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In a sense, this is a follow-up to this question.

By work of Freedman and Wall, it is known that if two simply-connected 4-manifolds $M$ and $N$ are homeomorphic, then there is $k \in \mathbb{N}$ such that $M \times \#^k (S^2 \times S^2)$ is diffeomorphic to $N \times \#^k (S^2 \times S^2)$. (By $\#^k$, we denote $k$-fold connected sum.)

Can this statement be generalised to non-simply-connected manifolds, possibly including Whitehead torsion?

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    $\begingroup$ Mark Powell gave a talk precisely about this last week; here's the abstract: home.kias.re.kr/MKG/h/invaiantsLDT/?pageNo=2725 . I haven't taken notes and I can't help you with details (or even precise statements), but it was largely in the direction of "if certain invariants agree, and under certain assumptions on pi_1, then you have stable diffeomorphism (in the sense of Wall)". $\endgroup$ – Marco Golla May 18 '17 at 16:55
  • $\begingroup$ What I remember: the assumption on $\pi_1$ is being good (in the sense of Freedman); among the characters playing a role there was $\pi_2(M)$ as $\pi_1(M)$-module, and whether $M$ and/or its universal cover are spin. $\endgroup$ – Marco Golla May 18 '17 at 16:57
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    $\begingroup$ Here's a link to a preprint of Kasprowski, Land, Powell, and Teichner (arxiv.org/abs/1511.01172) which treats a special class of fundamental groups, but also gives a good idea of the general machinery that goes into solving such stable classification problems. And there's probably a non-empty intersection with Mark's talk that Marco mentioned. $\endgroup$ – Stefan Behrens May 18 '17 at 17:38
  • $\begingroup$ Maybe I should point out that Marco hasn't quite remembered my talk accurately... My work with Daniel Kasprowski, Markus Land and Peter Teichner is about understanding when any two closed, orientable 4-manifolds are stably diffeomorphic. No assumption about being homeomorphic. $\endgroup$ – Dave Davidson Mar 17 at 21:56
  • $\begingroup$ There are restrictions on fundamental groups, but good groups are not relevant in the stable setting. I'll also mention that Gompf's result that homeomorphic 4-manifolds are stably diffeomorphic has an alternative proof via Kreck's theorem, using the fact that the forgetful map from smooth to topological bordism is injective in this dimension. Thanks to Jeff for pointing out this page to me! Mark Powell $\endgroup$ – Dave Davidson Mar 17 at 22:03
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Gompf showed that the statement can indeed be generalized in this 1984 paper. He proved that any two smooth structures on a compact 4-manifold become diffeomorphic after taking the connected sum with some number of copies of $S^2\times S^2$.

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For orientable 4-manifolds, it was shown by Gompf as Jeff says. For compact nonorientable 4-manifolds with universal cover not spin, homeomorphic also implies stably diffeomorphic.

On the other hand, Kreck showed that for every finitely presented group G with a surjective homomorphism $w \colon G \to C_2$, there is a nonorientable 4-manifold $M$ with 1-type $(G,w)$, with the following property. The 4-manifold ($M$ plus the K3 surface) is homeomorphic to ($M$ plus eleven copies of $S^2 \times S^2$), but these two manifolds do not become diffeomorphic after adding copies of $S^2 \times S^2$.

https://link.springer.com/content/pdf/10.1007/BFb0075570.pdf

Thus there is a sense in which Wall's theorem fails quite badly to generalise to nonorientable 4-manifolds.

Here 1-type means (fundamental group, orientation character).

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