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Simon Donaldson apparently made the following conjecture: Two closed symplectic 4-manifolds $(X_1,\omega_1)$ and $(X_2,\omega_2)$ are diffeomorphic if and only if $(X_1\times S^2,\omega_1\oplus\omega)$ is deformation-equivalent to $(X_2\times S^2,\omega_2\oplus\omega)$. Here $\omega$ is a symplectic structure on $S^2$, and a deformation-equivalence is a diffeomorphism $\phi:X_1\times S^2\to X_2\times S^2$ such that $\omega_1\oplus\omega$ and $\phi^*(\omega_2\oplus\omega)$ can be joined by a path of symplectic forms.

However, where I read this did not contain any background or the original source. Where did Donaldson make this claim? And why did he make this claim? What is the motivation / are there good examples where this holds? Ivan Smith showed (through examples) that this conjecture fails when we replace $S^2$ by $\mathbb{T}^2$, so the statement itself seems pretty rigid.

[Edit] Motivation and examples come from the 1994 paper "Symplectic Topology on Algebraic 3-Folds" of Ruan and the 1997 followup "Higher Genus Symplectic Invariants..." of Ruan-Tian, which states and proves the conjecture for simply-connected elliptic surfaces!

Simon Donaldson apparently made the following conjecture: Two closed symplectic 4-manifolds $(X_1,\omega_1)$ and $(X_2,\omega_2)$ are diffeomorphic if and only if $(X_1\times S^2,\omega_1\oplus\omega)$ is deformation-equivalent to $(X_2\times S^2,\omega_2\oplus\omega)$. Here $\omega$ is a symplectic structure on $S^2$, and a deformation-equivalence is a diffeomorphism $\phi:X_1\times S^2\to X_2\times S^2$ such that $\omega_1\oplus\omega$ and $\phi^*(\omega_2\oplus\omega)$ can be joined by a path of symplectic forms.

However, where I read this did not contain any background or the original source. Where did Donaldson make this claim? And why did he make this claim? What is the motivation / are there good examples where this holds? Ivan Smith showed (through examples) that this conjecture fails when we replace $S^2$ by $\mathbb{T}^2$, so the statement itself seems pretty rigid.

[Edit] Motivation and examples come from the 1994 paper "Symplectic Topology on Algebraic 3-Folds" of Ruan and the 1997 followup "Higher Genus Symplectic Invariants..." of Ruan-Tian, which states and proves the conjecture for simply-connected elliptic surfaces!

And now we're starting to get counterexamples!

Simon Donaldson apparently made the following conjecture: Two closed symplectic 4-manifolds $(X_1,\omega_1)$ and $(X_2,\omega_2)$ are diffeomorphic if and only if $(X_1\times S^2,\omega_1\oplus\omega)$ is deformation-equivalent to $(X_2\times S^2,\omega_2\oplus\omega)$. Here $\omega$ is a symplectic structure on $S^2$, and a deformation-equivalence is a diffeomorphism $\phi:X_1\times S^2\to X_2\times S^2$ such that $\omega_1\oplus\omega$ and $\phi^*(\omega_2\oplus\omega)$ can be joined by a path of symplectic forms.

However, where I read this did not contain any background or the original source. Where did Donaldson make this claim? And why did he make this claim? What is the motivation / are there good examples where this holds? Ivan Smith showed (through examples) that this conjecture fails when we replace $S^2$ by $\mathbb{T}^2$, so the statement itself seems pretty rigid.

Simon Donaldson apparently made the following conjecture: Two closed symplectic 4-manifolds $(X_1,\omega_1)$ and $(X_2,\omega_2)$ are diffeomorphic if and only if $(X_1\times S^2,\omega_1\oplus\omega)$ is deformation-equivalent to $(X_2\times S^2,\omega_2\oplus\omega)$. Here $\omega$ is a symplectic structure on $S^2$, and a deformation-equivalence is a diffeomorphism $\phi:X_1\times S^2\to X_2\times S^2$ such that $\omega_1\oplus\omega$ and $\phi^*(\omega_2\oplus\omega)$ can be joined by a path of symplectic forms.

However, where I read this did not contain any background or the original source. Where did Donaldson make this claim? And why did he make this claim? What is the motivation / are there good examples where this holds? Ivan Smith showed (through examples) that this conjecture fails when we replace $S^2$ by $\mathbb{T}^2$, so the statement itself seems pretty rigid.

[Edit] Motivation and examples come from the 1994 paper "Symplectic Topology on Algebraic 3-Folds" of Ruan and the 1997 followup "Higher Genus Symplectic Invariants..." of Ruan-Tian, which states and proves the conjecture for simply-connected elliptic surfaces!

Simon Donaldson apparently made the following conjecture: Two closed symplectic manifolds $(X_1,\omega_1)$ and $(X_2,\omega_2)$ are diffeomorphic if and only if $(X_1\times S^2,\omega_1\oplus\omega)$ is deformation-equivalent to $(X_2\times S^2,\omega_2\oplus\omega)$. Here $\omega$ is a symplectic structure on $S^2$, and a $\textit{deformation-equivalence}$ is a diffeomorphism $\phi:X_1\times S^2\to X_2\times S^2$ such that $\omega_1\oplus\omega$ and $\phi^*(\omega_2\oplus\omega)$ can be joined by a path of symplectic forms.

However, where I read this did not contain any background or the original source. Where did Donaldson make this claim? And why did he make this claim? What is the motivation / are there good examples where this holds? Ivan Smith showed that this conjecture fails when we replace $S^2$ by $\mathbb{T}^2$, so the statement of this conjecture seems pretty rigid.

Simon Donaldson apparently made the following conjecture: Two closed symplectic 4-manifolds $(X_1,\omega_1)$ and $(X_2,\omega_2)$ are diffeomorphic if and only if $(X_1\times S^2,\omega_1\oplus\omega)$ is deformation-equivalent to $(X_2\times S^2,\omega_2\oplus\omega)$. Here $\omega$ is a symplectic structure on $S^2$, and a deformation-equivalence is a diffeomorphism $\phi:X_1\times S^2\to X_2\times S^2$ such that $\omega_1\oplus\omega$ and $\phi^*(\omega_2\oplus\omega)$ can be joined by a path of symplectic forms.

However, where I read this did not contain any background or the original source. Where did Donaldson make this claim? And why did he make this claim? What is the motivation / are there good examples where this holds? Ivan Smith showed (through examples) that this conjecture fails when we replace $S^2$ by $\mathbb{T}^2$, so the statement itself seems pretty rigid.