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I have a basic question which I am not able to figure out. If we do a Gluck twist on a nullhomologous 2-sphere in a 4-manifold, it is said that it does not change its intersection form. But as far as I understand the Gluck twist changes the framings and knotting of the link components which represent the second homology. So why doesn't the intersection form change?

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  • $\begingroup$ Have you tried computing the homology before and after the twist? I think that will answer your question. $\endgroup$ Mar 30, 2013 at 17:37
  • $\begingroup$ the second homology will not change, I think because we are not adding or deleting any 2-handles. but the Gluck twist changes framings of 2-handles. So I don't see why still the intersection form is unchanged. $\endgroup$
    – nikita
    Mar 30, 2013 at 18:49

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Basically this is Lefschetz duality. Chopping out the neighbourhood of the 2-sphere gives a 4-manifold $X$ with boundary $Y = S^2 \times S^1$. You can think about the intersection form here and how this will change after the surgery.

The point of $S^2$ being null-homologous is that if you look at the long exact sequence on homology (say) of the pair $(X,Y)$ you will see that every element of $H_2(X,Y)$ is represented by a unique element of $H_2(X)$.

I think it's necessary that the $S^2$ be null-homologous, I would imagine that otherwise, a priori, the intersection form may change.

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