Constructive Differential Algebraic Framework for Differential and Exterior Differential Topology: A Comprehensive Treatment with Certified Computations

This paper establishes a comprehensive constructive differential algebraic framework for both differential topology and exterior differential topology, extending the methodology previously developed for exterior differential equations and partial differential equations. We define the exterior differential topological algebraic closure KEDT, a differentially closed structure constructed through a recursive adjunction process that incorporates solutions to geometric partial differential equations, constructively defined harmonic forms, characteristic classes, Poisson structures, Lie algebroids, and other exterior differential topological invariants with certified error bounds. Within this closure, we prove that solutions to fundamental problems in both differential topology and exterior differential topology—including the construction of harmonic forms, characteristic class representatives, Poisson structures, symplectic foliations, and Lie algebroid cohomology—admit unified constructive representations with explicit convergence rates and error estimates. The framework rigorously addresses the challenges of combining local coordinate descriptions with global topological constraints while preserving the geometric and algebraic structures inherent in both domains. We provide detailed constructive proofs with complete error analysis, derive explicit expressions for geometric objects with rigorous bounds, and establish convergence criteria in appropriate Sobolev spaces on manifolds. Detailed algorithms with precise complexity analysis and stability guarantees are presented, including adaptive precision control with certified error bounds derived from a posteriori error estimation. A comprehensive validation framework is established, employing discrete exterior calculus and numerical verification of topological invariants with mathematically rigorous error certification. This work demonstrates that explicit constructive representations of fundamental objects in both differential topology and exterior differential topology exist within the appropriately extended and constructively defined differential topological algebraic closure KEDT. The framework is shown to be consistent with classical differential topology while extending the constructive power to include harmonic forms, characteristic classes, Poisson structures, Lie algebroids, and other geometric structures that respect both differential and topological properties. Extensive numerical experiments validate the theoretical results and demonstrate the practical effectiveness of the proposed approach, including applications to sphere geometry, complex projective spaces, Poisson-Lie groups, and moduli spaces in gauge theory.