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Let $f\colon X\to X$ be a homeomorphism between a CW-complex $X$ and iteself.

Let $M_f=X\times [0,1]/(x,0)\sim (f(x),1)$, mapping torus of $X$ from $f$.

I want to calculate the fundamental group $\pi_1(M_f)$ of $M_f$ in terms of $\pi_1(X)$ and $f_*\colon \pi_1(X)\to \pi_1(X)$.

Are there any hint to do this?

p.s: This is not the homework problem.

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1 Answer

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You can use van Kampen's Theorem. The upshot is that you get a semi-direct product:

$\pi_1M_f\cong\pi_1X\rtimes_{f_*}\mathbb{Z}$.

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I took $U=\textrm{image of } X\times [0,1/2]$ and $V=\textrm{image of } X\times [1/2,1]$ but it doesn't work because $U\cap V$ is not connected. – Topologieeeee May 17 '11 at 13:20
Indeed, you need to be a little careful. The cleanest way is to use the the fundamental groupoid. – HJRW May 17 '11 at 13:24
Frankly, I'm not familiar with the usage of the fundamental groupoid. Can you suggest me some useful reference? – Topologieeeee May 17 '11 at 13:57
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I wouldn't use May's book as a first look at the fundamental groupoid (or anything else for that matter). It's a great book, but not good for an introduction. Brown's Topology and Groupoids would probably be better. – Josh May 17 '11 at 17:16
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You don't need to use the fundamental groupoid for for this computation. As long as $f$ is a cellular map you have a natural CW-structure on $M_f$, so you can compute $\pi_1 M_f$ from that CW-structure. Seifert-Van Kampen kicks in and you get the desired result. – Ryan Budney May 17 '11 at 22:39
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