# Fundamental group of a genus-$2$ surface using van Kampen

I am trying to compute the fundamental group of a genus-$$2$$ surface using van Kampen. Let $$U_1$$ and $$U_2$$ be the component tori with $$U_1 \cap U_2 = U_0$$ homotopically equivalent to a circle. I am convinced that $$U_1$$ is homotopically equivalent to a torus missing a point, which has fundamental group $$\mathbb{Z} * \mathbb{Z}$$.

So the fundamental group of the genus-2 surface is $$\pi_1(U_1) \cdot \pi_1(U_2) / N = \left \langle a, b, c, d \right \rangle / N$$, where: $$N = \{ \left \langle i_{1}(w) i_{2}(w^{-1}) \right \rangle : w \in \pi_1(U_0) \cong \mathbb{Z} \}$$

Here, $$i_{1}:\pi_1(U_0) \rightarrow \pi_1(U_1)$$ and $$i_{2}:\pi_1(U_0) \rightarrow \pi_1(U_2)$$ are induced by the inclusions $$U_0 \hookrightarrow U_1$$ and $$U_0 \hookrightarrow U_2$$. I have seen the solution so I know what $$N$$ should be, but I am confused why this is the case. I think my confusion lies in the descriptions of $$i_1$$ and $$i_2$$. Does anyone have an intuitive explanation for what is going on here? Any help / advice is appreciated. Thanks

In the identification of $$U_1$$ as a punctured torus, the boundary circle is a simple loop around the puncture. Now to identify the class of this loop as an element of $$\mathbb{Z} * \mathbb{Z}$$, you need to follow this loop in the identification $$\pi_1(\text{punctured torus}) = \mathbb{Z} * \mathbb{Z}$$. For instance, if you do this by taking a deformation retract to a wedge of two circles, then applying the deformation retract to this specific loop should produce a loop that traverses each circle twice in opposite orientations, and equivalent to the commutator of the generators.
• I appreciate the response. You explained the part I did not initially understand: that I should find the equivalence class of the loop around the puncture in terms of the generators of $\mathbb{Z} * \mathbb{Z}$. However, I agree with @RyleeLyman. I believe a nice way to see this is to start with a rectangle (as in the usual construction of a torus) with a puncture in the middle. Then the loop around the puncture is simply a loop around the edges of the rectangle: $aba^{-1} b^{-1}$. Jan 12, 2020 at 23:14