On a Class of Orientation-Preserving Maps of $$\pmb {\mathbb {R}}^4$$R4

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 matrix $$J_f$$ of a slice regular function f proving in particular that $$\det (J_f)\ge 0$$, 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.