Classification of vector bundles over the torus

Question

In M. Rieffel's paper "The Cancellation Theorem for projective modules over irrational rotation $C^*$-algebras", he classifies finitely generated projective modules over the $C^*$-algebra $C(\mathbb{T}^2)$, which by the Serre-Swan theorem should be equivalent to classify vector bundles over $\mathbb{T}^2$. Does anyone know a reference where this classification is stated in terms of vector bundles, and where the parameters are interpreted more explicitly as the rank and first Chern number of the vector bundle?

Answer 1

It was not me who posted in the comments (that got deleted), but let me answer your questions in the comments.

Hatcher's vector bundles and $K$ theory discusses clutching functions.

Complex vector bundles over a torus minus a point correspond to complex vector bundles over the wedge of two circles. Such vector bundles are the same thing as two vector bundles over $S^1$. Complex vector bundles over a circle are trivial as $GL(\mathbb{C})$ is connected. Hence complex vector bundles over the torus minus a point are trivial. The bundle trivializes over a disc, and over the torus minus a point. The information of the bundle is then contained in how these trivializations are matched in their intersection, which is homotopy equivalent to a circle $S^1$. The matching can be understood as a map $S^1\rightarrow GL_n(\mathbb{C})$. There are $\mathbb{Z}$ such maps, and I believe these correspond to the first chern class of your bundle.