Mathematical Sciences: Homogeneous Embeddings In Manifolds

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For a compact set K and a manifold M, the three properties of K being homogeneously embedded in M, K being isotopically homogeneous in M, and K being extendibly embedded in M are closely related but not well studied. Lewis will investigate these properties primarily in the case where K is a closed manifold with dimension less than that of M, but also for K being a continuum or a Cantor set. Two types of questions are of interest in this area: 1.) for a given manifold M, which continua admit homogeneous embeddings in M, and 2.) which embeddings of a given continuum K in a manifold M are homogeneous. For two-dimensional manifolds, both questions are completely solved, but in higher dimensions the information known consists mostly of examples demonstrating existence of certain types of embeddings.