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?
[Math] Classification of vector bundles over the torus
algebraic-topologynoncommutative-geometryoperator-algebrastopological-k-theory
Best Answer
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.