TITLE: Presentations of exotic 4-manifolds
QUESTION [5 upvotes]: TLDR I want to see more examples of exotic $4$-manifold (hopefully connected, simply connected, oriented, and closed).
Are there known presentations of $4$-manifolds $M$ with exotic structures, whether in terms of Kirby linky data, PL-triangulations, or any other constructions? There are a few given in Akbulut's book [1], but they are $4$-manifolds with boundaries.
Mainly, I am looking for those $M$ that are oriented, connected, simply-connected and closed. But really, any pointers to any example with exotic smooth structures are appreciated.
[1] 4-Manifolds - Selman Akbulut

REPLY [8 votes]: I guess that this is as explicit and low-tech as it gets: if $X$ is a K3 surface (i.e. a non-singular quartic hypersurface in $\mathbb{CP}^3$, with the complex orientation), then $X \# \overline{\mathbb{CP}}{}^2$ and $3\mathbb{CP}^2 \# 20\overline{\mathbb{CP}}{}^2$ are an exotic pair.
To see that they are homeomorphic, we use that odd indefinite forms are diagonalisable and then Freedman's theorem. (Ok, and that complex projective hypersurfaces are simply-connected, by Lefschetz's theorem.) To see that they are not diffeomorphic, we use that Kähler surfaces have (some) non-zero Seiberg–Witten invariant, while anything written as a connected sum of indefinite pieces doesn't. (In particular, the same argument applies to any hypersurface of $\mathbb{CP}^3$ of degree at least 4; in this case, blowing up/connected summing with $\overline{\mathbb{CP}}{}^2$ is only needed in even degrees.)
This is just the tip of the iceberg of the tip of the iceberg that Anubhav mentioned in his comment.