As in the question, let $X$ be the $3$-manifold bounded by $\partial X$ which is a torus of genus $2$.What is $\pi_1(X)$?

I noted that $\pi_1(\mathbb{R} ^3\backslash X) \cong \langle a, b | aba^{-1} b^{-1} \rangle$ but I cannot see any commutator relation in $\pi_1(X) $. However I also know that $\pi_1(\partial X) $ has a relation between its four generators. I just cannot see why that does not translate to $\pi_1(X)$.

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    $\begingroup$ Your calculation of $\pi_1(\mathbb R^3\setminus X)$ is not correct. How did you come to this conclusion? $\endgroup$ – Cheerful Parsnip Feb 27 at 6:43
  • $\begingroup$ Any representative of either of the two generators $[\gamma]$ of $\pi_1(\partial X)$ that goes through one of the "holes" of $\partial X$ is a contractible path in $X$, i.e. the homomorphism $\pi_1(\partial X) \to \pi_1(X)$ induced by the inclusion $\partial X \hookrightarrow X$ is trivial. $\endgroup$ – Travis Willse Feb 27 at 7:23
  • $\begingroup$ I figured out a mistake in ny drawing. I understand the problem now. $\endgroup$ – wilsonw Feb 27 at 11:18

Your manifold deformation retracts to a figure-eight (or, perhaps more easily, to a figure-$\theta$). Meaning it has fundamental group $\langle a,b\rangle$.


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