Geometric topology of generalized 3-manifolds

In this paper, we describe the history and the present status of one of the main classical problems in low-dimensional geometric topology-the recognition of topological 3-manifolds in the class of all generalized 3-manifolds (i.e., ANR homology 3-manifolds). This problem naturally splits into the cell-like resolution problem for 3-manifolds by means of homology 3-manifolds and the general-position problem for topological 3-manifolds. We have also included some open problems. © 2007 Springer Science+Business Media, Inc.