Why is it useful to study vector bundles?
I have this question coming from an earlier Qiaochu's post. Some answers there, especially David Lehavi's one, were drawing the analogy bundles and varieties versus modules and rings. So I am just wondering, is there any big reason why the study of bundles would give information about varieties? (I suppose that for this matter, I should actually replace varieties by manifolds?)
I have heard about some invariants, like the Picard group for complex manifolds. But given my inexperience in these concepts, I don't really know why they should be important. So for those who are thinking about "cooking up invariants", some more detailed explanations on why they are useful (and hopefully some elementary examples!) would be appreciated. Thanks!