Quaternionic Cartan coverings and applications

TL;DR

This paper develops the topological and geometric foundations for solving multiplicative Cousin problems on axially symmetric quaternionic domains, introducing quaternionic Cartan coverings that account for symmetry constraints.

Contribution

It introduces quaternionic Cartan coverings tailored for symmetric domains, extending complex Cartan coverings to accommodate quaternionic symmetry requirements.

Findings

Constructed quaternionic Cartan coverings for symmetric domains.

Proved the vanishing of antisymmetric cohomology groups in planar symmetric domains.

Established geometric methods for solving multiplicative Cousin problems in quaternionic analysis.

Abstract

We present the topological foundations for the solvability of Multiplicative Cousin problems formulated on an axially symmetric domain $\Omega \subset \mathbb H.$ In particular, we provide a geometric construction of quaternionic Cartan coverings, which are generalizations of (complex) Cartan coverings as presented in Section 4 of [FP]. Because of the requirements of symmetry inherent to the domains of definition of quaternionic regular functions, the existence of quaternionic Cartan coverings of $\Omega$ is not a consequence of the existence of complex Cartan coverings because, for the latter, there are no requirements for the symmetries with respect to the real axis. Due to the real axis's special, also the covering restricted to $\Omega \cap \mathbb R$ must have additional properties. All these required properties were achieved by starting from a particular symmetric tiling of the symmetric set $\Omega \cap (\mathbb R + i\mathbb R)$. Finally, we apply these results to prove the vanishing of 'antisymmetric' cohomology groups of planar symmetric domains for $n \geq 2$.