Use van Kampens theorem to compute the fundamental group of the following space: $A$ is a torus with an open disk $D$ removed. Let $f:\partial B \rightarrow \partial A$ be a map from the boundary of a 2-ball $B$ to the boundary of $A$ winding twice (that is, double coverin gmap from a circle to a circle). Let $X$ be the space joining the 2-cell $B$ by the map $f$. What is the fundamental group of $X$?
Edit: Using proposition 1.26 from Hatcher it says that if $Y$ is obtained from $X$ by attaching a 2-cell, then the inclusion $X \rightarrow Y$ induces a surjection $\pi_{1}(X,x_{0}) \rightarrow \pi_{1}(Y,x_{0})$ with kernel $N$ (which should be the attaching map) in this case, so $\pi_{1}(Y) \simeq \pi_{1}(X)/N$.
So my thought is that since the torus with a disk removed is homotopy equivalent to the wedge of two circles, then we will mod by the attaching loop (say $w^{2}$) so the fundamental group is $(\mathbb{Z}*\mathbb{Z})/<w^{2}>$, where $<w^{2}>$ denotes the normal subgroup generated by $w^{2}$ ? My solution doesn't use van Kampens theorem directly though.