Quaternionic Cartan Coverings and Applications

We present the topological foundations for solvability of multiplicative Cousin problems formulated on an axially symmetric domain Ω ⊂ 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 Forstneriˇc and Prezelj (Math. Ann. 322(4), 633-666 (2002)). Because of the requirements of symmetry inherent to the domains of definition of quaternionic regular functions, the existence of quaternionic Cartan coverings of Ω is not a consequence of the existence of complex Cartan coverings; for the latter, there are no requirements for the symmetries with respect to the real axis. Due to the real axis’s special role, also the covering restricted to Ω ∩ R must have additional properties. All these required properties were achieved by start ing from a particular symmetric tiling of the symmetric set Ω∩ (R + iR). Finally, we apply these results to prove the vanishing of ’antisymmetric’ cohomology groups of planar symmetric domains for n ≥ 2.