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Aug 13, 2020 at 17:36 vote accept Cihan
Jun 28, 2020 at 18:03 comment added Cihan I see. It is a theorem of Massey that $w_{d-1}(M) = 0$ whenever $d \equiv 3 \pmod{4}$ with $M$ is orientable and closed. I think we can drop the closed assumption by arguing that $w_{d-1}$ still vanishes because it vanishes on compact submanifolds. Thus there is always a $2$-field in this case.
Jun 28, 2020 at 4:42 history edited Panagiotis Konstantis CC BY-SA 4.0
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Jun 28, 2020 at 3:27 history edited Panagiotis Konstantis CC BY-SA 4.0
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Jun 27, 2020 at 21:11 history answered Panagiotis Konstantis CC BY-SA 4.0