  In this paper we construct a coordinate atlas in an exotic $\mathbb{R}^{4}$
using Bizaca's construction. The main source for such an atlas is the handle
body decomposition of a Casson handle, which of course is an infinite, but
periodic, process. The immersion of the end-periodic manifold into
$\mathbb{R}^{4}$ is directly related to the exoticness of the $\mathbb{R}^{4}$
and also gives rise naturally to a spinor field. Thus we obtain the interesting
result that the simplest exotic $\mathbb{R}^{4}$ generates an extra spinor
field by exoticness.
