5
$\begingroup$

I am working on this problem:

The mapping torus $T_f$ of a map $f : X → X$ is the quotient of $X×I$ obtained by identifying each point $(x,0)$ with $(f(x),1)$. In the case $X = S_1∨S_1$ with $f$ basepoint-preserving, compute a presentation for $π_1(T_f)$ in terms of the induced map $f_∗ : π_1(X) → π_1(X)$. Do the same when $X = S_1 ×S_1$.

There are 3 different versions of solutions that I find: version 1:

enter image description here

version 2: enter image description here

version 3: enter image description here

Now I am confused. By different methods they gives different conclusions. I think all of them sounds reasonable. So which is correct? Or are they equivalent? Why?

Thanks for any help!

$\endgroup$
1
  • $\begingroup$ The last one is correct, and in general this is what happens. The interval that closes to a circle in the gluing becomes a new generator $g$, and there are new 2-cells with boundary $g^{-1}xgf_\ast(x)$ for original generators $x$. $\endgroup$
    – Steve D
    Oct 13, 2020 at 20:07

1 Answer 1

1
$\begingroup$

Edited:

I was wrong. The last one is right by tracing the attaching $1-$cells. You can verify that by doing the case when $X = S^1\wedge S^1$ with $f=id$ easily, which you have probably done in question $1.2.8.$ In that case, the fundamental group should be $\mathbb{Z}\oplus(\mathbb{Z} * \mathbb{Z})$, which is given by the last solution, while with the first and second solution, the answer is $\mathbb{Z} * \mathbb{Z} * \mathbb{Z}$.

My Original Answer:

In comparison with the work that I have done with this problem, this first solution is right. As for the second one, for the case when $X = S^1 \wedge S^1$, the answer is the same as the first solution. The author did not do the case when $X= S^1\times S^1$. According to the third solution, there were some kind of conjugates involved, which I think might be a mistake due to messing up with the base-point, but I am not a hundred percent sure, since it is kind of hard to prove two group presentations are different. I hope someone could help with this.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.