Vanishing of quaternionic cohomology groups and applications

We present solutions to additive and multiplicative Cousin problems formulated on an axially symmetric domain $\Omega \subset \mathbb H$ for slice--regular functions starting from the solutions for subclasses, namely slice--regular slice--preserving functions and functions in a given vectorial class. Consequently, we prove the vanishing of the corresponding cohomology groups with respect to axially symmetric open coverings (Theorems 1.1, 4.1, 4.2). The primary tool used in the proofs of these theorems is the existence of quaternionic Cartan coverings and Cartan's splitting lemmas. As an application, we prove a jet interpolation theorem (Theorem 1.2) and show that every divisor is principal (Theorem 5.1).