Unified Analytic Representation of Topological Invariants in Exterior Algebraic Topology

This paper establishes a rigorous topological algebraic framework for constructing explicit analytic representations of topological invariants in both classical algebraic topology and exterior algebraic topology. We prove that homotopy groups, homology groups, and cohomology rings of finite CW complexes, as well as de Rham cohomology classes, Hodge decompositions, and harmonic forms of compact smooth manifolds, can be analytically expressed within a topological algebraic closure Ktop and its extension exterior algebraic closure K∧, which extend the coefficient field with topological and exterior algebraic operators and their evaluations.We provide complete constructive proofs, derive combinatorial expressions for correction coefficients γ(n)m from Morse theory, CW structure, and Hodge theory, and present detailed O(N2) algorithms for computational implementation. Extensive theoretical validation across classical examples demonstrates consistency with established results while providing new explicit representations.This work reconciles with classical impossibility results by demonstrating that while elementary closed-form solutions may not exist for general topological spaces, explicit analytic solutions exist in the appropriately extended topological algebraic closure Ktop and exterior algebraic closure K∧.