This is related to my older question Tools for constructing homeomorphisms between 4-manifolds , but it is much more down to earth. It is known since 1960s that for homotopy equivalent simply connected smooth 4-manifolds $M$ and $N$ there is $k$ such that $M\#k(\mathbb{S}^2\times \mathbb{S}^2)$ is diffeomorphic to $N\#k(\mathbb{S}^2\times \mathbb{S}^2)$. Nowadays, this diffeomorphism may be constructed very explicitly using Kirby calculus, at least in theory.

However, there is a potential problem. This classical result of Wall [Wall, C.T.C., On simply-connected 4-manifolds. J. London Math. Soc. Vol. 39 (1964), pp. 141–149.] consists of two parts. The first part (Theorem 2) is "homotopy equivalent" $\Rightarrow$ "h-cobordant", the second part (Theorem 3)
is "h-cobordant" $\Rightarrow$ "stably diffeomorphic". As I understand it, the proof of the latter is not constructive, and gives no a priory upper bound on the number $k$. In practical terms it means no guarantee that you can produce a genuine stable diffeomorphism even in simplest cases.

So, I have two questions. Did anyone construct a stable diffeomorphism beteween non-diffeomorphic closed simply connected 4-manifolds using Kirby calculus? (Or, at least, tried to.) And, is there an effective version of the Wall's theorem about h-cobordism?

[EDIT] This is nearly a duplicate of (Non-trivial examples of Stably diffeomorphic 4-manifolds), as Danny Ruberman pointed out. (That's embarrassing, I really should have checked this!) Maybe it is better to close it, I am not sure.

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    $\begingroup$ This is basically a duplicate of mathoverflow.net/questions/264701/…. If you want to look at some nice handle pictures, try D. Auckly, ‘Families of four-dimensional manifolds that become mutually diffeomorphic after one stabilization’, Topol. Appl. 127 (2003) 277–298. In all of the examples that anyone [that I know] knows, k can be taken to be 1, but there is no known upper bound. $\endgroup$ – Danny Ruberman Sep 30 '18 at 15:01

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