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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• 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 $H^\bullet(B U(n))\simeq \mathbb{Z}[ c_1,\cdots , c_n ]$

• 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 $B U(1)^n \to B U(n)$ 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 $BT \to BG$, for reasonable $G$

• 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 $H^*(BG) \to H^*(BT)$ 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 $G/T$.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeJun 27th 2016

Thanks. Do you have the energy left to make a note about this on some $n$Lab 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.