Timeline for Why do Chern classes and Stiefel-Whitney classes satisfy the "same" Whitney sum formula?
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| Mar 21, 2021 at 15:58 | comment | added | Jeremy Hahn | I think perhaps that this can be considered a "calculation-free" construction, of the kind you might have in mind, as long as you only ask for an E_2 ring map (an E_2 map is more than enough to get the formula for the chern class of a sum). The input is that the double delooping of BU (i.e. BSU) has even cells. It looks likely to me that the same argument goes through word for word in the Real setting (using that BSU_{R} has a nice equivariant cell structure), which develops the story for stiefel whitney and chern class in the same construction. | |
| Mar 21, 2021 at 15:42 | comment | added | Tim Campion | Welcome to MathOverflow, Jeremy! Thank you for sorting this out -- I was very disappointed when I read the title of Totaro's paper (I admit I did not look into it in detail, though), so it is very good to know that the total Chern / SW class is part of an $E_\infty$ map. The idea in the present context was that the existence of such a map, if true, would likely come from some sort of "formal" construction which could be be done without computing anything -- without delving into the details of the many-author paper, I suppose it's unclear whether that's the case here. | |
| Mar 21, 2021 at 13:49 | comment | added | Jeremy Hahn | Perhaps it is also worth remarking that to get the double loop map from BU to GL_1 that suffices for your question, one can use classical obstruction theory. Indeed, the double delooping of BU is BSU, which has even cells, and the double delooping of GL_1 has even homotopy groups. | |
| Mar 21, 2021 at 13:41 | review | First posts | |||
| Mar 21, 2021 at 13:55 | |||||
| Mar 21, 2021 at 13:38 | history | answered | Jeremy Hahn | CC BY-SA 4.0 |