# Fundamental Group of Genus 2 Surface

1. Aug 15, 2011

### jgens

1. The problem statement, all variables and given/known data

Given two tori, the two-holed torus can be formed by removing the interior of a small disk from each and identifying the boundaries. Compute the fundamental group of the two torus.

2. Relevant equations

$$\pi_1(T^2) = \mathbb{Z} \times \mathbb{Z}$$

The van Kampen Theorem

3. The attempt at a solution

So, I know the standard way of tackling this problem, by decomposing the torus into a fundamental polygon and applying van Kampen's Theorem from there. However, I was wondering if there was some way to tackle the problem which appealed more to the definition of the connected sum, rather than the fundamental polygon.

For example, since the disk is homotopy equivalent to a point, would we obtain a homotopy equivalent structure by simply removing a point from each torus and identifying these points?