In a sense, this is a follow-up to [this question](https://mathoverflow.net/questions/180723/simply-connected-4-manifolds-can-be-blown-up-and-down-to-complex-projective-plan).


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?



How can Wall's theorem be generalised to non-simply connected manifolds?
Gompf showed that the statement can indeed be generalized in [this 1984 paper](https://www.sciencedirect.com/science/article/pii/016686418490004X). 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 $S^2\times S^2$.


