# What is the fundamental group of the $3$-manifold bounded by a genus-$2$ torus?

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 $$\pi_1(X)$$.

• Your calculation of $\pi_1(\mathbb R^3\setminus X)$ is not correct. How did you come to this conclusion? – Cheerful Parsnip Feb 27 at 6:43
• 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. – Travis Willse Feb 27 at 7:23
• I figured out a mistake in ny drawing. I understand the problem now. – 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 $$\langle a,b\rangle$$.