Can some one tell me what are the prerequisites for learning characteristic classes as they are in book Foundations of Differential geometry by Kobayashi and Nomizu.


I only read first two chapters of that book which covers details about principal bundles and connections on principal bundles. I started reading this chapters on characteristic classes. It starts section on chern classes saying that



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> We consider the category of differential complex vector bundles over differentiable manifolds.
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> 


But they have not many details about complex vector bundles in any of previous chapters. So, I am not able to understand anything about this characteristic classes.


Any suggestion on references for complex vector bundles is welcome. I have seen Kobayashi’s another book On complex vector bundles. He never explains in detail what they are in that book as well. Let me know if there are any other prerequisites.



Prerequisites for reading characteristic classes
*The topology of fibre bundles* by Norman Steenrod tells you everything you want to know from the beginning, in a way that is appealing if you like to see a mathematical object as a patchwork of elementary pieces glued together via chart maps.


It does define characteristic classes as well, but for this part I would also definitely recommend the Milnor and Stasheff book.


