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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeAug 18th 2010
• (edited Aug 18th 2010)

tried to polish a bit the matrial at Chern-Weil theory.

(not that there is much, yet, but still)

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeSep 19th 2010

I pasted some historical comments and references (but still incomplete currently) provided by Jim Stasheff into Chern-Weil theory. More to come.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeSep 19th 2010

Yet more ancient references, this time kindly dug out by Alan Hatcher.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeMay 9th 2011

I have expanded the Idea-section at Chern-Weil theory, giving a more detailed survey of the main constructions.

(this is a spin-off of me staring to write an “Introduction and Survey”-section on my personal web at infinity-Chern-Simons theory (schreiber))

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeAug 27th 2020

I have finally dug through the original citations for the “Chern-Weil” constructions. Have added the following to the entry:

The differential-geometric “Chern-Weil”-construction (evaluating curvature 2-forms of connections in invariant polynomials) is due to

• Henri Cartan, Section 7 of: Notions d’algébre différentielle; applications aux groupes de Lie et aux variétés où opère un groupe de Lie, Coll. Topologie Algébrique Bruxelles (May 1950) 15-28 (numdam:SHC_1949-1950__2__A18_0)

and around equation (10) of:

• {#Chern50} Shiing-shen Chern, Differential geometry of fiber bundles, in: Proceedings of the International Congress of Mathematicians, Cambridge, Mass., (August-September 1950), vol. 2, pages 397-411, Amer. Math. Soc., Providence, R. I. (1952) (pdf)

It is the independence of this construction under the choice of connection which Chern 50 atributes (below (10)) to

• André Weil, Géométrie différentielle des espaces fibres, unpublished

But the main result of Chern 50 is that this differential-geometric “Chern-Weil” construction is equivalent to the topological (homotopy theoretic) construction of pulling back the universal characteristic classes from the classifying space $B G$ along the classifying map of the given principal bundle:

This claim is equation (15) in Chern 50, using (quoting from the same page):

methods initiated by E. Cartan and recently developed with success by H. Cartan, Chevalley, Koszul, Leray, and Weil [13]

Here reference 13 is:

• Jean-Louis Koszul, Homologie et cohomologie des algebres de Lie, Bull. Soc. Math. France vol. 78 (1950) pp. 65-127

Later, an independent proof of the universal topological “Chern-Weil”-construction $inv(\mathfrak{g}) \to H^\bullet(B G)$ is given in:

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeMar 24th 2021

added pointer to:

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