(I am a complete amateur in topology, so this is a question out of curiosity.)   
The  question was inspired by this post 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 (which I have no clue about).