The 4-sphere $S^4$ cannot be a symplectic manifold. In particular, it does not admit an atlas whose transition maps are symplectomorphisms $(\mathbb{R}^4,\omega_0)\to(\mathbb{R}^4,\omega_0)$. A theorem of Eliashberg-Gromov states that if the $C^0$-limit of symplectomorphisms is a diffeomorphism, then it is a symplectomorphism. We should then 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 should be credited to Alan Weinstein, who I heard it from after conversations about symplectic rigidity.)

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.)