xvi Introduction: The Main Problem The problems listed above are the main ones that will occupy attention in this text. They all can be viewed as uniqueness questions in the sense that they ask whether given embeddings are equivalent. There are also existence questions for embeddings, which will be studied alongside the uniqueness questions. We identify two such: one global, the other local. Existence Problem. Given a map f : X Y, is f homotopic to a topo- logical embedding or a PL embedding? Approximation Problem. Which topological embeddings of X in Y can be approximated by PL embeddings? The flatness concept has a local version. A topological embedding e : M N of a k-dimensional manifold M into an n-dimensional manifold N is locally flat at x M if there exists a neighborhood U of e(x) in N such that (U, U e(M)) (Rn, Rk). An embedding is said to be locally flat if it is locally flat at each point x of its domain. The last two problems have local variations: for example, one can ask whether a map of manifolds is homotopic to a locally flat embedding or whether a topological embedding of manifolds can be approximated by locally flat embeddings. When considering an embedding e : X Y , the dimension of Y is called the ambient dimension. Almost all of the examples and theorems in this book involve embeddings in manifolds of ambient dimension three or more. We skip dimension two because classical results like the famous Sch¨onflies theorem (Theorem 0.11.1) imply that no nonstandard local phenomena arise in conjunction with embeddings into manifolds of that dimension. While isolated examples of wild embeddings were discovered earlier, the work of R. H. Bing in the 1950s and 1960s revealed the pervasiveness of wildness in dimensions three and higher. His pioneering work led to a pro- liferation of embedding results, first concentrating on dimension three, but soon expanding to include higher dimensions as well. The subject of topo- logical embeddings is now a mature branch of geometric topology, and this book is meant to be a summary and exposition of the fundamental results in the area. Organization. As mentioned earlier, the initial Chapter 0 addresses back- ground matters. The real beginning, Chapter 1, treats knottedness, tame- ness and local flatness it provides examples of knotted, PL codimension-two sphere pairs in all sufficiently large dimensions, and it delves into the local homotopy properties of nicely embedded objects. Chapter 2 presents the basic examples that motivate the study and offers context for theorems to come later it also includes several flatness theorems that can be proved without the use of engulfing. Engulfing the fundamental technical tool for
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