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added brief definition/characterization to Chern class
have spelled out a proof here, via induction over the Thom-Gysin sequence, of the basic fact $H^\bullet(B U(n))\simeq \mathbb{Z}[ c_1,\cdots , c_n ]$
I have spelled out the proof of the splitting principle for Chern classes here (modulo the lemma that pullback in cohomology along $B U(1)^n \to B U(n)$ is injective).
The proof that it is injective could go at a page dealing with maximal tori, presumably? Should hold for all $BT \to BG$, for reasonable $G$…
Do you have a pointer to a proof (in more generality or not)? That argument in Kochmann’s book is a little shaky.
I emailed you, but for others: Johannes Ebert, in this MO answer argues (following Dupont) that $H^*(BG) \to H^*(BT)$ is, under the Chern-Weil isomorphism for compact (connected?) $G$, $Sym^{\ast} \mathfrak{g}^{\vee} \to Sym^{\ast} \mathfrak{t}^{\vee}$, and in fact just multiplication by $\chi(G/T)$. This Euler characteristic is non-zero by a Lefshetz fixed-point argument involving the action of $G$ on $G/T$.
Thanks. Do you have the energy left to make a note about this on some $n$Lab page? Best place might be splitting principle.
Can do.
I’ve added a little something to splitting principle, more just a record of the argument, with a citation of Dupont.
Thanks! Here is a pointer to the remark that you added.
In your paragraph I have made “transfer” point to Becker-Gottlieb transfer and changed the pointer Euler characteristic to Euler class.
Hmm, I should point out at Euler class that the Euler characteristic is actually a number, not just a cohomology class.
I have also added proof of the Whitney sum formula for Chern classes, here.
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