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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

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.

Luigi M
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  • 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
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    @MikhailKatz The Enriques surface is not simply connected, so that won't work. – Danny Ruberman Mar 15 '17 at 16:39
  • @DannyRuberman, take its double cover then. – Mikhail Katz Mar 15 '17 at 16:42
  • Ok, but is there a way to prove that they are stably diffeomorphic without using Kreck's theorem (which tell us to compute the relevant bordism classes basically?) a more direct way, this is what I'm really interested – Luigi M Mar 15 '17 at 16:44
  • @MikhailKatz That double cover is a K3 surface and so is spin. If you blow up once to make it non-spin, then that is stably diffeomorphic (add one copy of $S^2\times S^2$) to a connected sum projective spaces (with both orientations). This follows from Mandelbaum-Moishezon as in my answer below. – Danny Ruberman Mar 15 '17 at 16:52
  • @LuigiM Take a look at Wall's 1964 paper "On simply-connected 4-manifolds". This tells you that for simply connected manifolds it's all about the intersection form. Unfortunately, the machinery used to prove this type of result (surgery theory) gets considerably more complicated for non-trivial $\pi_1$. – Stefan Behrens Mar 15 '17 at 16:53
  • But precisely you say that if they are homeomorphic and non-diffeomorphic then they are "stably diffeomorphic" and ask for other examples. So your question is about the non-homeomorphic case. – YCor Mar 15 '17 at 17:40
  • @YCor that was an example of two non diffeo mflds which are stably diffeo. my apologises if it confuses my request, – Luigi M Mar 15 '17 at 17:46
  • It's more and more confusing. You want stably diffeomorphic 4-manifolds that non-diffeomorphic, which implies either (a) homeomorphic but not diffeomorphic (b) non-homeomorphic. You say that in case (a), stably diffeomorphic follows automatically, so the case (a) is clear-cut. Thus only the case (b) remains. – YCor Mar 15 '17 at 17:55

2 Answers2

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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 $S^2 \times S^2$.

Danny Ruberman
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  • 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)? – mme Mar 15 '17 at 19:33
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    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. – Danny Ruberman Mar 16 '17 at 00:53
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Since the examples given above are smooth, simply connected, and homotopy equivalent, they are also homeomorphic. You might like the following examples. Let $L$ and $L'$ be 3-dimensional lens spaces that are homotopy equivalent but not homeomorphic. Then $L \times S^1$ and $L' \times S^1$ are simple homotopy equivalent and stably diffeomorphic, but not homeomorphic.

Dave Davidson
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