  We present the topological foundation for 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
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
special role of the real axis, also the covering restricted to $\Omega \cap
\mathbb R$ has to 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 provide an application
of these results to prove the vanishing of 'antisymmetric' cohomology groups of
planar symmetric domains for $n \geq 2$.
