# Examples of fundamental group of mapping torus

The original question was askes here.

I donot know how to apply or compute any example. I think a specified explanation will be helpful.

Let $M=\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$ and $f=A\in\mathrm{SL}(2,\mathbb{Z})$ be the quotient action on $\mathbb{T}^2$ induced from $f(x)=Ax$. Since $A$ preserves the lattice $\mathbb{Z}^2$, $f$ is well defined.

We know that $\pi_1(\mathbb{T}^2)=\langle\alpha,\beta\rangle\cong\mathbb{Z}^2$. What can we see about $\pi_1(M_f)$?

HJRW answered there that $\pi_1(M_f)\cong\pi_1(X)\rtimes_{f_*}\mathbb{Z}$. Would you write down explicitly the multiplication of this semiproduct in this example?

Thanks!

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It is exactly as you expect: if $t$ generates your copy of $\mathbb{Z}$, then $t^{-1}\alpha t = A\alpha$, etc. – user641 Oct 11 '11 at 6:06
Let $u,v\in\pi_1(X)\cong\mathbb{Z}^2$ and $i,j\in \mathbb{Z}$. Then $(u,i)+(v,j)=(u+A^iv,i+j)$. Hope some someone can help me to move your comment to an answer. Thank you! – Pengfei Oct 11 '11 at 12:55