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A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different.

13 votes
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Gromov-Witten invariants counting curves passing through two points

Let us say that a closed symplectic manifold $X$ is $GW_g$-connected if there is a nonvanishing Gromov-Witten invariant of the form $GW_{g,n}^{X,A}(\beta,point, point,\alpha_3,\ldots,\alpha_n)$ --in o …
Mike Usher's user avatar
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7 votes
Accepted

Minimal genus, adjunction inequality

Regarding Question 2, Corollary 2 of this paper by Li shows that any symplectic four-manifold which contains a smoothly embedded homologically essential sphere with nonnegative intersection is obtaine …
Mike Usher's user avatar
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6 votes

Compactify S^2\times S^2-\Delta

Any compactification of $S^2\times S^2\setminus\Delta$ which is simply connected and has the same homology as $S^2\times S^2$ will be at least homeomorphic to $S^2\times S^2$ (in particular it can't b …
Mike Usher's user avatar
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