Let $f\colon X\to X$ be a map and $X$ simply connected. Let $M_f=X\times [0,1]/(x,0)\sim (f(x),1)$ be the mapping torus of $X$ from $f$.
I want to calculate the fundamental group $\pi_1(M_f)$. Note that this is not a homework.
My intuition tells me that the fundamental group is trivial since $X$ is simply connected. This is my approach:
I want to apply van Kampen, so I define $U= \text{image of } [0, \frac 1 2+\epsilon)$ and $V= \text{image of } (\frac 1 2 , 1]$. We set $W = U \cap V = (\frac 1 2 , \frac 1 2 +\epsilon)$. We have that $U,V$ and $W$ are path connected and $\mathring U \cup \mathring V = M$, so we are allowed to apply van Kampen.
Now comes the part where I'm not sure.
I think we have $\pi_1(U) = 1$ and $\pi_1(V)=1$ simply because $\pi_1(X)=1$. Therefore, by van Kampen, we get $\pi_1(M_f)=1$.
Is this correct? I have a gut feeling that something is wrong. This seems clueless and too simple.
If not, how to apply van Kampen then?