Locally Flat Embeddings of 3-Manifolds in S4

An up-to-date treatment of an important area of research in low-dimensional topology.

This book addresses the challenges of embedding 3-manifolds in 4-space, focusing on locally flat embeddings. It introduces the main tools and results, with a focus on Seifert manifolds. Supported by figures throughout, it will be of interest to mathematicians and graduate students interested in 3- and 4-dimensional geometric topology.The study of smooth embeddings of 3-manifolds in 4-space has been hampered by difficulties with the simplest case, that of homology spheres. This book presents some advantages of working with locally flat embeddings. The first two chapters outline the tools used and give general results on embeddings of 3-manifolds in S4. The next two chapters consider which Seifert manifolds may embed, with criteria in terms of Seifert data. After summarizing results on those Seifert manifolds that embed smoothly, the following chapters determine which 3-manifolds with virtually solvable fundamental groups embed. The final three chapters study the complementary regions. When these have 'good' fundamental groups, topological surgery may be used to find homeomorphisms. Figures throughout help illustrate links representing embeddings and open questions are further discussed in the appendices, making this a valuable resource for graduate students and research workers in geometric topology.