Locally flat embeddings of 3-manifolds in $S^4$

TL;DR

This paper surveys the current knowledge on locally flat embeddings of 3-manifolds in 4-spheres, focusing on Seifert fibred manifolds and properties of their complements, highlighting differences from smooth embeddings.

Contribution

It provides a comprehensive overview of known results on locally flat embeddings of 3-manifolds in $S^4$, emphasizing Seifert fibred manifolds and the topology of their complements.

Findings

Freedman's result on homology 3-sphere embeddings

Use of 4-dimensional surgery to analyze embeddings with specific fundamental groups

Discussion of open questions in the field

Abstract

M. Freedman showed that every homology 3-sphere embeds as a locally flat submanifold of $S^4$. This is in striking contrast to the state of our knowledge of smooth embeddings of homology spheres. This book surveys what is presently known about the topic of its title, with emphasis on embeddings of Seifert fibred 3-manifolds and on distinguishing embeddings by properties of the complementary regions. In the final chapters 4-dimension surgery is used to study embeddings in which the complementary regions have abelian or nilpotent fundamental groups, and the final appendix contains a number of open questions. Work on smooth embeddings is only described briefly, as current techniques in this area are beyond the author's expertise.