Quaternionic Cartan coverings and applications

TL;DR

The paper addresses the solvability of multiplicative Cousin problems for slice--regular quaternionic functions on axially symmetric domains by constructing quaternionic Cartan coverings via symmetric tilings and Cartan strings. It proves the existence of coverings subordinate to a given symmetric cover and a discrete obstruction set, with a compatible one-parameter family of coverings. A key application shows the vanishing of antisymmetric cohomology groups $H^n_a$ on planar symmetric domains for $n\ge 2$, enabling a cohomological foundation for quaternionic Cousin-type results. These results extend complex Cartan theory to the quaternionic, symmetry-constrained setting and provide topological tools for gluing local slice--regular solutions.

Abstract

We present the topological foundations for the solvability of Multiplicative Cousin problems formulated on an axially symmetric domain $Ω\subset \mathbb H.$ 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 $Ω$ 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 $Ω\cap \mathbb R$ must have additional properties. All these required properties were achieved by starting from a particular symmetric tiling of the symmetric set $Ω\cap (\mathbb R + i\mathbb R)$. Finally, we apply these results to prove the vanishing of 'antisymmetric' cohomology groups of planar symmetric domains for $n \geq 2$.

Figures (5)

• Figure 1: Tiles (left) and tiles with neighbourhoods (right).
• Figure 2: (a) Symmetric tilings of the square and (b) the annulus $\overline{A(0;1,3)}$ with $c_0 = 1,c_1 =2$ and $c_2 = 3$.
• Figure 3: (a) Pair of squares and (b) tiling of the model square $Q=[-1,1] \times [1,3]\qquad\qquad$
• Figure 4: Symmetric $\mathcal{D}$-and $\mathcal{B}$-necklaces and their tilings.
• Figure 5: (a) the set $B \cap K_{n+1, \nu}$, (b) the set $K_{n}\cap B$ (blue), already tiled, with a hole $D$ (light blue centred at the origin) that intersects the real axis and with a pair of symmetric holes (light blue), filled components of $K_n$ (dark blue), domain to be tiled (light blue) and holes of $K_{n+1}$ (white), (c) the symmetric tiling of the complete symmetric necklace.

Theorems & Definitions (61)

• Theorem 1.1: Main Theorem
• Definition 2.1: Closure, interior, complex conjugation and complex symmetrization
• Definition 2.2: $\varepsilon$-neighbourhoods
• Lemma 2.3
• Definition 2.4: Tubular $\varepsilon$-neighbourhoods
• Remark 2.5
• Definition 2.6
• Proposition 2.7
• Definition 2.8
• Definition 2.9