A Geometric Interpretation of Ranicki Duality

Our main theorem provides an $(R,K)$ chain isomorphism: $T\Delta^*X\cong C(X_K)$. Here $T$ is the Ranicki Duality functor; $\Delta^*X$ is the simplicial cochain complex of the simplicial complex $X$, with control map $\pi:X \to K$ and $C(X_K)$ is the cellular chain complex of a CW complex $X_K$.