# Definition The g-handled torus $\Sigma_g$ is a torus with $g$ handles (i.e. $g$ holes). It can be written as the [[Connected Sum]] of the regular 2-[[torus]] $T^2$ as: $ \Sigma_g = \underbrace{T^2 \sharp \ldots \sharp T^2}_{\text{g times}} $ ![[Triple_torus_illustration.png|300]] The figure shows $\Sigma_3$