On a class of orientation-preserving maps of $\mathbb R^4$

The purpose of this paper is to present several new, sometimes surprising, results concerning a class of hyperholomorphic functions over quaternions, the so-called slice regular functions. The concept of slice regular function is a generalization of the one of holomorphic function in one complex variable. The results we present here show that such a generalization is multifaceted and highly non-trivial. We study the behavior of the Jacobian $J_f$ of a slice regular function $f$ proving in particular that $\det(J_f)\geq0$, i.e. $f$ is orientation-preserving. We give a complete characterization of the fibers of $f$ making use of a new notion we introduce here, the one of wing of $f$. We investigate the singular set $N_f$ of $f$, i.e. the set in which $J_f$ is singular. The singular set $N_f$ turns out to be equal to the branch set of $f$, i.e. the set of points $y$ such that $f$ is not a homeomorphism locally at $y$. We establish the quasi-openness properties of $f$. As a consequence we deduce the validity of the Maximum Modulus Principle for $f$ in its full generality. Our results are sharp as we show by explicit examples.