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i am looking fo books on characteristic classes as main argument. I'd appreciate if you add a little comment on why you would study on it/them. Thank you.

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  • $\begingroup$ why not add a reference-request tag? $\endgroup$ – awllower Jun 30 '17 at 9:20
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    $\begingroup$ Just edited thank you! $\endgroup$ – user459067 Jun 30 '17 at 9:22
  • $\begingroup$ You probably need to indicate your background. Are you strong in algebraic topology, differential geometry, etc.? $\endgroup$ – Ted Shifrin Jun 30 '17 at 17:01
  • $\begingroup$ Well i am mainly interested to their applications to complex differential geometry. I've already attended courses on basic algebraic geometry and riemannian/kähler geometry $\endgroup$ – user459067 Jun 30 '17 at 17:22
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The following is a celebrated classic. J. Milnor is a Fields medalist, famous for the power of his mathematical thinking and the clarity and precision of his style.

Milnor, John W.; Stasheff, James D., Characteristic classes, Annals of Mathematics Studies. No.76. Princeton, N.J.: Princeton University Press and University of Tokyo Press. VII, 331 p. (1974). ZBL0298.57008.

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    $\begingroup$ This is the correct answer. $\endgroup$ – Nefertiti Jun 30 '17 at 9:25
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    $\begingroup$ @Nefertiti: I would disagree. I love Milnor and his expository style, but I do not think about characteristic classes at all the way he develops them, and I've used them my entire mathematical career. $\endgroup$ – Ted Shifrin Jun 30 '17 at 17:03
  • $\begingroup$ I like this book a lot, except the chapter that uses obstruction theory. $\endgroup$ – Thomas Rot Jul 3 '17 at 8:11
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For someone coming from the complex geometry perspective, I would suggest reading some combination of Chern (Complex Manifolds without Potential Theory), Wells (Differential Analysis on Complex Manifolds), and particularly Griffiths and Harris. Only the latter discusses the important interpretation of the $j$th Chern class of a vector bundle of rank $k$ in terms of the locus where $k-j+1$ generic smooth sections of the bundle become linearly dependent.

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Also D. Husemoller - Fibre Bundles, 3rd edition, part III is a good reference.

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    $\begingroup$ It's also worth looking at the true classic, Steenrod's The Topology of Fiber Bundles. $\endgroup$ – Ted Shifrin Jun 30 '17 at 18:15

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