2
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ Thanks. Do you know some good references on clutching functions? $\endgroup$ – Ulrik Mar 21 '16 at 20:47
  • $\begingroup$ 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? $\endgroup$ – Ulrik Mar 22 '16 at 15:35
2
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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? $\endgroup$ – Ulrik Apr 10 '16 at 12:42
  • $\begingroup$ 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. $\endgroup$ – Thomas Rot Apr 11 '16 at 21:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.