Timeline for Why do Chern classes and Stiefel-Whitney classes satisfy the "same" Whitney sum formula?
Current License: CC BY-SA 4.0
12 events
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| Mar 21, 2021 at 14:50 | answer | added | Neil Strickland | timeline score: 13 | |
| Mar 21, 2021 at 13:38 | answer | added | Jeremy Hahn | timeline score: 11 | |
| Mar 21, 2021 at 4:36 | history | became hot network question | |||
| Mar 21, 2021 at 2:17 | answer | added | Nicholas Kuhn | timeline score: 11 | |
| Mar 21, 2021 at 1:13 | comment | added | Tom Goodwillie | I wonder about Real vector bundles in the sense of Atiyah. I suppose that there are characteristic classes living in equivariant cohomology of the base, and I wonder if the theory of SW classes and the theory of Chern classes are both special cases or reflections of some bigger story. | |
| Mar 21, 2021 at 0:35 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Mar 21, 2021 at 0:29 | history | edited | Tim Campion | CC BY-SA 4.0 |
added 877 characters in body
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| Mar 21, 2021 at 0:22 | comment | added | Tim Campion | @OscarRandal-Williams Thanks -- already Totaro's title resolves that part of the question in the negative! | |
| Mar 21, 2021 at 0:21 | comment | added | Tim Campion | @IgorBelegradek Thanks, but it still looks to me like an "accident" that the coproduct on $H^\ast(BU;\mathbb Z)$ turns out to be given by such a similar formula to the coproduct on $H^\ast(BO;\mathbb F_2)$ (perhaps you're arguing that this is indeed just a "brute fact", a coincidence?). The fact that both classes satisfy the same axioms is question-begging, because one of the axioms is this coproduct / Whitney sum formula! A priori, it should indeed be clear that they can be defined in the same way, but then it's still a "miracle" that their Whitney sum formluas turn out to be the same. | |
| Mar 21, 2021 at 0:07 | comment | added | Igor Belegradek | I think the proof in Milnor-Stasheff of the Whitney sum formula for Chern classes works for Stiefel-Whitney (SW) classes. The proof is a computation in $H^*(BU(n))$ and one just has to run it in $H^*(BO(n);\mathbb Z_2)$. The analogy between SW and Chern classes is also explained in Exercise 14.E of Milnor-Stasheff. Both classes satisfy the same axioms and can be constructed in the same way inductively starting with the Euler class (taken mod 2 in the SW case), and without Steenrod squares. | |
| Mar 20, 2021 at 23:29 | comment | added | Oscar Randal-Williams | Regarding your last point, see B. Totaro "The total Chern class is not a map of multiplicative cohomology theories". | |
| Mar 20, 2021 at 16:36 | history | asked | Tim Campion | CC BY-SA 4.0 |