Relational Singularities: Bitopological Rings and Obstruction Geometry

Classical topology and algebraic geometry describe singular behavior through points, neighborhoods, and local structures. Yet many fundamental phenomena in algebra—such as decomposition defects, instability under completion, persistence under quotients, and non-catenary behavior—are not governed by pointwise data. They arise instead from how elements interact.This monograph develops a geometric framework in which such phenomena are treated as relational singularities. Its central objects are bitopological rings, in which a topology is placed on pairs of elements rather than on elements alone. Relations, rather than points, become the primary carriers of geometric structure. Regularity properties such as decomposability, symmetry, reflexivity, and transitivity are reformulated as geometric constraints on relational neighborhoods, and their failure produces intrinsic singular structure.These failures are organized algebraically by radicals and geometrically by obstruction spectra, which replace classical prime spectra as carriers of singular data. Completion plays a decisive role, separating intrinsic obstructions from removable artifacts and leading to stable notions of depth, dimension, and stratification. The resulting theory—obstruction geometry—is developed as a self-contained geometric framework, independent of sheaves, pointwise localization, or analytic structure.The later parts of the monograph introduce obstruction homology and cohomology, spectral sequences, and deformation theory arising directly from relational geometry. Homological invariants measure interaction between singularities rather than their mere existence, revealing new structural phenomena invisible to classical methods.This work presents the first systematic theory of bitopological rings and establishes obstruction geometry as a foundational language for persistent relational failure. It is intended for researchers and advanced graduate students in algebra, topology, and geometry who seek geometric frameworks beyond classical locality and point-based methods. Published : January 2026 Available from : https://www.amazon.com/dp/B0GG5MH3G4