A Geometric Interpretation of Ranicki Duality

TL;DR

This paper presents a geometric interpretation of Ranicki duality, establishing an isomorphism between the Ranicki dual of a simplicial cochain complex and a cellular chain complex of a CW complex, linking algebraic and geometric perspectives.

Contribution

It introduces a main theorem that provides a chain isomorphism connecting Ranicki duality with cellular chain complexes via a geometric framework.

Findings

Establishes an explicit chain isomorphism involving Ranicki duality and cellular chain complexes.

Bridges algebraic duality with geometric chain complexes in a new interpretative framework.

Provides tools for translating between simplicial cochain complexes and CW complex chain complexes.

Abstract

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 $X_K$.