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The 4-sphere cannot be a symplectic manifold. In particular, it does not admit an atlas whose transition maps are symplectomorphisms $(\mathbb{R}^4,\omega_\text{std})\to(\mathbb{R}^4,\omega_\text{std})$. Eliashberg-Gromov proved that if the $C^0$-limit of symplectomorphisms is a diffeomorphism, then it is a symplectomorphism. We can now ask about the next best thing:

Does $S^4$ have a symplectohomeomorphic structure, i.e. are there charts (equipped with the standard symplectic form on $\mathbb{R}^4$) whose transition maps are homeomorphisms that are $C^0$-limits of symplectomorphisms?

(This question may have originated from Alan Weinstein, who told me it after conversations concerning symplectic rigidity.)

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    $\begingroup$ I'm pretty sure this is an open question--or at least I've heard it referred to as such during this calendar year by multiple people who work on C^0 symplectic topology. $\endgroup$
    – Mike Usher
    Sep 2, 2014 at 2:29
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    $\begingroup$ I completely agree with what Mike said. $\endgroup$ Sep 12, 2014 at 14:44

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