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

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

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.

  • $\begingroup$ I have asked question in stack exchange but I have not received any reply $\endgroup$ Jan 26, 2018 at 11:38
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    $\begingroup$ Probably worth reading Characteristic Classes by Milnor and Stasheff: citeseerx.ist.psu.edu/viewdoc/… $\endgroup$
    – Neal
    Jan 26, 2018 at 14:10
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    $\begingroup$ I have had a good experience with Chapter 2 of "Complex Geometry" by Huybrechts for laying the groundwork of complex vector bundles. If the style of Kobayashi and Nomizu becomes difficult, a great alternative is Capter 4 of "Differential forms in algebraic topology" by Bott and Tu. $\endgroup$ Jan 26, 2018 at 14:19
  • $\begingroup$ @Neal I have seen Milnor and Stasheff book. They do not define characteristic classes as in Kobayashi book. In Kobayashi they define chern classes using some axioms saying, chern classes are elements of cohomology ring of base manifold of vector bundle satisfying some conditions. I would be interested in some reference which follow this idea. In Milnors book they define chern class of a vector bundle to be inverse image of chern clas of another vector bundle.. $\endgroup$ Jan 26, 2018 at 14:53
  • $\begingroup$ @JānisLazovskis Hello. In their book Bott and Tu, they assume so much. I have not even heard of something called Leray Hirsch theorem or something called universal bundle.. I will definitely see that complex geometry book, thank you $\endgroup$ Jan 26, 2018 at 15:05

1 Answer 1


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.

  • $\begingroup$ Hello, what do you suggest me to read in Steenrod book specifically? $\endgroup$ Jan 26, 2018 at 18:34
  • $\begingroup$ It depends on your current knowledge on fibre bundles, so I think the most efficient way is to try and read how Steenrod defines characteristic classes, and then if there is anything that is unclear you can be sure that it is explained in detail in a previous section. $\endgroup$ Jan 26, 2018 at 18:42
  • $\begingroup$ Just had a look, the Stiefel classes are section 39 while the Chern classes are in section 41.3. $\endgroup$ Jan 26, 2018 at 18:49

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