ copies of $\mathbb{S}^1$.
To visualize the homotopy equivalence, let the identified points be $p_0,\ldots,p_{n-1} \in X$. Embed trees $T_A \subset A$ and $T_B \subset B$ each with a vertex of valence $n-1$ at $p_0$ and $n-1$ vertices of valence $1$ at $p_1,\ldots,p_{n-1}$. So $T_A \cup T_B$ is homeomorphic to a wedge $\vee^{n-1} \mathbb{S}^1$ of $n-1$ copies of $\mathbb{S}^1$. Then consider the wedge sum $A \vee \bigl( \vee^{n-1} \mathbb{S}^1 \bigr) \vee B$ which is defined by identifying all copies of $p_0$. Define a map $A \vee \bigl( \vee^{n-1} \mathbb{S}^1 \bigr) \vee B \to X$, which restricts to the ``identity maps'' $A \mapsto A$, $B \mapsto B$, and which maps $\vee^{n-1} \mathbb{S}^1$ homeomorphically to $T_A \cup T_B$. This map is a homotopy equivalence.