How does one introduce, or how were you introduced to characteristic classes?

You can assume that the student is comfortable with principal bundles and connections on principal bundles.

I am not asking for references for characteristic classes, I have references but I am not able to really understand them.

What I am requesting is your way of introducing characteristic classes.

  • $\begingroup$ There is no tag for singular cohomology so could not add it here. Wikipedia article is not doing any good for me. I can make it community wiki if necessary. $\endgroup$ Feb 13, 2018 at 13:57
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    $\begingroup$ math.stackexchange.com/questions/2648635/… $\endgroup$
    – Thomas Rot
    Feb 13, 2018 at 14:19
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    $\begingroup$ Please do not cross-post simultaneously to MO and MSE. You should wait at least a week for answers on MSE before considering asking again on MO. $\endgroup$ Feb 13, 2018 at 15:45
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    $\begingroup$ My preference is the obstruction theoretic techniques -- chapter X in the Milnor-Stasheff notes. If that is a person's first exposure to characteristic classes, perhaps Whitehead's Algebraic Topology textbook would be one of the friendlier introductions. $\endgroup$ Feb 13, 2018 at 16:03
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    $\begingroup$ This question seems vague and with the assumption that connections are a prerequisite for characteristic classes. I, for example, learnt about connections years after characteristic classes (that I learned via the study of the cohomology of classifying spaces) $\endgroup$ Feb 13, 2018 at 16:41

1 Answer 1


I would follow Chern in his short paper Vector bundles with a connection, where he introduces characteristic classes as polynomials in the curvature of a connection, and then shows (easily) their basic properties, and applies them to prove Gauss-Bonnet on surfaces. A student who has seen principal bundles and connections can grasp this in under an hour. (The main problem is to get to an application before too long, so that students are not confused as to the purpose of these strange expressions.) Then you can demonstrate the general properties of the classes. In a subsequent lecture, show that vector bundles are pulled back from Grassmannians, so that you can explain how to define integer characteristic classes, and show that over complex number coefficients they agree with the Chern classes as defined by Chern. Finally, you might discuss classifying spaces, depending on the students' backgrounds.

  • $\begingroup$ I apologize to give you feel that I am a teacher and I want to teach characteristic classes. I am not a teacher, I am a student who is trying to learn characteristic classes. Can you give link for that paper you are talking about. Google does not give anything on searching vector bundles with a connection. Thanks for the answer. :) $\endgroup$ Feb 13, 2018 at 16:12
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    $\begingroup$ @cello S. Chern, Vector bundles with a connection, in Global Differential Geometry, Studies in Mathematics, Mathematical Association of America, 1989, Vol. 27, pp. 1-26. 3. $\endgroup$ Feb 13, 2018 at 16:41