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(I am a complete amateur in topology, so this is a question out of curiosity.)
The question was inspired by this post Fake versus Exotic . What methods can, realistically, be used to construct a homeomorphism between, for example, $\mathbb{C}\mathbb{P}^2\#9\overline{\mathbb{C}\mathbb{P}^2}$ and the Dolgachev surface $E(1)_{2,3}$? In the $Diff$ category there is the Kirby calculus which seems to be efficient enough. (Some very nontrivial diffeomorphisms were found using it, e.g. by Gompf.) My question is, essentially, how things are with $Top$ in comparison.

Also, a more specific question: is it possible to use for this purpose the old result of Wall that for simply connected h-cobordant 4-manifolds $M$ and $N$ there is $k$ such that $M\#k(\mathbb{S}^2\times \mathbb{S}^2)\cong N\#k(\mathbb{S}^2\times \mathbb{S}^2)$ (a stable diffeomorphism)? Simply adding $\mathbb{S}^2\times \mathbb{S}^2$ is too bold a move in this case, but maybe it can be useful if done carefully. To make it more clear, I mean $M\#k(\mathbb{S}^2\times \mathbb{S}^2)$ with some additional structure.

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Sadly, it seems that you need pretty much the full strength of Freedman's disc embedding theorem to construct such homeomorphisms. Wall's theorem builds an h-cobordism, essentially starting with the stabilization you mention, and then regluing by a diffeomorphism to get handles to algebraically cancel. But to get them to geometrically cancel, you need to do isotopies guided by Whitney disks. In some circumstances, you can see explicit Casson handles where those disks should go. But you still have to hit those with Freedman's theorem to get the homeomorphism.

At some level, Freedman's theorem is based on several complicated limiting arguments (`Bing topology') that produce homeomorphisms rather than diffeomorphisms. It is interesting to compare this with the situation in higher dimensions. Siebenmann's article, Topological manifolds, in the Proceedings of the 1970 ICM (you'll find a pdf readily online) comes as close as one might hope to explaining how this works for a particular homeomorphism that is not isotopic to a PL homeomorphism. Again, a limit is taken that produces homeomorphisms that fail to be PL (and hence smooth). It would be great to see something so explicit in the 4-dimensional case.

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    $\begingroup$ Thank you. Siebenmann's article is quite a nice reading. By the way, one quote from it: "One can expect that mathematicians will consequently come to use freely the notions of homeomorphism and topological manifold untroubled by the frustrating difficulties that worried their early history". It was 50 years ago! $\endgroup$ – Alex Gavrilov Aug 29 '18 at 16:00
  • $\begingroup$ If you liked that one, you might like to look at a couple of other articles of Siebenmann from that era: Disruption of low-dimensional handlebody theory by Rohlin's theorem, and Are non triangulable manifolds triangulable. The first one explains a lot of the relation between Rohlin's theorem and the Kirby-Siebenmann invariant. The second explains a lot of the background to the triangulation conjecture (disproved by Manolescu). $\endgroup$ – Danny Ruberman Aug 29 '18 at 20:54

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