Homology manifolds

The study of the local-global geometric topology of homology manifolds has a long history. Homology manifolds were introduced in the 1930s in attempts to identify local homological properties that implied the duality theorems satisfied by manifolds [23, 56]. Bing’s work on decomposition space theory opened new perspectives. He constructed important examples of 3-dimensional homology manifolds with non-manifold points, which led to the study of other structural properties of these spaces, and also established his shrinking criterion that can be used to determine when homology manifolds obtained as decomposition spaces of manifolds are manifolds [4]. In the 1970s, the fundamental work of Cannon and Edwards on the double suspension problem led Cannon to propose a conjecture on the nature of manifolds, and generated a program that culminated with the Edwards-Quinn characterization of higher-dimensional topological manifolds [15, 24, 21]. Starting with the work of Quinn [44, 46], a new viewpoint has emerged. Recent advances [10] use techniques of controlled topology to produce a wealth of previously unknown homology manifolds and to extend to these spaces the Browder-Novikov-Sullivan-Wall surgery classification of compact manifolds [53], suggesting a new role for these objects in geometric topology, and tying together two strands of manifold theory that have developed independently. In this article, we approach homology manifolds from this perspective. We present a summary of these developments and discuss some of what we consider to be among the pressing questions in the subject. For more detailed treatments, we refer the reader to article [10] by Bryant, Ferry, Mio and Weinberger, and the forthcoming lecture notes by Ferry [26]. The survey papers by Quinn [45] and Weinberger [54] offer overviews of these developments.