Non-trivial examples of Stably diffeomorphic 4-manifolds

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

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.

• So you mean you want "non-homeomorphic" $M,N$ (rather than "non-trivial examples"). Also of course you mean "after connected with the same number of copies of $S^2\times S^2$". – YCor Mar 15 '17 at 16:08
• @YCor, I wrote that I want $M$ and $N$ non-diffeomorphic (but examples where $M$ and $N$ are not even homeo are good. The connected sum of copies of $S^2\times S^2$ is not required to be the same in my definition – Luigi M Mar 15 '17 at 16:10
• Have you tried an obvious candidate like the Enriques surface and the appropriate blow-up of $CP^2$ of the same signature? – Mikhail Katz Mar 15 '17 at 16:15
• @MikhailKatz to be honest no. I am pretty new to these kind of techniques, I learnt not a lot ago about blow-ups so I am not even sure how to start. I will think about it now – Luigi M Mar 15 '17 at 16:17
• @MikhailKatz The Enriques surface is not simply connected, so that won't work. – Danny Ruberman Mar 15 '17 at 16:39

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 $S^2 \times S^2$.