An Intrinsic Domain Exchange Involution via Dual Morse Flows and Point-Loop Duality
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.
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BibTeX
@misc{aitelhad2026intrinsic,
title = {{An Intrinsic Domain Exchange Involution via Dual Morse Flows and Point-Loop Duality}},
author = {Abdellatif Aitelhad},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.13140/rg.2.2.19734.51529},
url = {https://hal.science/hal-05840149v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An Intrinsic Domain Exchange Involution via Dual Morse Flows and Point-Loop Duality
AU - Abdellatif Aitelhad
PY - 2026
DA - 2026/10/26
PB - HAL
DO - 10.13140/rg.2.2.19734.51529
UR - https://hal.science/hal-05840149v1
AB - 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.
ER -