Singularities, double points, controlled topology and chain duality (Q1283859)

This paper provides homological criteria for recognizing when an orientable polyhedral Poincaré duality complex is a homology manifold and when a degree 1 PL map between two orientable polyhedral homology manifolds has acyclic point inverses. These questions are discussed in the more general context of controlled simplicial complexes. If \(X\) is a simplicial complex with barycentric subdivision \(X'\) an \(X\)-controlled simplicial complex is a simplicial map \(p_M:M\to X'\). A simplicial map \(f:M\to N\) is a map of \(X\)-controlled simplicial complexes if \(p_Nf=p_M\). The algebraic setting is the category of \((R,X)\)-module chain complexes, which admits a chain duality functor. Every \(X\)-controlled simplicial complex determines such a \((R,X)\)-module chain complex. After reviewing these and related notions in \S 1-\S 5, the main results are proven in \S 6 and \S 7. These are Theorem A: An \(n\)-dimensional polyhedral Poincaré complex is an \(n\)-dimensional homology manifold if and only if there is defined a Lefshetz duality isomorphism \(H^n(X\times X,\Delta_X)\cong H_n(X\times X-\Delta_X)\); and Theorem B: A simplicial map \(f:M\to N\) of \(n\)-dimensional polyhedral homology manifolds has acyclic point inverses if and only if it has degree 1 and \(H_n((f\times f)^{-1}\Delta_N,\Delta_M)=0\). These are reinterpreted in terms of the Spivak fibration and tangent block bundles in \S 8 and related to the total surgery obstruction in \S 9. Combinatorial analogues of controlled topology are developed further in \S 10--\S 13 and finally some standard constructions in high dimensional knot theory are given new interpretations in \S 14.