Abstract
This chapter presents the work on geometric topology that has been carried out in a seminar at the Steklov Institute. It discusses the structure of the homeomorphism group of Rn and locally flat embeddings of manifolds in Rn; topological embeddings of manifolds, polyhedra and compacta in Rn; and monotone mappings of manifolds. A homeomorphism h of Rn is said to be stable if it is a finite product of homeomorphisms, each of which leaves fixed all points of an open set. At present, a most important problem in geometric topology is to decide whether every orientation-preserving homeomorphism of Rn is stable. The only known orientation-preserving homeomorphisms are stable. A pseudoisotopy of a space X is a one-parameter continuous collection of maps of X onto itself, gt, where gt is a homeomorphism for all t < 1. A mapping of a manifold is point-like if the inverse images of points are cellular. A mapping of a manifold onto a manifold may be called cellular if the inverse images of cellular sets are cellular.
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