Fundamental Group of Genus 2 Surface

[jgens]

Homework Statement

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.

Homework Equations

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

The van Kampen Theorem

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?

 

[lippyka]

If so, then would the fundamental group be that of a connected sum of two tori, i.e. $\pi_1(T^2 \# T^2) = \mathbb{Z} \times \mathbb{Z}$? I am looking for any insight on this approach, and whether or not it is valid. Thanks!