Let $f\colon X\to X$ be a map and $X$ simply connected. Let $M=X\times [0,1]/(x,0)\sim (f(x),1)$ be the mapping torus of $X$ from $f$. Calculate the fundamental group $\pi_1(M)$.
Here is my proof, but it is incomplete at parts and I need help to complete it:
Let $U$ be the image of $X \times (0,1)$ and $V$ be the image of $X\times [0,\frac 1 4) \cup X \times (\frac 3 4 , 1] \cup N \times [0,1]$ where $N$ is a contractible neighbourhood of our basepoint $x_0$ in $X$. (The basepoint can be chosen arbitrarily because the mappign torus is path connected)
Question 1: How can I argue that $N$ exists?
Let $W=U \cap V$, so $W$ is the image of $X\times (0,\frac 1 4) \cup X \times (\frac 3 4 , 1) \cup N \times (0,1)$. We see $U,V,W$ are path connected and $\mathring U \cup \mathring V = M$. So we can apply van Kampen.
Furthermore, we have $\pi_1(U)=1$ since $\pi_1(X)=1$ (note the "critical points" $0$ and $1$ are not in the interval).
Also, we see that $V$ is homotopy equivalent to $X \vee S^1$, so we get $\pi_1(V)=\mathbb Z$ by applying van Kampen (decompose it to $N \vee S^1$ and $X$).
Question 2: How to actually proof that $V$ is homotopy equivalent to $X \vee S^1$?
Also, we see that $\pi_1(W)=1$.
Question 3: How to proove $\pi_1(W)=1$?
Now we apply van Kampen again and get a pushout of groups:
$$\require{AMScd} \begin{CD} 1 @>{}>> 1\\ @V{}VV @VV{}V\\ \mathbb Z @>>{}> \pi_1(M)\end{CD}$$
So finally we get: $\pi_1(M)=\mathbb Z$.
