Singularities, Double Points, Controlled Topology and Chain Duality

A manifold is a Poincaré duality space without singularities. McCrory obtained a homological criterion of a global nature for deciding if a polyhedral Poincaré duality space is a homology manifold, i.e. if the singularities are homologically inessential. A homeomorphism of manifolds is a degree 1 map without double points. In this paper combinatorially controlled topology and chain complex methods are used to provide a homological criterion of a global nature for deciding if a degree 1 map of polyhedral homology manifolds has acyclic point inverses, i.e. if the double points are homologically inessential.

1991 Mathematics Subject Classification: Primary 55N45, 57R67; Secondary 55U35.

Keywords and Phrases: manifold, Poincar\'e space, singularity, controlled topology, chain duality.

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