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.

7more comments