Characteristic classes in term of cocycles

Question

Giving a vector (principal) bundle is equivalent to give a family of cocycles ${g_{\beta \alpha}: U_\alpha\cap U_\beta \to G}$ where $G$ is the structure group of the bundle. Chern classes are powerful invariants of complex vector bundles, that can be described as

1. the pullback of characteristic elements via the classifying map of the bundle,
2. cohomology class of invariant polynomials in the curvature of any connection by Chern-Weyl theory.

Curiously, I am having an hard time in finding a clear definition of Chern Classes in term of the Cech cocycles of the bundle, which is one of the most practical ways of defining a bundle.

Is there any definition of the $k$-Chern class in term of the cocycles of the vector bundle?

Answer 1

A possible solution to the problem is to use the fact that cocycles representing characteristic classes can be seen as obstruction cocycles (see, e.g., Part III, "The Cohomology Theory of Bundles", in "The Topology of fiber bundles" by Steenrod). Unfortunately, these formulae seem to be non-algorithmic: for example, in the case of Stiefel–Whitney classes, one must be able to tell whether a certain cycle in a Stiefel manifold is $0$ in homology.

Other formulae can be found in "Čech Cocycles for Characteristic Classes", by Brylinski and McLaughlin, and, for the case of real vector bundles, in "Local formulae for Stiefel–Whitney classes" by McLaughlin. I am not too familiar with the first reference, but the formulae in the second one seem to also be non-algorithmic.