Q:

Find the fundamental group of the torus with an open disc removed

I'm trying to find a fundamental group of $\mathbb{T} \setminus \mathbb{D}$, the $2$-torus $\mathbb{T}$ with an open disc $\mathbb{D}$ removed. Any help would be really appreciated. 

A:

Big hint: View the torus as the square $I^2/\sim$ with the usual equivalence relation $\sim$ identifying opposite edges, and then removes a small disk from the interior of $I^2$. Can you see how this space is homotopy equivalent to a wedge of some number of circles? (I'll leave you to figure out how many)

