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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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have you looked at the problems in Milnor and Stashef? –  Eric O. Korman Apr 13 '12 at 1:20
    
Some of them yes, that's where I'm learning the basics. Do you know any good examples that aren't emphasized in this book? –  you Apr 13 '12 at 2:21
    
You can look through the chap IV, characteristic classes of Differential forms in Algebraic topology by R. Bott and L. W. Tu –  Ehsan M. Kermani Apr 13 '12 at 6:10

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