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I found the following in the wikipedia page for Euler class.

«If the rank $r$ is even, then this cohomology class $e(E) \cup e(E)$ equals the top Pontryagin class $p_{r/2}(E)$. Under the splitting principle, this corresponds to the square of the Vandermonde polynomial equaling the discriminant: the Euler class corresponds to the Vandermonde polynomial, the basic alternating polynomial, while the top Pontryagin class corresponds to the discriminant, a symmetric polynomial. More formally, the Euler class of a direct sum of line bundles is the Vandermonde polynomial (orientation determines the order of the line bundles up to sign), while top Pontryagin class is the discriminant.»

Can anybody clarify me this? In particular I don't understand the sense of "the Euler class of a direct sum of line bundles is the Vandermonde polynomial". I checked the references but there is nothing about that.

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up vote 6 down vote accepted

According to the Chern-Weil theory (see for example these lecture notes by Johan Dupont) , characteristic classes such as the Euler and Pontryagin classes correspond to invariant polynomials (under the bundle structure group) of the curvature form of the vector bundle. Basically, what the splitting principle tells us is that these polynomials can be "formally" written in terms of pullbacks of curvature forms of line bundles which correspond to the "eigenvalues" of the curvature form (viewed as an antisymmetric endomorphism).The invariant polynomial corresponding to the Euler class is the Vandermonede polynomial. For a given rank of the vector bundle, one can always write these polynomials in terms of the vector bundle curvature form and the use of the line bundle curvature forms is for simplifying the algebraic work.

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thank you for your clear answer. have you any reference for the splitting principle from this perspective? I have one remaining doubt: the square of the vandermonde polynomial is the discriminant, whereas the top pontryagin class should correspond to the top elementary symmetric polynomial, which is not the discriminant. Where is the missing link? – user1860 Sep 13 '10 at 12:45
A classical reference for the splitting principle is the book by Bott and Tu: differential forms in algebraic topology. Another reference containing many examples without proofs is the review article by: Eguchi, Gilkey and Hanson PHYSICS REPORTS 66. No. 6 (1980) 213—393. For your second question, I think that these polynomials coincide for Lie algebra valued differential forms, but I haven't done the calculation myself. – David Bar Moshe Sep 13 '10 at 13:16
As far as I know that statement about the Vandermonde polynomial is not true. See…. – Qiaochu Yuan Oct 17 '14 at 4:48

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