Is there an exotic $\mathbb{R}^4$ admitting an integrable almost complex structure?



Exotic $\mathbb{R}^4$ with a complex structure?
It is a result of Gromov that an open manifold of dimension six or less admits a complex structure if and only if it admits an almost complex structure; see the corollary on page 103 of his book *Partial Differential Relations*. As $\mathbb{R}^4$ is contractible, every exotic $\mathbb{R}^4$ is parallelisable. Therefore, they all admit almost complex structures, and hence complex structures by Gromov's result.


