A geometric interpretation of Ranicki duality

Abstract

Consider a commutative ring $R$ and a simplicial map, $X\mathop {\longrightarrow }\limits ^{\pi }K,$ of finite simplicial complexes. The simplicial cochain complex of $X$ with $R$ coefficients, $\Delta ^*X,$ then has the structure of an $(R,K)$ chain complex, in the sense of Ranicki . Therefore it has a Ranicki-dual $(R,K)$ chain complex, $T \Delta ^*X$ . This (contravariant) duality functor $T:\mathcal {B} R_K\to \mathcal {B} R_K$ was defined algebraically on the category of $(R,K)$ chain complexes and $(R,K)$ chain maps. Our main theorem, 8.1, provides a natural $(R,K)$ chain isomorphism: \[ T\Delta^*X\cong C(X_K) \] where $C(X_K)$ is the cellular chain complex of a CW complex $X_K$ . The complex $X_K$ is a (nonsimplicial) subdivision of the complex $X$ . The $(R,K)$ structure on $C(X_K)$ arises geometrically.