I would like to construct 2D vector bundles over the punctured torus, but I don't know a lot of K-theory. Over the square, there can only be the trivial bundle, but now since $\pi_1(\mathbb{T}^2\backslash \{ pt\})= \langle x,y \rangle $ (the free group on two elements) I simply need two matrices $x,y \in GL(V)$ where $V = \mathbb{R}^2$ is the fiber over the bundle.

It looks like I have shown that vector bundles over a surface are parameterized by representations of the fundamental group $\pi_1(\mathbb{T}^2 \backslash \{ pt\})$ over $V$. I wonder that many of these representations are the same... that you can find some element $U \in GL(V)$ so that if $x,y$ determine a vector bundle the, $UxU^{-1}, UyU^{-1}$ define an equivalent bundle.

In turn, this looks like I am asking about representations of a quiver $\circ \to^2 \circ$ of pairs of matrices up to simultaneous similarity.

Is this the right way to classify spaces vector bundles over Riemann surfaces? What is the relationship between vector bundles over Riemann surfaces, the representation variety and quiver representations?