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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!

Alessandro Cigna
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1 Answers1

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

markvs
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  • And how can I use the action of $(f)$ on $\pi_1(X)$ for $\pi_1(Y)$? Thanks. – Alessandro Cigna Jul 23 '20 at 16:09
  • Take any loop from the fundamental group, and apply $f$ to it. You get another loop. – markvs Jul 23 '20 at 16:11
  • Of course you need to use the fact that the space $X$ is path connected. That construction is called mapping torus, see this question: https://math.stackexchange.com/questions/39589/fundamental-group-of-mapping-torus – markvs Jul 23 '20 at 16:28