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nLab > Latest Changes: Chern-Weil theory

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  1.  
    • 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    • André Weil, Géométrie différentielle des espaces fibres, unpublished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 BGB 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(𝔤)→H •(BG)inv(\mathfrak{g}) \to H^\bullet(B G) is given in:

    diff, v23, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMar 24th 2021

    added pointer to:

    diff, v29, current

1 to 6 of 6

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