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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 15th 2011

    added brief definition/characterization to Chern class

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 29th 2014
    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 22nd 2016

    have spelled out a proof here, via induction over the Thom-Gysin sequence, of the basic fact H (BU(n))[c 1,,c n]H^\bullet(B U(n))\simeq \mathbb{Z}[ c_1,\cdots , c_n ]

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJun 24th 2016

    I have spelled out the proof of the splitting principle for Chern classes here (modulo the lemma that pullback in cohomology along BU(1) nBU(n)B U(1)^n \to B U(n) is injective).

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 24th 2016

    The proof that it is injective could go at a page dealing with maximal tori, presumably? Should hold for all BTBGBT \to BG, for reasonable GG

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJun 27th 2016

    Do you have a pointer to a proof (in more generality or not)? That argument in Kochmann’s book is a little shaky.

    • CommentRowNumber7.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 27th 2016

    I emailed you, but for others: Johannes Ebert, in this MO answer argues (following Dupont) that H *(BG)H *(BT)H^*(BG) \to H^*(BT) is, under the Chern-Weil isomorphism for compact (connected?) GG, Sym *𝔤 Sym *𝔱 Sym^{\ast} \mathfrak{g}^{\vee} \to Sym^{\ast} \mathfrak{t}^{\vee}, and in fact just multiplication by χ(G/T)\chi(G/T). This Euler characteristic is non-zero by a Lefshetz fixed-point argument involving the action of GG on G/TG/T.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJun 27th 2016

    Thanks. Do you have the energy left to make a note about this on some nnLab page? Best place might be splitting principle.

    • CommentRowNumber9.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 27th 2016

    Can do.

    • CommentRowNumber10.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 28th 2016

    I’ve added a little something to splitting principle, more just a record of the argument, with a citation of Dupont.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJun 28th 2016

    Thanks! Here is a pointer to the remark that you added.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJun 28th 2016

    In your paragraph I have made “transfer” point to Becker-Gottlieb transfer and changed the pointer Euler characteristic to Euler class.

    • CommentRowNumber13.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 28th 2016

    Hmm, I should point out at Euler class that the Euler characteristic is actually a number, not just a cohomology class.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJul 7th 2016

    I have also added proof of the Whitney sum formula for Chern classes, here.

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