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Let $M$ be a manifold of symplectic type (we do not fix a particular symplectic form) and $N \subset M$ a submanifold of half the dimension. Is there a topological obstruction on the pair $(M, N)$ for the existence of a symplectic form on $M$ for which $N$ is Lagrangian?

One idea is that if $N$ is Lagrangian for some symplectic form, then its normal bundle $TM|_N / TN$ is isomorphic to its cotangent bundle $T^*N$. Is that enough to detect a pair $(M, N)$ for which $N$ is never Lagrangian?

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  • $\begingroup$ Haven't you answered your own question here? Or are you looking for an example where your obstruction applies? e.g. take M = S^2 \times S^2 and N to be one of the two factors $\endgroup$
    – Jonny Evans
    Commented May 9, 2023 at 20:22
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    $\begingroup$ @JonnyEvans I guess OP here is asking if there are other known obstructions that can be expressed in algebraic topological terms. $\endgroup$
    – Overflowian
    Commented May 9, 2023 at 21:35
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    $\begingroup$ @JonnyEvans One way to answer my question would indeed be to give an example where the obstruction is non-trivial. It seems to me that your example does not work: the normal bundle is $TS^2$ which is isomorphic to $T^*S^2$. $\endgroup$
    – Mattis Bakken
    Commented May 10, 2023 at 10:48
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    $\begingroup$ @MattisBakken No, the normal bundle of either of the two factors is trivial. $\endgroup$
    – Jonny Evans
    Commented May 10, 2023 at 18:14

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