Unified Analytic Representation of Topological Invariants in Topological Algebraic Closure

This paper establishes a rigorous topological algebraic framework for constructing explicit analytic representations of topological invariants in algebraic topology. We prove that homotopy groups, homology groups, and cohomology rings of finite CW complexes can be analytically expressed within a topological algebraic closure K top , which extends the coefficient field with topological operators and their evaluations. We provide complete constructive proofs, derive combinatorial expressions for correction coefficients γ(n) m from Morse theory and CW attachment maps, 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.