The Topology of Manifolds and Cell-Like Maps

1. Introduction. The focus of this expository article will be on the notion of a cell-like set and a cell-like map (definitions below). It will be discussed how these notions arise naturally in the study of certain problems in topology, and how some solutions to these problems have been achieved. It should be emphasized at the outset that the problems discussed here are all topological in nature, and so in particular there will be a minimum of extra global structure on the various spaces at hand. The principal questions to be discussed (and motivated) are the following: 1. PoiNfT-LiKE QUESTION. Which compact subsets of the /w-sphere have the property that their complements are homeomorphic to euclidean /77-space R m l (Such subsets are called point-like, for the natural reason.) II. POLYHEDRAL MANIFOLD QUESTION. When is a polyhedron a topological manifold? In particular, are there any "unexpected" examples of such polyhedra? (i.e., examples which do not locally polyhedrally embed in the euclidean space of the same dimension.) III. MANIFOLD FACTOR QUESTION. When is a space X a factor of a manifold, i.e., when is it the case that XXY is a manifold for some space Yl (Usually Y is taken itself to be some euclidean space.)