An Intrinsic Domain Exchange Involution via Dual Morse Flows and Point-Loop Duality
Year: 2026
Type: preprint
Abstract: We construct an explicit, globally continuous domain exchange involution F : S 2 → S 2 that maps a closed Jordan domain into its complementary region on the two-dimensional sphere. The transformation is uniquely anchored via a projected spherical center of mass computed from the boundary embedding, guaranteeing global existence and uniqueness for any non-origin-symmetric closed Jordan curve. Governed by this reference pole and its antipode, the transformation operates locally as a homeomorphism between the punctured complementary domains and globally as an intrinsic inversion of the entire manifold S 2 . This concurrent deformation is formulated as a dual family of synchronized gradient homotopies under a canonical Morse height function. By exploiting the structural cut loci of the complementary domains, we prove that the global mapping extends continuously across the singular subsets to form a global bijection. The resulting operator preserves the topological invariants of the sphere with a stable topological degree of deg(F ) = +1. Finally, the global deformation is formalized as an inversion homotopy restricted entirely to the intrinsic geometry of the two-dimensional manifold, establishing a structural reciprocity between the contraction of a 1-manifold loop and its dual 1-dimensional cut locus complexes. Finally, the framework is extended to higher-dimensional spheres S n (n ≥ 3), formalizing a generalized Point-Hyperloop Duality where the topological degree deg(F ) = +1 remains stable due to orientation consistency under opposite driving elds.
Source: HAL
Author: Abdellatif Aitelhad
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221780398