(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](https://mathoverflow.net/questions/108631/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.



Tools for constructing homeomorphisms between 4-manifolds
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. 


