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

## TL;DR

This paper explores the properties of slice regular quaternionic functions, demonstrating they are orientation-preserving, characterizing their fibers and singular sets, and establishing a maximum modulus principle with sharp results and explicit examples.

## Contribution

It introduces new concepts like the 'wing' of a function and provides a comprehensive analysis of the Jacobian, fibers, and singular sets of slice regular functions, extending classical complex analysis results.

## Key findings

- Slice regular functions are orientation-preserving.
- The singular set equals the branch set where the function is not locally a homeomorphism.
- The maximum modulus principle holds for these functions in full generality.

## Abstract

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.