Topology of Homology Manifolds

Abstract

ANR homology n-manifolds are nite-dimensional absolute neighborhood retracts X with the property that for every x 2 X, Hi(X;X fxg) is 0 for i 6= n and Z for i = n. Topological manifolds are natural examples of such spaces. To obtain nonmanifold examples, we can take a manifold whose boundary consists of a union of integral homology spheres and glue on the cone on each one of the boundary components. The resulting space is not a manifold if the fundamental group of any boundary component is a nontrivial perfect group. It is a consequence of the double suspension theorem of Cannon and Edwards that, as in the examples above, the singularities of polyhedral ANR homology manifolds are isolated. There are, however, many examples of ANR homology manifolds which have no manifold points whatever. See [12] for a good exposition of the relevant theory. The purpose of this paper is to begin a surgical classi cation of ANR homology manifolds, sometimes referred to in the sequel, simply as homology manifolds. One way to approach this circle of ideas is via the problem of characterizing topological manifolds among ANR homology manifolds. In Cannon's work on the double suspension problem [6], it became clear that in dimensions greater than 4, the right transversality hypothesis is the following (weak) form of general position.