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I tagged this as "homework" because my supervisor told me I need to be better at computing characteristic classes.

The classic examples I can think of are tangent bundles and tautological line bundles. I'm wondering if people can suggest some illustrative problems to solve, computing characteristic classes of fibre bundles.

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    You can look through the chap IV, characteristic classes of *Differential forms in Algebraic topology* by R. Bott and L. W. Tu2012-04-13

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