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> I am looking for some non-trivial examples of (smooth) 4-mflds $M,N$ such that $M$ and $N$ are STABLY diffeomorphic. I.e. $$M\sharp\_n (S^2\times S^2) \cong N \sharp\_r (S^2\times S^2)$$ for $r,n$ not necessarily the same
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By non trivial I mean that $M$ and $N$ are not diffeomorphic or that $M \cong N\sharp S^ 2 \times S^2$ or similar situation to these two. 


Clearly the two manifolds have to have the same fundamental group and the same signature at least. What I am looking for is a concrete/explicit example.


By results of Gompf we know that if $M$ is orientable carries two non-equivalent smooth structures ($M\_1,M\_2$) then $M\_1$ and $M\_2$ are stably diffeomorphic (but clearly non-diffeomorphic). I would like to have a less exotic example as main example in mind when speaking about stable diffeomorphism.


I found abstract criterions for certain families of manifolds but I'm unable to cook up an example.



Non-trivial examples of Stably diffeomorphic 4-manifolds
The main examples of this go back to work of Moishezon and Mandelbaum in the late 1970s. For instance, if a simply-connected elliptic surface E(2n+1) is homotopy equivalent to the connected sum of $4n+1$ copies of $\mathbb{C}P^2$ blown up $20n +9$ times, but those manifolds are not diffeomorphic for $n>1$. This is a theorem of Donaldson. But after a single connected sum with $S^2 \times S^2$ they become diffeomorphic, as proved by Moishezon (with a simplified proof by Mandelbaum-Moishezon).


These matters are described pretty well in the book of Gompf and Stipsicz. As far as I know, there is no known example of a simply connected pair of this sort where you have to add more than one copy of $S^2 \times S^2$.


