A Geometric Interpretation of Ranicki Duality
Frank Connolly
TL;DR
The paper provides a geometric reinterpretation of Ranicki duality by proving the central isomorphism $$T\Delta^*X\cong C(X_K)$$ for a $K$-space $(X,\pi)$, where $X_K$ is a ball complex encoding the $K$-blocking. It defines and analyzes the Ranicki duality functor $T$ and the natural transformation $e: T^2\to \mathrm{id}$, showing each $e_C$ is an $(R,K)$-chain equivalence, and constructs the derived/combinatorial machinery that links $\Delta^*X$, $C(X_K)$, and the derived complex $\Delta X'$. The McCrory cap product is used to relate $T\Delta^*X$ to the subdivision $\Delta X'$ and to prove chain homotopy equivalences among the three $(R,K)$-complexes, providing a transparent geometric interpretation of duality. These results pave the way for applying blocked-surgery frameworks to questions of topological rigidity for groups with finite-order elements, by making Ranicki duality accessible in geometric terms. Overall, the work unifies the algebraic Ranicki duality with explicit geometric constructions, enabling concrete computations and potential applications in geometric topology.
Abstract
Our main theorem provides an $(R,K)$ chain isomorphism: $ TΔ^*X\cong C(X_K) $. Here $T$ is the Ranicki Duality functor; $Δ^*X$ is the simplicial cochain complex of the simplicial complex $X$, with control map $π:X \to K$ and $C(X_K) $ is the cellular chain complex of a CW complex $X_K$.