Let $( \mathbb{S}^{1} × \mathbb{S}^{1} )∪e^2$ be the adjunction space of the torus and a 2-cell, where the attachment is given by the attaching map $f:\mathbb{S}^{1} \to ( \mathbb{S}^{1} × \mathbb{S}^{1} )$ such that$f(ζ)= (ζ^{2},ζ^{3})$. Compute the fundamental group of this space.
By a consequence of Seifert-Van Kampen theorem I know that the fundamental group of this space is the fundamental group of the quotient space of the torus (which I know is $\mathbb{Z}$x$ \mathbb{Z}$), divided by its kernel (which is the normal subgroup generated by $\gamma$ where $\gamma=f_{*} (\alpha)$ and $\alpha$ is the generator of the infinitely cyclic group of the fundamental group of $\mathbb{S}^{1}$).
I'm having trouble with determining what is this last kernel, or at least how to simplify this last expression. Also, I don't know what am I supposed to obtain as a result, so any help would be appreciated.
Thank you so much!