Embeddings of (๐-1)-spheres in Euclidean ๐-space
Robert J. Daverman ยท Bulletin of the American Mathematical Society ยท 1978
Introduction.The program described in this report revolves around the basic embedding question of geometric topology: under what conditions are two embeddings j\ andf 2 of a space X in a space Y equivalent in the sense that there exists a self-homeomorphism F of the ambient space Y for which Ff { = f {I This article focuses, in particular, on information and questions concerning embeddings of the (n -l)-sphere in Euclidean ยซ-space E n 9 a specific problem that serves as a convenient abbreviation for discussing the broader category of embeddings of (n -l)-manifolds in w-manifolds, and the article emphasizes a comparison between information about codimension one embeddings in high-dimensional ^-manifolds, high usually requiring n to be at least 5, with the extensive collection of known information pertaining to embeddings of 2-manifolds in 3-manifolds.First, we fix some indispensible notation and conventions.We use B n to denote the standard ยซ-cell in E" consisting of all points in E" having norm 0. We say that a ยฃ-cell or a (k -l)-sphere X in E n is flat if there exists a homeomorphism of E n to itself that takes X to the standard object of its type.In this language, our fundamental concern is the question: what conditions is an (n -1)-sphere in E n flaft Equivalently, under what conditions is an embedding of S n ~x in E n equivalent to the inclusion S n ~x -โข> E n 1Flatness questions for spheres and cells form the prototype of questions concerning local matters.Let e denote an embedding of an m-manifold M (a metric space locally homeomorphic to either E m or E+) in the interior of an ยซ-manifold N (henceforth to be written as Int TV).One says that e is locally flat at x EL M (and that e(M) is locally flat at e( x)) if there exists a An expanded version of an invited address delivered to the American Mathematical Society at Blacksburg, Virginia, on November 7, 1975; received by the editors February 5, 1977.