Calculating fundamental group of adjunction space with linear transformation.

Question

$X = D^{2} \times S^{1} \cup_{f} S^{1} \times D^{2}$, where $f : S^{1} \times S^{1} \to S^{1} \times S^{1}$ is a map induced by the linear map on $\mathbf{R}^{2}$ given by the matrix $$\left( \begin{array} {cc} a & b \\ c & d \end{array} \right)$$ where $a, b, c, d$ are integers. How can I represent the fundamental group $\pi_{1}(X)$ in terms of $a, b, c, d$? Please give me some hints.

Answer 1

Hint: $(D^2\times S^1)\cap(S^1\times D^2)$ is a torus which we know the fundamental group of. We also know the fundamental groups of each component of the adjunction. Try using Van-Kampen's theorem - the amalgamation of the free product will be dependent on the map $f$ so be careful when considering the quotient.