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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.
added pointer to
added pointer to
Someone on Twitter was asking about the geometric meaning of Chern classes. I mentioned the nPOV on characteristic classes. Now, I get the idea that unitary group bundles over a space $X$ are mapped to ordinary cohomology: $[X, B U] \mapsto [X, B^{2i} \mathbb{Z}]$ by cocycle composition. But I hadn’t thought what the latter is qua principal $B^{2i} \mathbb{Z}$-bundle.
Is there a geometric picture of this, e.g., in the case $i = 1$, the $B^2 \mathbb{Z}$-bundle associated to a circle-bundle via $c_1$?
I see that in the area of enumerative geometry, I’d need to be thinking about Chow rings
Chern classes are elements of the Chow ring.
Maybe it’s not what you are looking for, but Chern classes are basically algebraic/geometrical versions of topological homology classes: i.e. instead of looking at topological subspaces, one looks at (a generalisation of) sub-varieties. This is why Chow groups are closely related to motivic phenomena: in some sense, motives are an attempt to build a ’new geometry’ out of these algebraic analogues of homology classes, or more precisely to obtain/express results which hold in general for them.
One can always ’forget’ some information, and that is how one ends up with cycle maps, i.e. cohomological versions of Chern classes. Thus, for a smooth complex variety, we can ’forget’ the geometry of the ’algebraic/geometrical homology classes’ and just remember their topology, and this gives a cycle map to (topological) singular cohomology. Over a finite field, one uses étale, crystalline, etc cohomology instead. Of course forgetting information often leads to something more computable. One can forget ’a little less’, and one then ends up with cycle maps into things like Deligne cohomology.
In short, though, one is just looking at how a variety can be built up from sub-varieties of it (up to an algebraic analogue of homotopy, namely rational equivalence). Riemann-Roch says that one can equivalently speak of (algebraic) vector bundles.
Not sure if I understand yet properly what the question is after (is it about finding intuitively accessible geometric representatives of objects classified by the universal Chern classes?), but here are two quick comments:
at least the second universal Chern class on $B SU(2) \simeq B Sp(1)$ may be understood as being “like” the first is for complex line bundles, but now for quaternionic line bundles. This is the eventual content of the discussion (or monologue) on quaternionic orientation from complex orientation here.
one bit of algebraic topology analogous to Chow groups is (not $BU$ but) MU (which is “close”, of course, in a sense), complex cobordism: By the article and its MO commentary briefly recorded there.
Yes, complex cobordism is definitely related: from the point of view of my comment, one is still forgetting something, but in a slightly different way. Hence there is for example a ’motivic cobordism theory’ (i.e. a generalised motivic cohomology’) which ’lifts’ the algebro-topological cobordism theory into the motivic world.
I can maybe add that the following…
But I hadn’t thought what the latter is qua principal $B^{2i} \mathbb{Z}$-bundle.
…is basically the analogue in a algebraic/geometrical setting of finding ’nice’ representatives of a homology class. This is a difficult/interesting problem in general, it is like asking for an analogue of the description of $K(\mathbb{Z}, 2)$ as $\mathbb{C}\mathbb{P}^{\infty}$. This is the kind of thing Urs was getting at in 1. in one case, I think.
Thanks both! Plenty of useful ideas there.
Just as a sanity check, the shift to universal Chern classes on $B SU(n)$ rather than $B U(n)$ in #19 point 1 is made because we care about $SU(2)$ rather than $U(2)$, and is just mediated by the inclusion of $SU(2)$ into $U(2)$?
Yes, I considered restricting $c_2$ along $SU(2) \overset{i}{\hookrightarrow} U(n)$, or rather along $B SU(2) \overset{B i}{\longrightarrow} B U(n)$ only to make use of the exceptional isomorphism $SU(2) \simeq Sp(1)$, since the latter group Sp(1) is for quaternionic line bundles what U(1) is for complex line bundles.
By the way, not to forget that $c_2$ in general (i.e. without such restrictions) also “is” the universal Chern-Simons circle 3-bundle.
Thanks! And of course instantons make us care about $SU(2)$.
Oh, we care about $c_2$ on $B SU(2)$ for many reasons, also for chiral perturbation theory and Skyrmions and heterotic line bundles and … all appearing at once in the Hypothesis H M5-brane model (see on the right of the big diagram here).
Added:
A MathOverflow question discussing references: https://mathoverflow.net/questions/345437/canonical-reference-for-chern-characteristic-classes
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