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@abx Could you please clarify why $y$ is 4-torsion? Also, in the last line you say $H^2(N,\mathbb Z)$, shouldn't it me $M$ instead of $N$? Sorry for not getting something very basic...
Thank you very much for your contribution! Still I think advising on useful direction to look for an answer should be made as a comment, not as an answer. Then we can upvote the comment if it proved to be useful. An answer is as a proof as a theorem: you don't say "PROOF. I think one could look for a proof along these lines... QED"
