Quaternionic Cartan coverings and applications

Prezelj, Jasna (author), Vlacci, Fabio (author)

We present the topological foundations 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 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 $\Omega$ 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 $\Omega \cap {\mathbb R}$ 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 $n \ge 2$.

• Language: English
• Keywords: quaternionic Cartan coverings, antisymmetric cohomology groups
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.03.2025
• Year: 2025
• Number of pages: 25 str.
• Numbering: Vol. 35, iss. 3, [article no.] 80
• PID: 20.500.12556/RUL-167120
• UDC: 517.5
• ISSN on article: 1050-6926
• DOI: 10.1007/s12220-025-01900-0
• COBISS.SI-ID: 225630467
• Publication date in RUL: 10.02.2025

Record is a part of a journal

• Title: The journal of geometric analysis
• Shortened title: J. geom. anal.
• Publisher: Springer Nature, Mathematica Josephina
• ISSN: 1050-6926
• COBISS.SI-ID: 30685696

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.