Singularities, double points, controlled topology and chain duality. (English) Zbl 0917.57018

Ranicki, Andrew

Doc. Math. 4, 1-59 (1999).

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 §1-§5, the main results are proven in §6 and §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 §8 and related to the total surgery obstruction in §9. Combinatorial analogues of controlled topology are developed further in §10–§13 and finally some standard constructions in high dimensional knot theory are given new interpretations in §14.

Reviewer: Jonathan Hillman (Sidney)

MSC:

57P10	Poincaré duality spaces
57Q99	PL-topology
57R67	Surgery obstructions, Wall groups
57Q45	Knots and links in high dimensions (PL-topology) (MSC2010)

Keywords:

chain duality; controlled topology; homology manifold; Poincaré duality; polyhedral; singularity