Mathematical Sciences: Homogeneous Embeddings In Manifolds

对于紧集K和流形M，K齐次嵌入M、K均匀嵌入M和K可拓嵌入M的三个性质是密切相关的，但研究得不多。Lewis将主要在K是维度小于M的闭流形的情况下，以及K是连续统或Cantor集的情况下研究这些性质。这一领域涉及两类问题：1)对于给定的流形M，它连续地允许齐次嵌入到M中，和2。)其中给定的连续统K在流形M中的嵌入是齐次的。对于二维流形，这两个问题都是完全解决的，但在高维中，已知的信息主要由证明某些类型嵌入的存在的例子组成。

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.