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Does there exist a smooth simply connected closed 4-manifold $X$ with the property below?

Every smooth basis for $H_2(X)$ contains a surface with genus $\geq 1$.

I understand that in general the genus problem is hard to tackle, and so I would not be surprised if there's not an example of this yet, but it would be interesting if there were.

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    $\begingroup$ This is on the Kirby problem list (Problem 4.27 (A)) where an update claims: "No in the smooth case using gauge theory." I think that this is an inaccurate update and the problem is still open. My guess is that whoever wrote this update was thinking about the fact that for 4-manifolds with non-zero Donaldson invariant, some specific bases (eg if they include classes of non-negative square) wouldn't be realized by spheres. But there is typically some basis consisting of classes of negative square for which there's no obstruction. $\endgroup$
    – Danny Ruberman
    Sep 6, 2019 at 18:20

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