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.

  • $\begingroup$ 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 $S^2\times S^2$". $\endgroup$ – YCor Mar 15 '17 at 16:08
  • $\begingroup$ @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 $\endgroup$ – Luigi M Mar 15 '17 at 16:10
  • $\begingroup$ Have you tried an obvious candidate like the Enriques surface and the appropriate blow-up of $CP^2$ of the same signature? $\endgroup$ – Mikhail Katz Mar 15 '17 at 16:15
  • $\begingroup$ @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 $\endgroup$ – Luigi M Mar 15 '17 at 16:17
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    $\begingroup$ @MikhailKatz The Enriques surface is not simply connected, so that won't work. $\endgroup$ – 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 $S^2 \times S^2$.

  • $\begingroup$ Is there any heuristic reason to believe you might only need one (other than the fact that gauge theory can't tell apart things after a single connected sum)? $\endgroup$ – Mike Miller Eismeier Mar 15 '17 at 19:33
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    $\begingroup$ No good reason whatsoever to believe this, other than there are quite a few examples known (cf. papers by Auckly, Akbulut, Baykur-Sunukjian and probably others) and no good gauge theory tools to tell manifolds apart after one stabilization. $\endgroup$ – Danny Ruberman Mar 16 '17 at 0:53

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