In my mind, algebraic topology is comprised of two components:
- Chain complex information, which is intrinsic information concerning how your object may be built up out of simple "lego blocks".
- Characteristic classes (bundle information) which give information on how your object might stably embed in some sufficiently big standard object.
I have no corresponding intuitive understanding of what the "natural setting" for characteristic classes should be. The classical theory looks to me like a concession to the sad fact that, at its very basis, manifolds are locally modeled on Euclidean space and are not intrinsically defined objects. This is reflected in the central role played by specific concrete spaces such as the Thom spaces MSO(n), the real and complex Grassmanians used to define Wu classes, and the classifying spaces BU and BO concerning which we have Bott periodicity.
I have no corresponding intuitive understanding of what the "natural setting" for characteristic classes should be. The classical theory looks to me like a concession to the sad fact that, at its very basis, manifolds are locally modeled on Euclidean space and are not intrinsically defined objects. This is reflected in the central role played by specific concrete spaces such as the Thom spaces MSO(n), the real and complex Grassmanians used to define Wu classes, and the classifying spaces BU and BO concerning which we have Bott periodicity.
I realize that I have no understanding of any of this. Part of this feeling is because I really don't understand what forces us to consider these specific concrete spaces, to the exclusion of all others. If constants appearing in physics ought to be conceptually explained, I'd like to understand these "constants" in mathematics. Can one work with characteristic classes in a more general setting, to parallel abelian categories? What about over number fields, over arbitrary rings, or in finite characteristic? Can I replace Lie groups such as SO(n), U(n), and O(n) by groups of Lie type for instance, and still have a "useful" theory?
My question is then:
What is the most general categorical setting for a "useful" theory of characteristic classes? In particular, are all of those special concrete spaces really necessary, and if so, why?
