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Question 11 of Section 1.2 of Hatcher asks us to compute the fundamental group of $T_f$, where $T_f$ is the mapping torus. In particular, when $X = S^1 \vee S^1$ and $f_{\star} : \pi_1(X) \to \pi_1(X)$.

It is assumed that $f: X \to X$ preserves base points. We therefore have that the unit interval $[0,1]$ becomes a loop that is homeomorphic to $S^1$. Since the base point is preserved, it follows that it is attached at the base point of $S^1 \vee S^1$. This yields that $S^1 \vee S^1 \vee S^1$ is part of $T_f$.

I'm unsure of how to proceed with obtaining the rest of $T_f$ however.

  • There is the assumption on $f_*$ missing in the second sentence. – user39082 Aug 11 '16 at 06:31
  • The question in generality has already been answered in http://math.stackexchange.com/questions/39589/fundamental-group-of-mapping-torus?rq=1 – user39082 Aug 11 '16 at 06:32
  • @user39082 This doesn't help. I do not want to consider the problem using a semidirect product. I want to use my current solution. I just need guidance as to how to construct the rest of the mapping torus. –  Aug 11 '16 at 07:17
  • Am I being dense, or having you not actually said what $f$ is? – Dan Rust Aug 11 '16 at 12:24
  • You may have misread the question. I quote exactly from Hatcher (my emphasis added): "Compute a presentation for $\pi_1(T_f)$ in terms of the induced map $f_\star : \pi_1(X) \to \pi_1(X)$". – Lee Mosher Aug 11 '16 at 14:43
  • Have a look at pg4 of http://www.math.northwestern.edu/~pgoerss/math440/probset2/probset2.pdf – James Aug 19 '16 at 11:19

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