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:
version 2:
version 3:
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!