Prove that a fundamental group is equal to a semidirect product.

Question

Let $X$ be a connected space that admits universal covering $E$ and $f:X \to X $ an homeomorphism . Now let’s call $Y=(X\times [0,1])/\sim$ where $\sim$ is the relation generated by $(0,x)\sim(1,f(x))$ for all $x\in X$. The request is to show that the fundamental group of $Y$ is a semidirect product between the fundamental group of $X$ and $\Bbb Z$.

I tried to demonstrate the normality of $\Bbb Z$ in $\pi_1 (Y)$ but I didn’t reach any result. Any hint or solution will be very appreciated!

Answer 1

$\mathbb{Z}$ is not normal, $\pi_1(X)$ is. And the cyclic group acts on $\pi_1(X)$ naturally using $f$.