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The first Chern number $\cal C$ is known to be related to various physical objects. Gauge fields are known as connections of some principle bundles. In particular, principle $U(1)$ bundle is said to be classified by first Chern number. In terms of electromagnetic field, ${\cal C} \neq 0$ is equivalent to the existance of monopoles. In the case of integer quantum Hall states, Chern number is simply the Hall conductance up to a constant.

In both physical problems, Chern number is related to vorticity -- a quantized value (first case, Dirac's string argument and second, vortices in magnetic Brillion zone).

Then my questions:

  1. What was the "physical" picture in Chern's mind when he originally "dreamed up" the theory? (Maybe knots, but how?)
  2. If I want to learn how Chern classified $U(1)$ bundles using integers (first Chern number), which books or papers should I refer to?

Notes:

My point is that mathematical theorems are not God-given but arose from concrete problems. I was asking what was the original problem that Chern solved, from which he codified the general theorems?

And Chern number seems related to vorticity and then what are the corresponding vortices in his problem?

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    $\begingroup$ Regarding 1., in the early 1990s, Chern gave an undergraduate lecture at Berkeley on characteristic classes. (Stated prerequisites: "Multivariable calculus.") Faculty made up the majority of the audience, graduate students most of the remainder. To quote Chern, explaining the origins of the characteristic forms of a connection in an Hermitian vector bundle over a CW complex, "I made the elementary observation that [the cohomology ring of] the complex Grassmannian has no torsion." $\endgroup$ – Andrew D. Hwang Dec 7 '16 at 0:07
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Chern classes (which are more general then Chern numbers and as well necessary for their construction)are particular cases of characteristic classes. Generally speaking characteristic classes are ways to associate to each vector bundle (complex or real) some classes in cohomologies of the base. They show, roughly speaking, how non-trivial the bundle is (one should be rather cautious about this interpretation: for example vanishing of characteristic classes do not always imply triviality of the bundle).

Chern classes are characteristic classes for complex vector bundles (and complex line bundles can always be regarded as principal $U(1)$-bundles).

The classical (and amazing) introduction to the theory is J.Milnor & J.D. Stasheff's book "Characteristic classes".

I believe characteristic classes where first introduced by Stiefel and Whitney in the middle of 30's -- they probably studied vector fields on manifolds. I'm not sure who and when introduced Chern classes or developed the general theory of characteristic classes via classyfying maps, but I suspect that in some form they where already known to Chern's teacher Èlie Cartan and/or his co-author André Weil. The Chern's development to this theory is giving purely differentialy geometric description of Chern classes as integrals of the curvature form. This approach is now sometimes called Chern-Weil theory.

Answering your second question -- you can find it in almost any basic book on either algebraic geometry, differential geometry or algebraic topology (with possibly different points of view for different choice of subject). I'm sure the already mentioned Milnos&Stasheff's contains it.

Indeed the construction of Chern classes is rather simple, inspite of the fact that one should be familiar (at least on basic properties-level) with sheaves and their cohomologies.

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