# Classification of vector bundles over the torus

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?

• Thanks. Do you know some good references on clutching functions? – Ulrik Mar 21 '16 at 20:47
• By the way in your first comment, did you mean that we start with some bundle over the torus, then form the pullback along the inclusion of the torus with a point removed into the torus? In that case, why is this pullback bundle trivial? – Ulrik Mar 22 '16 at 15:35

## 1 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.

• There are a number of steps here that I don't follow because of my limited knowledge of vector bundle theory. Is the torus minus a point homotopy equivalent to the wedge of two circles, and is it true in general that a vector bundle over a wedge of two spaces is the same as one vector bundle over each of them? – Ulrik Apr 10 '16 at 12:42
• Yes the torus minus a point is homotopic to the wedge of two circles. Yes a vector bundle over the wedge of two spaces can be identified with a vector bundle over each one of them. Maybe there is a small issue with contractability of a neighborhood around the basepoints, I would have to think about that. – Thomas Rot Apr 11 '16 at 21:38