Timeline for Realizing Stiefel-Whitney classes via vector bundles
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Jan 15, 2022 at 17:20 | comment | added | Tim Campion | @AleksandarMilivojevic Ah, thanks, that's helpful! | |
| Jan 15, 2022 at 9:57 | comment | added | Aleksandar Milivojević | What does hold only for closed manifolds, and also goes under the name "Wu's formula", at least in Milnor-Stasheff, is the formula Sq(total Wu class) = total Stiefel-Whitney class. This restriction to closed manifolds is to be expected, since the Wu classes use the (mod 2) Poincare duality structure in their definition. However, the "Wu's formula" linked to in the question is the one concerning the Steenrod algebra action on H*(BO;Z/2). | |
| Jan 15, 2022 at 9:26 | comment | added | Aleksandar Milivojević | The Wu formula does hold for arbitrary vector bundles. I am having trouble locating Wu's original paper at the moment, but see for example May's Concise p.197. The Wu formula describes the action of the Steenrod algebra on the mod 2 cohomology of BO (generated as a Z/2-algebra by the universal Stiefel-Whitney classes) | |
| Jan 15, 2022 at 1:00 | comment | added | Tim Campion | This answer reflects a fundamental confusion on my part: The Wu formula is a formula satisfied by the Stiefel-Whitney classes of the tangent bundle of a smooth manifold. Presumably it is not satisifed by an arbitrary vector bundle. So the case considered here, where the multiplication is trivial, is not terribly relevant. | |
| Jan 14, 2022 at 18:22 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| S Jan 14, 2022 at 1:12 | history | answered | Tim Campion | CC BY-SA 4.0 | |
| S Jan 14, 2022 at 1:12 | history | made wiki | Post Made Community Wiki by Tim Campion |