Mathematical Sciences: Homogeneous Embeddings In Manifolds

对于一个紧集K和一个流形M， K均匀地嵌入M中，K是同位素 在M中是齐次的，并且K可扩展地嵌入M中， 密切相关，但没有得到很好的研究。 刘易斯会调查 这些性质主要是在K是封闭的情况下， 维数小于M的流形，但K也是 连续统或康托集。 两种类型的问题 在这个领域的兴趣：1。对于给定的流形M， 连续体允许在M中的齐次嵌入，并且2.）这 给定连续统K在流形M中的嵌入是 同质的 对于二维流形，这两个问题都是 完全解决了，但在更高的维度， 主要包括一些例子，证明存在某些 嵌入的类型。

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.