Quaternionic Cartan Coverings and Applications

Abstract We present the topological foundations for solvability of multiplicative Cousin problems formulated on an axially symmetric domain $$\Omega \subset \mathbb {H}.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>H</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> In particular, we provide a geometric construction of quaternionic Cartan coverings, which are generalizations of (complex) Cartan coverings as presented in Section 4 of Forstnerič 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 $$\Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> 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 $$\Omega \cap \mathbb {R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>∩</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> must have additional properties. All these required properties were achieved by starting from a particular symmetric tiling of the symmetric set $$\Omega \cap (\mathbb {R}+ i\mathbb {R})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>∩</mml:mo> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>+</mml:mo> <mml:mi>i</mml:mi> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . Finally, we apply these results to prove the vanishing of ’antisymmetric’ cohomology groups of planar symmetric domains for $$n \ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> .